Planet Musings

The Whole Foods in Madison used to be on University Avenue where Shorewood Boulevard empties onto it. That’s about a 7-minute drive from my house. I stopped in there all the time, just because I was driving by and I needed something that was easy to find there. Last year, Whole Foods moved over to the new Hilldale Yards development, still on University but just across Midvale. It’s now an 8-minute drive from my house. And I haven’t been there since. Somehow, everything east of Midvale is “my zone” — a place I a place I might pass by, and go without planning to go. And anything west of Midvale is only accessible by a dedicated trip.

By the way, the old Whole Foods location is now Fresh Mart, an international (but mostly Middle Eastern and Eastern European) grocery store. It’s awesome to have an international store the size of a Whole Foods but it is really hard to imagine how they’re making rent on a space that big. So please go there, buy some actually sharp feta and freekeh and ajvar and marinated sprats in a can! They have an extremely multicultural hot bar with great grape leaves and a coffeeshop with multiple kinds of baklava! I’m glad it’s in my zone.

“Never doubt that the master’s tools will dismantle the master’s house; indeed, they’re the only thing that ever have.”

The word “abstemious” is one that usually comes to mind only because (apparently sharing this feature only with “facetious”) it has all five English vowels, once each, in alphabetical order. I had always assumed it derived from the word “abstain.” But no! “Abstain” is the Latin negative prefix ab- followed by tenere, “to have” — you don’t have what you abstain from. But to be abstemious is specifically to abstain from alcohol. What comes after ab is actually temetum, a Latin word whose ultimate origin is the proto-Indo-European temH, or “darkness” (i.e. what you experience when you drink too much temetum.)

Anyway, no better time to listen to EBN OZN’s wondrous one hit, “AEIOU Sometimes Y,” which is apparently the first pop song ever recorded entirely on a computer. “Do I want to go out?”

When you’ve done a lot of science communication you start to see patterns. You notice the choices people make when they write a public talk or a TV script, the different goals and practical constraints that shape a piece. I’ve likened it to watching an old kung fu movie and seeing where the wires are.

I don’t have a lot of experience doing science journalism, I can’t see the wires yet. But I’m starting to notice things, subtle elements like notes at a wine-tasting. Just like science communication by academics, science journalism is shaped by a variety of different goals.

First, there’s the need for news to be “new”. A classic news story is about something that happened recently, or even something that’s happening right now. Historical stories usually only show up as new “revelations”, something the journalist or a researcher recently dug up. This isn’t a strict requirement, and it seems looser in science journalism than in other types of journalism: sometimes you can have a piece on something cool the audience might not know, even if it’s not “new”. But it shapes how things are covered, it means that a piece on something old will often have something tying it back to a recent paper or an ongoing research topic.

Then, a news story should usually also be a “story”. Science communication can sometimes involve a grab-bag of different topics, like a TED talk that shows off a few different examples. Journalistic pieces often try to deliver one core message, with details that don’t fit the narrative needing to wait for another piece where they fit better. You might be tempted to round this off to saying that journalists are better writers than academics, since it’s easier for a reader to absorb one message than many. But I think it also ties to the structure. Journalists do have content with multiple messages, it just usually is not published as one story, but a thematic collection of stories.

Combining those two goals, there’s a tendency for news to focus on what happened. “First they had the idea, then there were challenges, then they made their discovery, now they look to the future.” You can’t just do that, though, because of another goal: pedagogy. Your audience doesn’t know everything you know. In order for them to understand what happened, there are often other things they have to understand. In non-science news, this can sometimes be brief, a paragraph that gives the background for people who have been “living under a rock”. In science news, there’s a lot more to explain. You have to teach something, and teaching well can demand a structure very different from the one-step-at a time narrative of what happened. Balancing these two is tricky, and it’s something I’m still learning how to do, as can be attested by the editors who’ve had to rearrange some of my pieces to make the story flow better.

News in general cares about being independent, about journalists who figure out the story and tell the truth regardless of what the people in power are saying. Science news is strange because, if a scientist gets covered at all, it’s almost always positive. Aside from the occasional scandal or replication crisis, science news tends to portray scientific developments as valuable, “good news” rather than “bad news”. If you’re a politician or a company, hearing from a journalist might make you worry. If you say the wrong thing, you might come off badly. If you’re a scientist, your biggest worry is that a journalist might twist your words into a falsehood that makes your work sound too good. On the other hand, a journalist who regularly publishes negative things about scientists would probably have a hard time finding scientists to talk to! There are basic journalistic ethics questions here that one probably learns about at journalism school and we who sneak in with no training have to learn another way.

These are the flavors I’ve tasted so far: novelty and narrative vs. education, positivity vs. accuracy. I’ll doubtless see more over the years, and go from someone who kind of knows what they’re doing to someone who can mentor others. With that in mind, I should get to writing!

Update: In the comments, Craig Gidney points out that Ronit’s O(n log² n) quantum circuits for the exact QFT were already published by Cleve and Watrous in 2000 (in a paper whose main point was something else, parallelization). Ronit’s O(n (log log n)²) circuits for the approximate QFT still appear to be new (Gidney says he and others knew related techniques but had never explicitly combined them). Of course, while the exact result was Platonically “known,” it wasn’t sufficiently well known that any of the four quantum algorithms experts I’d consulted had heard of it! Hopefully this very post will go some way toward fixing the situation.

Another Update: Richard Cleve writes in to say that the approximate QFT circuits were known also—albeit, in an unpublished 2-page abstract by Ahokas, Hales, and himself from the 2003 ERATO conference, as well as a followup Master’s thesis by Ahokas. Unlike with the exact case, I’m not kicking myself trying to understand how I missed these.

Ironically, I hope this post helps get this prior work a well-deserved mention when the QFT is covered in introductory quantum information classes.

Meanwhile, my hope that Ronit returns to do more theory is undiminished! When I was a kid, I too started by rediscovering things (like the integral for the length of a curve) that were centuries old, then rediscovering things (like an efficient algorithm for isotonic regression) that were decades old, then rediscovering things (like BQP⊆PP) that were about a year old … until I finally started discovering things (like the collision lower bound) that were zero years old. This is the way.

In my last post, I tried to nudge the arc of history back onto the narrow path of reasoned dialogue, walking the mile-high tightrope between shrill, unsupported accusation and naïve moral blindness. For my trouble, I was condemned about equally by leftists for my right-wing sympathies and by rightists for my left-wing ones. So today, I’ll ignore the fate of civilization and return to quantum computing theory: a subject that’s reliably brought joy to my life for a quarter-century, and still does, even as my abilities fade. It turns out there is a consolation for advancing age and senility, and it’s called “students.”

This fall, I returned from my two-year leave at OpenAI to teach my undergrad Introduction to Quantum Information Science course at UT Austin. This course doesn’t pretend to bring students all the way to the research frontier, and yet sometimes it’s done so anyway. It was in my first offering of Intro to QIS, eight years ago, that I encountered the then 17-year-old Ewin Tang, who broke the curve and then wanted an independent study project. So I gave her the problem of proving that the Kerenidis-Prakash quantum algorithm achieves an exponential speedup over any classical algorithm for the same task, not expecting anything to come of it. But after a year of work, Ewin refuted my conjecture by dequantizing the K-P algorithm—a breakthrough that led to the demolition of many other hopes for quantum machine learning. (Demolishing people’s hopes? In complexity theory, we call that a proud day’s work.)

Today I’m delighted to announce that my undergrad quantum course has led to another quantum advance. One day, after my lecture, a junior named Ronit Shah came to me with an idea for how best to distinguish three possible states of a qubit, rather than only two. For some reason I didn’t think much of it at the time, even though it would later turn out that Ronit had essentially rediscovered the concept of POVMs, the Pretty Good Measurement (PGM), and the 2002 theorem that the PGM is optimal for distinguishing sets of states subject to a transitive group action.

Later, after I’d lectured about Shor’s algorithm, and one of its centerpieces, the O(n²)-gate recursive circuit for the Quantum Fourier Transform, Ronit struck a second time. He told me it should be possible to give a smaller circuit by recursively reducing the n-qubit QFT to two (n/2)-qubit QFTs, rather than to a single (n-1)-qubit QFT.

This was surely just a trivial confusion, perfectly excusable in an undergrad. Did Ronit perhaps not realize that an n-qubit unitary is actually a 2ⁿ×2ⁿ matrix, so he was proposing to pass directly from 2ⁿ×2ⁿ to 2^n/2×2^n/2, rather than to 2^n-1×2^n-1?

No, he said, he understood that perfectly well. He still thought the plan would work. Then he emailed me a writeup—claiming to implement the exact n-qubit QFT in O(n log²n) gates, the first-ever improvement over O(n²), and also the approximate n-qubit QFT in O(n (log log n)²) gates, the first-ever improvement over O(n log n). He used fast integer multiplication algorithms to make the new recursions work.

At that point, I did something I’m still ashamed of: I sat on Ronit’s writeup for three weeks. When I at last dug it out of my inbox and read it, I could discover no reason why it was wrong, or unoriginal, or unimportant. But I didn’t trust myself, so with Ronit’s permission I sent the work to some of my oldest quantum friends: Ronald de Wolf, Cris Moore, Andrew Childs, and Wim van Dam. They agreed, after some back-and-forth, that the new circuits looked legit. A keystone of Shor’s algorithm, of quantum computing itself, and of my undergrad class had seen its first real improvement since 1994.

Last night Ronit’s paper appeared on the arXiv where you can read it.

In case anyone asks: no, this probably has no practical implication for speeding up factoring on a quantum computer, since the QFT wasn’t the expensive part of Shor’s algorithm anyway—that’s the modular exponentiation—and also, the O(n log n) approximate QFT would already have been used in practice. But it’s conceivable that Ronit’s circuits could speed up other practical quantum computing tasks! And no, we have no idea what’s the ultimate limit here, as usual in circuit complexity. Could the exact n-qubit QFT even be doable in O(n) gates?

I’d love for Ronit to continue in quantum computing theory. But in what’s surely a sign of the times, he’s just gone on leave from UT to intern at an AI hardware startup. I hope he returns and does some more theory, but if he doesn’t, I’m grateful that he shared this little gem with us on his way to more world-changing endeavors.

In this post and the next, I want to try out a new idea and see where it leads. It goes back to where magnitude began, which was the desire to unify elementary counting formulas like the inclusion-exclusion principle and the simple formula for the number of orbits in a free action of a group on a finite set.

To prepare the ground for comagnitude, I need to present magnitude itself in a slightly different way from usual. I won’t assume you know anything about magnitude, but if you do, watch out for something new: a connection between magnitude and entropy (ordinary, relative and conditional) that I don’t think has quite been articulated before.

The inclusion-exclusion formula tells us the cardinality of a union, which in categorical terms is a pushout. The set of orbits of a free action of a group $G$ on a finite set is the colimit of the associated functor $B G \to FinSet$. (Here $B G$ is the one-object category whose maps are the elements of $G$.) Generally, we’re going to think about the cardinality of the colimit of a functor $X: A \to FinSet$, where $A$ is a finite category.

In both the examples I just mentioned, the cardinality $|colim X|$ of the colimit is a $\mathbb{Q}$-linear combination of the cardinalities $|X(a)|$ of the individual sets $X(a)$, taken over all objects $a \in A$. For inclusion-exclusion, we take

$$A = (0 \leftarrow 1 \rightarrow 2)$$

and get

$$|colim(X)| = |X(0)| - |X(1)| + |X(2)|.$$

For a free action of a group $G$, we take $A = B G$. If the single object of $A$ is called $a$, then

$$|colim(X)| = |X(a)|/o(G)$$

— the cardinality of the set $X(a)$ being acted on, divided by the order of $G$.

Generally, there’s no hope of computing the cardinality of the colimit of a functor $X: A \to FinSet$ in terms of the cardinalities of the sets $X(a)$ alone. The colimit usually depends on what the functor $X$ does to morphisms, as well as objects. So in order to pull this off, we’re going to have to confine ourselves to only those functors $X$ that are especially convenient. The hope is that for any $A$, there are rational numbers $w_A(a)$ ($a \in A$) such that for all “convenient” functors $X: A \to FinSet$,

$$|colim(X)| = \sum_{a \in A} w_A(a) |X(a)|.$$

Here’s how to simultaneously figure out what the coefficients $w_A(a)$ in this weighted sum must be and what “convenient” should mean.

The starting thought is that most notions of “nice enough” for set-valued functors include the representables. Since the colimit of any representable is the one-element set, for the last equation to hold for all representables $A(b, -)$ means that

$$1 = \sum_{a \in A} w_A(a) |A(b, a)|$$

for all $b \in A$.

And now we’re getting somewhere! This is a system of equations with the same number of equations as unknowns $w_A(a)$, namely, the number of objects of $A$. And in this situation, there’s typically a unique solution, at least if we work over the field $\mathbb{Q}$.

A family $(w_A(a))_{a \in A}$ of rational numbers satisfying

$$\sum_{a \in A} |A(b, a)| w_A(a) = 1$$

for all $b \in A$ is called a weighting on $A$, and $w_A(a)$ is called the weight of $a$. For the purposes of this post, I’ll assume there’s always exactly one weighting on $A$. That’s not entirely justified: there are examples of finite categories with no weighting or finitely many. (See Examples 1.11 of my paper The Euler characteristic of a category, where all the stuff I’m talking about right now is worked out.) But it’s safe enough.

So, take the weighting $w_A$ on $A$. By defintion, the class of functors $X: A \to FinSet$ satisfying

$$|colim(X)| = \sum_{a \in A} w_A(a) |X(a)|$$

contains the representables. But it’s easy to see that it’s also closed under coproducts. So, this equation — which I’ll call the colimit formula — holds for all coproducts of representables!

Which functors $A \to Set$ are coproducts of representables? They’re sometimes called the “familially representable” functors, the idea being that a coproduct of representables $\sum_{i \in I} A(a_i, -)$ is represented by a family of objects $(a_i)_{i \in I}$. But in my last post, I called them the “free functors”. The way I put it there was that the forgetful functor

$$Set^A \to Set^{ob\ A}$$

has both a left and a right adjoint, and a functor $X \in Set^A$ is free if and only if it’s in the image of the left adjoint. That means

$$X \cong \sum_{a \in A} S(a) \times A(a, -)$$

for some family of sets $(S(a))_{a \in A}$.

Examples

• When $A$ is the discrete two-object category $\{0, 1\}$, both weights are $1$, and the general colimit formula

  $$|colim(X)| = \sum_a w_A(a) |X(a)|$$

  reduces to something very basic:

  $$|X(0) + X(1)| = |X(0)| + |X(1)|$$

  — the cardinality of a coproduct is the sum of the individual cardinalities. When $A$ is a discrete category, all functors $A \to Set$ are coproducts of representables.

• When $A$ is the shape $(0 \leftarrow 1 \rightarrow 2)$ for pushouts, a functor $X: A \to FinSet$ is a diagram

  $$(X(0) \leftarrow X(1) \rightarrow X(2))$$

  of finite sets, and $X$ is a coproduct of representables if and only if both these maps are injections. Assume $X$ is a coproduct of representables. The weighting on $A$ is $(1, -1, 1)$, so the colimit formula says that the cardinality of a pushout is

  $$|X(0)| - |X(1)| + |X(2)|$$

  — the inclusion-exclusion formula.

• For a group $G$ (or more generally, a monoid), the weight of the unique object of $B G$ is $1/o(G)$, the reciprocal of the order. Given an action of $G$ on a a finite set, the corresponding functor $B G \to FinSet$ is a coproduct of representables if and only if the action is free. In that case, the colimit formula says that the number of orbits is the cardinality of the set divided by $o(G)$.

• Finally, when $A = (0 \rightrightarrows 1)$, the colimit formula says that the coequalizer of $X(0) \rightrightarrows X(1)$ has $|X(1)| - |X(0)|$ elements, provided that the two functions from $X(0)$ to $X(1)$ are injective with disjoint images.

Aside for category theorists   Under finiteness conditions, and assuming that $A$ is Cauchy complete, a functor $X: A \to Set$ is a coproduct of representables if and only if it is flat with respect to the class of finite connected limits. This might seem more abstract than the original condition, but it actually gives a concrete, testable condition for $X$ to be a coproduct of representables. (See Lemma 5.2 and Definition 3.2 here.) This result enables one to see rather quickly that in all the examples above, the functors arising as coproducts of representables are exactly what I said they are.

Summary so far  We’ve seen that a finite category $A$ typically has a unique weighting $w_A$ — meaning a family $(w_A(a))_{a \in A}$ of rational numbers satisfying

$$\sum_a |A(a, b)| w_A(a) = 1$$

for all $b \in A$. And we’ve seen that when $X: A \to FinSet$ is a coproduct of representables, the “colimit formula” holds:

$$|colim(X)| = \sum_a w_A(a) |X(a)|.$$

Now, what if $X$ isn’t a coproduct of representables? The colimit formula won’t usually hold. But the right-hand side of this non-equation still calculates something. What is it?

Let me define the magnitude $|X|$ of a functor $X: A \to FinSet$ by

$$|X| = \sum_{a \in A} w_A(a) |X(a)| \in \mathbb{Q}.$$

This is equal to $|colim(X)|$ if $X$ is a coproduct of representables, but not in general.

For example, if we take a monoid $M$ acting on a finite set, the magnitude of the corresponding functor $B M \to FinSet$ is the cardinality of the set divided by the order of $M$. Unless the action is free, this isn’t usually the number of orbits, and it might not even be an integer.

The functor $\Delta 1: A \to FinSet$ sending everything in $A$ to the one-element set has magnitude

$$|\Delta 1| = \sum_{a \in A} w_A(a),$$

which is called the magnitude or Euler characteristic of $A$. (Under finiteness hypothesis, it’s equal to the topological Euler characteristic of the classifying space in $A$.) Hence the magnitude of a category is a special case of the magnitude of a set-valued functor. Conversely, one can show that the magnitude of a functor $X: A \to FinSet$ is equal to the magnitude of its category of elements.

So, the concepts of magnitude of a category (already studied extensively) and the magnitude of a set-valued functor (a term I think I’m introducing for the first time here) are each special cases of the other. But in these posts, unlike everything I’ve written about magnitude before, I’m going to give prime position to magnitude of set-valued functors rather than of categories.

Does this definition of the magnitude of a set-valued functor have sensible properties? It does!

First, if $X, Y: A \to FinSet$ are isomorphic then certainly $|X| = |Y|$. Second, and a bit less trivially, magnitude is invariant under equivalence in the following sense: if we compose $X: A \to FinSet$ with an equivalence of categories $G: B \to A$ then

$$|X \circ G| = |X|.$$

In fact, this holds just as long as $G$ has a left adjoint.

The term “magnitude” suggests it should behave in a cardinality-like way, and it does that too. For example, given functors $X, Y: A \to FinSet$,

$$|X + Y| = |X| + |Y|,$$

where the left-hand side is the magnitude of the coproduct of our two functors. A bit more generally, suppose we have a diagram of functors and natural transformations

$$X \leftarrow Y \rightarrow Z$$

such that for each $a \in A$, the functions

$$X(a) \leftarrow Y(a) \rightarrow Z(a)$$

are injective with disjoint images (or more abstractly, the corresponding functor from $(0 \leftarrow 1 \rightarrow 2)$ to $FinSet$ is a coproduct of representables). Then one can show that the magnitude of the pushout $X +_Y Z$ is given by

$$|X +_Y Z| = |X| - |Y| + |Z|.$$

So magnitude in this sense obeys the inclusion-exclusion formula. More generally still, there’s a similar result for any shape of colimit, but I won’t write it out here. It’s exactly what the pushout case suggests.

I hope I’ve driven home the message that although the colimit formula

$$|colim(X)| = |X|$$

for functors $X: A \to FinSet$ holds when $X$ is a coproduct of representables, it doesn’t hold for arbitrary $X$. So perhaps it’s interesting to see how much the two sides of the equation differ when $X$ isn’t a coproduct of representables. In other words, let’s look at the difference

$$|colim(X)| - |X| = |colim(X)| - \sum_a w_A(a) |X(a)| \in \mathbb{Q}.$$

It seems to me that this quantity could be interesting. However, I don’t have a strong feel for what it means yet, so for now I’ll give it a bland name: the discrepancy of $X$.

Example  The discrepancy of a coequalizer

$$X_0 \rightrightarrows X_1 \to C$$

is

$$|C| - (|X_1| - |X_0|) = |X_0| - |X_1| + |C|.$$

We’ve already seen that this is $0$ if our two functions $f, g: X_0 \rightrightarrows X_1$ are injective with disjoint images. But in general, it looks rather like an Euler characteristic. In fact, from our coequalizer we can form a chain complex of free abelian groups

$$0 \to \mathbb{Z} X_0 \stackrel{\mathbb{Z}g - \mathbb{Z}f}{\to} \mathbb{Z} X_1 \stackrel{\mathbb{Z}h}{\to} \mathbb{Z} C \to 0$$

where $h$ is the original function $X_1 \to C$. Then the discrepancy of our coequalizer is equal to the Euler characteristic of this complex.

This complex is exact except maybe at $\mathbb{Z} X_0$. Let’s do a little check: if $f$ and $g$ are injective with disjoint images then $\mathbb{Z}g - \mathbb{Z}f$ is injective too, so the complex is exact everywhere, which implies that the Euler characteristic is $0$. Since the discrepancy is also $0$ in this case, we’ve just confirmed one case of the result that the discrepancy of the coequalizer is the Euler characteristic of the resulting complex.

Example  For a pushout square

$$\begin{array}{ccc} X_1 &\rightarrow &X_0 \\ \downarrow & &\downarrow \\ X_2 &\rightarrow &P, \end{array}$$

the discrepancy is

$$|P| - (|X_0| - |X_1| + |X_2|) = |X_1| - |X_0| - |X_2| + |P|.$$

I don’t know what to do with this formula, but it has a certain symmetry. Has anyone come across it before?

I’ll finish by explaining the entropy connection I promised at the start. Here we go!

Ordinary Shannon entropy  For this, we need to think about the Shannon entropy not just of probability measures — the usual setting — but of arbitrary measures. The sets we’re considering measures on will always be finite.

A probability measure $p$ on a finite set $\{1, \ldots, n\}$ is just an $n$-tuple $(p_1, \ldots, p_n)$ of nonnegative reals summing to $1$, and its entropy $H(p)$ is defined by

$$H(p) = -\sum_i p_i \log p_i,$$

with the convention that $0\ \log\ 0 = 0$. A measure $a = (a_1, \ldots, a_n)$ on $\{1, \ldots, n\}$ is any $n$-tuple of nonnegative reals, and the definition of entropy is extended from probability measures to arbitrary measures by homogeneity. That is, writing $\|a\| = \sum a_i$, the tuple $a/\|a\|$ is a probability measure, and we define

$$H(a) = \|a\| H(a/\|a\|)$$

(or $H(a) = 0$ if $\|a\| = 0$). This is the unique way of extending the definition so that

$$H(\lambda a) = \lambda H(a)$$

for all measures $a$ and real $\lambda \geq 0$.

Let’s now consider measures $a = (a_1, \ldots, a_n)$ where each $a_i$ is an integer. Such a thing amounts to a map

$$A \stackrel{\pi}{\to} I$$

of finite sets. To see this, take $I = \{1, \ldots, n\}$, and take $A \stackrel{\pi}{\to} I$ to be a function into $I$ whose $i$-fibre has cardinality $a_i$.

This categorification manoeuvre — upgrading from numbers to sets — brings maps into play. In particular, we get the monoid $End(A)$ of all functions $A \to A$, and its submonoid $End_I(A)$ consisting of the endomorphisms of $A$ over $I$. That is, the elements of $End_I(A)$ are the functions $f: A \to A$ such that $\pi \circ f = \pi$.

The inclusion of monoids

$$End_I(A) \hookrightarrow End(A),$$

like any other monoid homomorphism, induces an action of the domain on the underlying set of the codomain: applying an element $f \in End_I(A)$ to an element $g \in End(A)$ gives a new element $f \circ g \in End(A)$. And this monoid action corresponds to a functor

$$End_I(A) \to FinSet.$$

(Or “$B End_I(A) \to FinSet$” if you want to be fussy, but I’ll drop the $B$s.)

Question: what’s the magnitude of this functor?

Answer: it’s

$$\frac{|End(A)|}{|End_I(A)|} = \frac{\|a\|^{\|a\|}}{\prod_i a_i^{a_i}} = \prod_i \biggl( \frac{\|a\|}{a_i} \biggr)^{a_i}.$$

This is nothing but $\exp(H(a))$, the exponential of the entropy of our measure! So:

The exponential of entropy is a special case of magnitude.

(Sometimes it seems that the exponential of entropy is a more fundamental quantity than the entropy itself. Apart from anything else, it doesn’t depend on a choice of base. I like to call the exponential of entropy the diversity.)

At least, that’s true when the measure of each point is an integer. But from knowing the entropy of such measures, we can easily obtain the entropy of those where the entropy of each point is rational, assuming the homogeneity property $H(\lambda a) = \lambda H(a)$. And then one can get the completely general case of real measures by extending by continuity.

The basic calculation above was described in an earlier post of mine, but at that time I didn’t see clearly what it meant.

Relative entropy  Given two probability measures $p$ and $r$ on the same finite set, you can calculate their relative entropy. It’s a measure of how surprised you’d be to observe a distribution of $p$ in a sample drawn from a population with distribution $r$. (I think of $r$ as the “reference” distribution.) The formula for $H(p\|r)$, the entropy of $p$ relative to $r$, is

$$H(p\|r) = \sum_i p_i \log(p_i/r_i) \in [0, \infty].$$

Just as for ordinary entropy, we can extend it from probability measures to arbitrary measures. We’re now dealing with two measures $a$ and $b$ on the same finite set, and I’ll assume they have the same total mass: $\|a\| = \|b\|$. Again, we extend homogeneously, so that

$$H(\lambda a \| \lambda b) = \lambda H(a\|b)$$

whenever $\lambda \geq 0$.

And as before, we’ll focus on measures taking integer values. A pair of measures on a finite set $I$, both taking integer values and with the same total mass, amounts to a diagram

$$A \rightrightarrows I$$

in $FinSet$. The idea is that if we take $I = \{1, \ldots, n\}$ and write $a_i$ and $b_i$ for the cardinalities of the fibres of the $i$-fibres of these two maps, then one of our measures is $(a_1, \ldots, a_n)$ and the other is $(b_1, \ldots, b_n)$. They have the same total mass because

$$a_1 + \cdots + a_n = |A| = b_1 + \cdots + b_n.$$

Calling our two maps $\pi, \rho: A \to I$, we can consider the set

$$Hom_I(A \stackrel{\pi}{\to} I, A \stackrel{\rho}{\to} I)$$

of functions $g : A \to A$ such that $\rho \circ g = \pi$. This set is acted on by the monoid

$$End_I(A \stackrel{\pi}{\to} I)$$

of functions $f : A \to A$ such that $\pi \circ f = \pi$, by composition.

As we keep seeing, a monoid action corresponds to a functor from that monoid to $Set$, whose magnitude we can compute. In this case, the magnitude is

$$\frac{|Hom_I(A \stackrel{\pi}{\to} I, A \stackrel{\rho}{\to} I)|}{|End_I(A \stackrel{\pi}{\to} I)|} = \frac{\prod b_i^{a_i}}{\prod a_i} = \prod \biggl( \frac{a_i}{b_i} \biggr)^{a_i},$$

which is equal to

$$\exp(-H(a \| b))$$

—the negative exponential of relative entropy.

(We’re never going to be able to get rid of that negative and get $\exp(H(a \| b))$ itself as a magnitude, since $H(a\|b)$ can be infinite but that the magnitude of a functor can’t be, at least in the context at hand.)

As for ordinary entropy, if we know relative entropy for integer-valued measures then it’s only two short steps to the definition for arbitrary measures.

Conditional entropy  Whereas relative entropy takes as its input a pair of measures on the same set, conditional entropy takes as its input two measures on different sets.

It’s usually presented in terms of random variables: we have random variables $X$ and $Y$ taking values in sets $I$ and $J$ respectively (which for us will be finite), and the conditional entropy $H(X | Y)$ is defined by

$$H(X | Y) = \sum_{i \in I, j \in J} P(X = i, Y = j) \log \frac{1}{P(X = i | Y = j)}.$$

If we write $(p_{i j})_{i \in I, j \in J}$ for the joint distribution — so that the $p_{i j}$ are nonnegative reals summing to $1$ — then

$$H(X | Y) = \sum_{i, j} p_{i j} \log \biggl( \frac{p_j}{p_{i j}} \biggr).$$

Here $p$ is a probability measure on $I \times J$. But just as for ordinary and relative entropy, this definition extends to arbitrary measures $a$ on $I \times J$, by scaling homogeneously.

To see how conditional entropy is a special case of magnitude, we follow a path that should be familiar by now.

An integer-valued measure on $I \times J$ amounts to a map $A \to I \times J$ of finite sets, or equivalently a diagram

$$I \leftarrow A \rightarrow J$$

in $FinSet$. In the monoid of all endomorphisms of $A$, we can consider the endomorphisms over $I$, or over $J$, or most restrictively of all, over $I \times J$. In particular, there is an inclusion of monoids

$$End_{I \times J}(A) \hookrightarrow End_J(A).$$

As for ordinary entropy, this homomorphism induces an action of the domain on the underlying set of the codomain, giving a functor $End_{I \times J}(A) \to FinSet$ whose magnitude we can calculate. It’s

$$\frac{|End_J(A)|}{|End_{I \times J}(A)|} = \frac{\prod_j a_j^{a_j}}{\prod_{i, j} a_{i j}^{a_{i j}}} = \prod_{i, j} \biggl( \frac{a_j}{a_{i j}} \biggr)^{a_{i j}},$$

where I’m using $a_{i j}$ to mean the cardinality of the $(i, j)$-fibre of $A \to I \times J$ and $a_j$ to mean the cardinality of the $j$-fibre of $A \to J$. This is exactly the exponential of the conditional entropy.

Conclusion   The exponentials of

• ordinary entropy
• the negative of relative entropy
• conditional entropy

all arise as special cases of magnitude — at least for integer-valued measures, but it’s routine to derive the definition for general measures from that special case.

If you know lots about entropy, you might be wondering now: what about mutual information? Is that the magnitude of some set-valued functor too?

I don’t know! Given $A \to I \times J$, the quantity we’d like to obtain as a magnitude is

$$\frac{|End(A)| |End_{I \times J}(A)|}{|End_I(A)| |End_J(A)|}.$$

I can’t see how to do that in any natural way.

Maybe I’ll come back to this point next time, when I’ll start on the topic that’s actually the title of these posts: comagnitude.

Traditionally, mathematics research projects are conducted by a small number (typically one to five) of expert mathematicians, each of which are familiar enough with all aspects of the project that they can verify each other’s contributions. It has been challenging to organize mathematical projects at larger scales, and particularly those that involve contributions from the general public, due to the need to verify all of the contributions; a single error in one component of a mathematical argument could invalidate the entire project. Furthermore, the sophistication of a typical math project is such that it would not be realistic to expect a member of the public, with say an undergraduate level of mathematics education, to contribute in a meaningful way to many such projects.

For related reasons, it is also challenging to incorporate assistance from modern AI tools into a research project, as these tools can “hallucinate” plausible-looking, but nonsensical arguments, which therefore need additional verification before they could be added into the project.

Proof assistant languages, such as Lean, provide a potential way to overcome these obstacles, and allow for large-scale collaborations involving professional mathematicians, the broader public, and/or AI tools to all contribute to a complex project, provided that it can be broken up in a modular fashion into smaller pieces that can be attacked without necessarily understanding all aspects of the project as a whole. Projects to formalize an existing mathematical result (such as the formalization of the recent proof of the PFR conjecture of Marton, discussed in this previous blog post) are currently the main examples of such large-scale collaborations that are enabled via proof assistants. At present, these formalizations are mostly crowdsourced by human contributors (which include both professional mathematicians and interested members of the general public), but there are also some nascent efforts to incorporate more automated tools (either “good old-fashioned” automated theorem provers, or more modern AI-based tools) to assist with the (still quite tedious) task of formalization.

However, I believe that this sort of paradigm can also be used to explore new mathematics, as opposed to formalizing existing mathematics. The online collaborative “Polymath” projects that several people including myself organized in the past are one example of this; but as they did not incorporate proof assistants into the workflow, the contributions had to be managed and verified by the human moderators of the project, which was quite a time-consuming responsibility, and one which limited the ability to scale these projects up further. But I am hoping that the addition of proof assistants will remove this bottleneck.

I am particularly interested in the possibility of using these modern tools to explore a class of many mathematical problems at once, as opposed to the current approach of focusing on only one or two problems at a time. This seems like an inherently modularizable and repetitive task, which could particularly benefit from both crowdsourcing and automated tools, if given the right platform to rigorously coordinate all the contributions; and it is a type of mathematics that previous methods usually could not scale up to (except perhaps over a period of many years, as individual papers slowly explore the class one data point at a time until a reasonable intuition about the class is attained). Among other things, having a large data set of problems to work on could be helpful for benchmarking various automated tools and compare the efficacy of different workflows.

One recent example of such a project was the Busy Beaver Challenge, which showed this July that the fifth Busy Beaver number was equal to . Some older crowdsourced computational projects, such as the Great Internet Mersenne Prime Search (GIMPS), are also somewhat similar in spirit to this type of project (though using more traditional proof of work certificates instead of proof assistants). I would be interested in hearing of any other extant examples of crowdsourced projects exploring a mathematical space, and whether there are lessons from those examples that could be relevant for the project I propose here.

More specifically I would like to propose the following (admittedly artificial) project as a pilot to further test out this paradigm, which was inspired by a MathOverflow question from last year, and discussed somewhat further on my Mastodon account shortly afterwards.

The problem is in the field of universal algebra, and concerns the (medium-scale) exploration of simple equational theories for magmas. A magma is nothing more than a set equipped with a binary operation . Initially, no additional axioms on this operation are imposed, and as such magmas by themselves are somewhat boring objects. Of course, with additional axioms, such as the identity axiom or the associative axiom, one can get more familiar mathematical objects such as groups, semigroups, or monoids. Here we will be interested in (constant-free) equational axioms, which are axioms of equality involving expressions built from the operation and one or more indeterminate variables in . Two familiar examples of such axioms are the commutative axiom

To illustrate the project I have in mind, let me first introduce eleven examples of equational axioms for magmas:

• Equation1:
• Equation2:
• Equation3:
• Equation4:
• Equation5:
• Equation6:
• Equation7:
• Equation8:
• Equation9:
• Equation10:
• Equation11:

One can then ask which axioms imply which others. For instance, Equation1 implies all the other axioms in this list, which in turn imply Equation11. Equation8 implies Equation9 as a special case, which in turn implies Equation10 as a special case. The full poset of implications can be depicted by the following Hasse diagram:

This in particular answers the MathOverflow question of whether there were equational axioms intermediate between the constant axiom Equation1 and the associative axiom Equation10.

Most of the implications here are quite easy to prove, but there is one non-trivial one, obtained in this answer to a MathOverflow post closely related to the preceding one:

Proposition 1 Equation4 implies Equation7.

Proof: Suppose that obeys Equation4, thus

A formalization of the above argument in Lean can be found here.

I will remark that the general question of determining whether one set of equational axioms determines another is undecidable; see Theorem 14 of this paper of Perkins. (This is similar in spirit to the more well known undecidability of various word problems.) So, the situation here is somewhat similar to the Busy Beaver Challenge, in that past a certain point of complexity, we would necessarily encounter unsolvable problems; but hopefully there would be interesting problems and phenomena to discover before we reach that threshold.

The above Hasse diagram does not just assert implications between the listed equational axioms; it also asserts non-implications between the axioms. For instance, as seen in the diagram, the commutative axiom Equation7 does not imply the Equation4 axiom

• Equation2 does not imply Equation3.
• Equation3 does not imply Equation5.
• Equation3 does not imply Equation7.
• Equation5 does not imply Equation6.
• Equation5 does not imply Equation7.
• Equation6 does not imply Equation7.
• Equation6 does not imply Equation10.
• Equation7 does not imply Equation6.
• Equation7 does not imply Equation10.
• Equation9 does not imply Equation8.
• Equation10 does not imply Equation9.
• Equation10 does not imply Equation6.

As one can see, it is already somewhat tedious to compute the Hasse diagram of just eleven equations. The project I propose is to try to expand this Hasse diagram by a couple orders of magnitude, covering a significantly larger set of equations. The set I propose is the set of equations that use the magma operation at most four times, up to relabeling and the reflexive and symmetric axioms of equality; this includes the eleven equations above, but also many more. How many more? Recall that the Catalan number is the number of ways one can form an expression out of applications of a binary operation (applied to placeholder variables); and, given a string of placeholder variables, the Bell number is the number of ways (up to relabeling) to assign names to each of these variables, where some of the placeholders are allowed to be assigned the same name. As a consequence, ignoring symmetry, the number of equations that involve at most four operations is

It is not clear to me at all what the geometry of will look like. Will most equations be incomparable with each other? Will it stratify into layers of “strong” and “weak” axioms? Will there be a lot of equivalent axioms? It might be interesting to record now any speculations as what the structure of this poset, and compare these predictions with the outcome of the project afterwards.

A brute force computation of the poset would then require comparisons, which looks rather daunting; but of course due to the axioms of a partial order, one could presumably identify the poset by a much smaller number of comparisons. I am thinking that it should be possible to crowdsource the exploration of this poset in the form of submissions to a central repository (such as the github repository I just created) of proofs in Lean of implications or non-implications between various equations, which could be validated in Lean, and also checked against some file recording the current status (true, false, or open) of all the comparisons, to avoid redundant effort. Most submissions could be handled automatically, with relatively little human moderation required; and the status of the poset could be updated after each such submission.

I would imagine that there is some “low-hanging fruit” that could establish a large number of implications (or anti-implications) quite easily. For instance, laws such as Equation2 or Equation3 more or less completely describe the binary operation , and it should be quite easy to check which of the laws are implied by either of these two laws. The poset has a reflection symmetry associated to replacing the binary operator by its reflection , which in principle cuts down the total work by a factor of about two. Specific examples of magmas, such as the natural numbers with the addition operation, obey some set of equations in but not others, and so could be used to generate a large number of anti-implications. Some existing automated proving tools for equational logic, such as Prover9 and Mace4 (for obtaining implications and anti-implications respectively), could then be used to handle most of the remaining “easy” cases (though some work may be needed to convert the outputs of such tools into Lean). The remaining “hard” cases could then be targeted by some combination of human contributors and more advanced AI tools.

Perhaps, in analogy with formalization projects, we could have a semi-formal “blueprint” evolving in parallel with the formal Lean component of the project. This way, the project could accept human-written proofs by contributors who do not necessarily have any proficiency in Lean, as well as contributions from automated tools (such as the aforementioned Prover9 and Mace4), whose output is in some other format than Lean. The task of converting these semi-formal proofs into Lean could then be done by other humans or automated tools; in particular I imagine modern AI tools could be particularly valuable for this portion of the workflow. I am not quite sure though if existing blueprint software can scale to handle the large number of individual proofs that would be generated by this project; and as this portion would not be formally verified, a significant amount of human moderation might also be needed here, and this also might not scale properly. Perhaps the semi-formal portion of the project could instead be coordinated on a forum such as this blog, in a similar spirit to past Polymath projects.

It would be nice to be able to integrate such a project with some sort of graph visualization software that can take an incomplete determination of the poset as input (in which each potential comparison in is marked as either true, false, or open), completes the graph as much as possible using the axioms of partial order, and then presents the partially known poset in a visually appealing way. If anyone knows of such a software package, I would be happy to hear of it in the comments.

Anyway, I would be happy to receive any feedback on this project; in addition to the previous requests, I would be interested in any suggestions for improving the project, as well as gauging whether there is sufficient interest in participating to actually launch it. (I am imagining running it vaguely along the lines of a Polymath project, though perhaps not formally labeled as such.)

UPDATE, Sep 30 2024: The project is up and running (and highly active), with the main page being this Github repository. See also the Lean Zulip chat for some (also very active) discussion on the project.

Did Elon Musk make a Nazi salute? Well, not exactly. As far as I can tell, the truth is that he recklessly and repeatedly made a hand gesture that the world’s millions of Nazi sympathizers eagerly misinterpreted as a Nazi salute. He then (the worse part) declined to clarify or apologize in any way, opting instead for laugh emojis.

I hasten to add: just like with Trump’s Charlottesville dogwhistles, I find it ludicrous to imagine that Elon has any secret desire to reopen the gas chambers or whatever—and not only because of Elon’s many pro-Zionist and philosemitic actions, statements, and connections. That isn’t the issue, so don’t pretend I think it is.

Crucially, though, “not being a literal Nazi” isn’t fully exculpatory. I don’t want the overlords of the planet treating these matters as jokes. I want them to feel the crushing weight of history, exactly like I would feel it in their shoes.

Regardless of my distaste for everything that happened to reach this point, Elon is now in a unique position to nudge Trump in the direction of liberality and enlightenment on various issues.  And while I doubt Elon finds time to read Shtetl-Optimized between his CEOing, DOGEing, tweeting, and video game speedruns, I know for certain that there are multiple readers of this blog to whom Elon has listened in the past—and those people are now in a unique position too!

A public “clarification” from Elon—not an apology, not an admission of guilt, but just an acknowledgment that he knows why sleeping dragons like Nazism shouldn’t be poked for shits and giggles, that he’ll try to be careful in the future—would be a non-negligible positive update for me about the future of the world.

I understand exactly why he doesn’t want to do it: because he doesn’t want to grant any legitimacy to what he sees as the biased narrative of a legacy media that despises him. But granting some legitimacy to that narrative is precisely what I, a classically liberal Jewish scientist who bears the battle scars of attempted woke cancellation, am asking him to do. I’m asking him to acknowledge that he’s now by any measure one of the most powerful people on the planet, that with great power comes great responsibility, and that fascism is a well-known failure mode for powerful rightists, just like Communism is a well-known failure mode for leftists. I’m asking for reassurance that he takes that failure mode seriously, just like he correctly takes human extinction and catastrophic AI risk seriously.

Anyway, I figured it was worth a try, given how much I really believe might hinge on how Elon chooses to handle this. I don’t want to be kicking myself, for the rest of my life, that I had a chance to intervene in the critical moment and didn’t.

Err, happy MLK Day!

This week represents the convergence of so many plotlines that, if it were the season finale of some streaming show, I’d feel like the writers had too many balls in the air. For the benefit of the tiny part of the world that cares what I think, I offer the following comments.

My view of Trump is the same as it’s been for a decade—that he’s a con man, a criminal, and the most dangerous internal threat the US has ever faced in its history. I think Congress and Merrick Garland deserve eternal shame for not moving aggressively to bar Trump from office and then prosecute him for insurrection—that this was a catastrophic failure of our system, one for which we’ll now suffer the consequences. If this time Trump got 52% of some swing state rather than 48%, if the “zeitgeist” or the “vibes” have shifted, if the “Resistance” is so weary that it’s barely bothering to show up, if Bezos and Zuckerberg and Musk and even Sam Altman now find it expedient to placate the tyrant rather than standing up for what previously appeared to be their principles—well, I don’t see how any of that affects how I ought to feel.

All the same, I have no plans to flee the United States or anything, just like I didn’t the last time. I’ll even permit myself pleasure when the crazed strongman takes actions that I happen to agree with (like pushing the tottering Ayatollah regime toward its well-deserved end). And then I’ll vote for Enlightenment values (or the nearest available approximation) in 2026 and 2028, assuming the country survives until then.

The second plotline is the ceasefire in Gaza, and the beginning of the release of the Israeli hostages, in exchange for thousands of Palestinian prisoners. I have all the mixed emotions you might expect. I’m terrified about the precedent this reinforces and about the many mass-murderers it will free—as I was terrified in 2011 by the Gilad Shalit deal, the one that released Sinwar and thereby set the stage for October 7. Certainly World War II didn’t end with the Nazis marching triumphantly around Berlin, guns in the air, and vowing to repeat their conquest of Europe at the earliest opportunity. All the same, it’s not my place to be more Zionist than Netanyahu, or than the vast majority of the Israeli public that supported the deal. I’m obviously thrilled to see the hostages return, and even slightly touched by the ethic that would move heaven and earth to save these specific people, almost every consideration of game theory and utilitarianism be damned. I take solace that we’re not quite returning to the situation of October 6, since Hamas, Hezbollah, and Iran itself have all been severely degraded (and the Assad regime no longer exists). This is no longer 1944, when you can slaughter 1200 Jews without paying any price for it: that was the original promise of the State of Israel. All the same, I fear that bloodshed will continue from here until the Singularity, unless majorities on both sides choose coexistence—partition, the two-state solution, call it whatever you will. And that’s primarily a question of culture, and the education of children.

The third plotline was the end of TikTok, quickly followed by its (temporary?) return on Trump’s order. As far as I can tell, Instagram, Twitter/X, and TikTok have all been net negatives for the world; it would’ve been far better if none of them had been invented. But, OK, our society allows many things that are plausibly net-negative, like sports betting and Cheetos. In this case, however, the US Supreme Court ruled 9-0 (!!) that Congress has a legitimate interest in keeping Chinese Communist Party spyware off 170 million Americans’ phones—and that there’s no First Amendment concern that overrides this security interest, since the TikTok ban isn’t targeting speech on the basis of its content. I found the court’s argument convincing. I hope TikTok goes dark 90 days from now—or, second-best, that it gets sold to some entity that’s merely bad in the normal ways and not a hostile foreign power.

The fourth plotline is the still-ongoing devastation of much of Los Angeles. I heard from friends at Caltech and elsewhere who had to evacuate their homes—but at least they had homes to return to, as those in Altadena and the Palisades didn’t. It’s a sign of the times that even a disaster of this magnitude now brings only partisan bickering: was the cause climate change, reshaping the entire planet in terrifying ways, just like all those experts have been warning for decades? Or was it staggering lack of preparation from the California and LA governments? My own answers to these questions are “yes” and “yes.”

Maybe I’ll briefly highlight the role of the utilitarianism versus deontology debate. According to this article from back in October, widely shared once the fires started, the US Forest Service halted controlled burns in California because it lacked the manpower, but also this:

“I think the Forest Service is worried about the risk of something bad happening [with a prescribed burn]. And they’re willing to trade that risk — which they will be blamed for — for increased risks on wildfires,” Wara said. In the event of a wildfire, “if something bad happens, they’re much less likely to be blamed because they can point the finger at Mother Nature.”

We saw something similar with the refusal to allow challenge trials for the COVID vaccines, which could’ve moved the approval date up by months and saved millions of lives. Humans are really bad at trolley problems, at weighing a concrete, immediate risk against a diffuse future risk that might be orders of magnitude worse. (Come to think of it, Israel’s repeated hostage deals are another example—though that one has the defense that it demonstrates the lengths to which the state will go to protect its people.)

Oh, and on top of all the other plotlines, today—January 20th—is my daughter’s 12th birthday. Happy birthday Lily!!

Last August, Toronto’s Centre for Quantum Information and Quantum Control (CQIQC) gave me 35 minutes to make fun of John Preskill in public. CQIQC was hosting its biannual conference, also called CQIQC, in Toronto. The conference features the awarding of the John Stewart Bell Prize for fundamental quantum physics. The prize derives its name for the thinker who transformed our understanding of entanglement. John received this year’s Bell Prize for identifying, with collaborators, how we can learn about quantum states from surprisingly few trials and measurements.

The organizers invited three Preskillites to present talks in John’s honor: Hoi-Kwong Lo, who’s helped steer quantum cryptography and communications; Daniel Gottesman, who’s helped lay the foundations of quantum error correction; and me. I believe that one of the most fitting ways to honor John is by sharing the most exciting physics you know of. I shared about quantum thermodynamics for (simple models of) nuclear physics, along with ten lessons I learned from John. You can watch the talk here and check out the paper, recently published in Physical Review Letters, for technicalities.

John has illustrated this lesson by wrestling with the black-hole-information paradox, including alongside Stephen Hawking. Quantum information theory has informed quantum thermodynamics, as Quantum Frontiers regulars know. Quantum thermodynamics is the study of work (coordinated energy that we can harness directly) and heat (the energy of random motion). Systems exchange heat with heat reservoirs—large, fixed-temperature systems. As I draft this blog post, for instance, I’m radiating heat into the frigid air in Montreal Trudeau Airport.

So much for quantum information. How about high-energy physics? I’ll include nuclear physics in the category, as many of my Europeans colleagues do. Much of nuclear physics and condensed matter involves gauge theories. A gauge theory is a model that contains more degrees of freedom than the physics it describes. Similarly, a friend’s description of the CN Tower could last twice as long as necessary, due to redundancies. Electrodynamics—the theory behind light bulbs—is a gauge theory. So is quantum chromodynamics, the theory of the strong force that holds together a nucleus’s constituents.

Every gauge theory obeys Gauss’s law. Gauss’s law interrelates the matter at a site to the gauge field around the site. For example, imagine a positive electric charge in empty space. An electric field—a gauge field—points away from the charge at every spot in space. Imagine a sphere that encloses the charge. How much of the electric field is exiting the sphere? The answer depends on the amount of charge inside, according to Gauss’s law.

Gauss’s law interrelates the matter at a site with the gauge field nearby…which is related to the matter at the next site…which is related to the gauge field farther away. So everything depends on everything else. So we can’t easily claim that over here are independent degrees of freedom that form a system of interest, while over there are independent degrees of freedom that form a heat reservoir. So how can we define the heat and work exchanged within a lattice gauge theory? If we can’t, we should start biting our nails: thermodynamics is the queen of the physical theories, a metatheory expected to govern all other theories. But how can we define the quantum thermodynamics of lattice gauge theories? My colleague Zohreh Davoudi and her group asked me this question.

I had the pleasure of addressing the question with five present and recent Marylanders…

…the mention of whom in my CQIQC talk invited…

How did we begin establishing a quantum thermodynamics for lattice gauge theories?

Someone who had a better idea than I, when I embarked upon this project, was my colleague Chris Jarzynski. So did Dvira Segal, a University of Toronto chemist and CQIQC’s director. So did everyone else who’d helped develop the toolkit of strong-coupling thermodynamics. I’d only heard of the toolkit, but I thought it sounded useful for lattice gauge theories, so I invited Chris to my conversations with Zohreh’s group.

Strong-coupling thermodynamics concerns systems that interact strongly with reservoirs. System–reservoir interactions are weak, or encode little energy, throughout much of thermodynamics. For example, I exchange little energy with Montreal Trudeau’s air, relative to the amount of energy inside me. The reason is, I exchange energy only through my skin. My skin forms a small fraction of me because it forms my surface. My surface is much smaller than my volume, which is proportional to the energy inside me. So I couple to Montreal Trudeau’s air weakly.

My surface would be comparable to my volume if I were extremely small—say, a quantum particle. My interaction with the air would encode loads of energy—an amount comparable to the amount inside me. Should we count that interaction energy as part of my energy or as part of the air’s energy? Could we even say that I existed, and had a well-defined form, independently of that interaction energy? Strong-coupling thermodynamics provides a framework for answering these questions.

Little is more basic than the first law of thermodynamics, synopsized as energy conservation. The first law governs how much a system’s internal energy changes during any process. The energy change equals the heat absorbed, plus the work absorbed, by the system. Every formulation of thermodynamics should obey the first law—including strong-coupling thermodynamics. 

Which lattice-gauge-theory processes should we study, armed with the toolkit of strong-coupling thermodynamics? My collaborators and I implicitly followed

and

We don’t want to irritate experimentalists by asking them to run difficult protocols. Tom Rosenbaum, on the left of the previous photograph, is a quantum experimentalist. He’s also the president of Caltech, so John has multiple reasons to want not to irritate him.

Quantum experimentalists have run quench protocols on many quantum simulators, or special-purpose quantum computers. During a quench protocol, one changes a feature of the system quickly. For example, many quantum systems consist of particles hopping across a landscape of hills and valleys. One might flatten a hill during a quench.

We focused on a three-step quench protocol: (1) Set the system up in its initial landscape. (2) Quickly change the landscape within a small region. (3) Let the system evolve under its natural dynamics for a long time. Step 2 should cost work. How can we define the amount of work performed? By following

John wrote a blog post about how the typical physicist is a one-trick pony: they know one narrow subject deeply. John prefers to know two subjects. He can apply insights from one field to the other. A two-trick pony can show that Gauss’s law behaves like a strong interaction—that lattice gauge theories are strongly coupled thermodynamic systems. Using strong-coupling thermodynamics, the two-trick pony can define the work (and heat) exchanged within a lattice gauge theory. 

An experimentalist can easily measure the amount of work performed,¹ we expect, for two reasons. First, the experimentalist need measure only the small region where the landscape changed. Measuring the whole system would be tricky, because it’s so large and it can contain many particles. But an experimentalist can control the small region. Second, we proved an equation that should facilitate experimental measurements. The equation interrelates the work performed¹ with a quantity that seems experimentally accessible.

My team applied our work definition to a lattice gauge theory in one spatial dimension—a theory restricted to living on a line, like a caterpillar on a thin rope. You can think of the matter as qubits² and the gauge field as more qubits. The system looks identical if you flip it upside-down; that is, the theory has a symmetry. The system has two phases, analogous to the liquid and ice phases of HO. Which phase the system occupies depends on the chemical potential—the average amount of energy needed to add a particle to the system (while the system’s entropy, its volume, and more remain constant).

My coauthor Connor simulated the system numerically, calculating its behavior on a classical computer. During the simulated quench process, the system began in one phase (like HO beginning as water). The quench steered the system around within the phase (as though changing the water’s temperature) or across the phase transition (as though freezing the water). Connor computed the work performed during the quench.¹ The amount of work changed dramatically when the quench started steering the system across the phase transition. 

Not only could we define the work exchanged within a lattice gauge theory, using strong-coupling quantum thermodynamics. Also, that work signaled a phase transition—a large-scale, qualitative behavior.

What future do my collaborators and I dream of for our work? First, we want for an experimentalist to measure the work¹ spent on a lattice-gauge-theory system in a quantum simulation. Second, we should expand our definitions of quantum work and heat beyond sudden-quench processes. How much work and heat do particles exchange while scattering in particle accelerators, for instance? Third, we hope to identify other phase transitions and macroscopic phenomena using our work and heat definitions. Fourth—most broadly—we want to establish a quantum thermodynamics for lattice gauge theories.

Five years ago, I didn’t expect to be collaborating on lattice gauge theories inspired by nuclear physics. But this work is some of the most exciting I can think of to do. I hope you think it exciting, too. And, more importantly, I hope John thought it exciting in Toronto.

I was a student at Caltech during “One Entangled Evening,” the campus-wide celebration of Richard Feynman’s 100th birthday. So I watched John sing and dance onstage, exhibiting no fear of embarrassing himself. That observation seemed like an appropriate note on which to finish with my slides…and invite questions from the audience.

Congratulations on your Bell Prize, John.

¹Really, the dissipated work.

²Really, hardcore bosons.

Happy New Year!

I recently wrote about how the Parker Solar Probe crossed the Sun’s ‘Alfvén surface’: the surface outside which the outflowing solar wind becomes supersonic.

This is already pretty cool—but even better, the ‘sound’ here is not ordinary sound: it consists of vibrations in both the hot electrically conductive plasma of the Sun’s atmosphere and its magnetic field! These vibrations are called ‘Alfvén waves’.

To understand these waves, we need to describe how an electrically conductive fluid interacts with the electric and magnetic fields. We can do this—to a reasonable approximation—using the equations of ‘magnetohydrodynamics’:

• Wikipedia, Magnetohydrodynamics: equations.

These equations also describe other phenomena we see in stars, outer space, fusion reactors, the Earth’s liquid outer core, and the Earth’s upper atmosphere: the ionosphere, and above that the magnetosphere. These phenomena can be very difficult to understand, combining all the complexities of turbulence with new features arising from electromagnetism. Here’s an example, called the Orzag–Tang vortex, simulated by Philip Mocz:

I’ve never studied magnetohydrodynamics—I was afraid that if I learned a little, I’d never stop, because it’s endlessly complex. Now I’m getting curious. But all I want to do today is explain to you, and myself, what the equations of magnetohydrodynamics are—and where they come from.

To get these equations, we assume that our system is described by some time-dependent fields on 3-dimensional space, or whatever region of space our fluid occupies. I’ll write vector fields in boldface and scalar fields in non-bold:

• the velocity of our fluid,

• the density field of our fluid,

• the pressure field of our fluid,

• the electric current,

• the electric field,

• the magnetic field,

You may have noticed one missing: the charge density! We assume that this is zero, because in a highly conductive medium the positive and negative charges quickly even out unless the electric field is changing very rapidly. (When this assumption breaks down we use more complicated equations.)

So, we start with Maxwell’s equations, but with the charge density set to zero:

I’m writing them in SI units, where they include two constants:

• the electric permittivity of the vacuum,

• the magnetic permeability of the vacuum,

The product of these two constants is the reciprocal of the square of the speed of light:

This makes the last term Maxwell added to his equations very small unless the electric field is changing very rapidly:

In magnetohydrodynamics we assume the electric field is changing slowly, so we drop this last term, getting a simpler equation:

We also assume a sophisticated version of Ohm’s law: the electric current is proportional to the force on the charges at that location. But here the force involves not only the electric field but the magnetic field! So, it’s given by the Lorentz force law, namely

where notice we’re assuming the velocity of the charges is the fluid velocity Thus we get

where is the electrical conductivity of the fluid.

Next we assume local conservation of mass: the increase (or decrease) of the fluid’s density at some point can only be caused by fluid flowing toward (or away from) that point. So, the time derivative of the density is minus the divergence of the momentum density

This is analogous to the equation describing local conservation of charge in electromagnetism: the so-called continuity equation.

We also assume that the pressure is some function of the density:

This is called the equation of state of our fluid. The function depends on the fluid: for example, for an ideal gas is simply proportional to We can use the equation of state to eliminate and work with

Last—but not least—we need an equation of motion saying how the fluid’s velocity changes with time! This equation follows from Newton’s law or more precisely his actual law

where is momentum. But we need to replace by the momentum density and replace by the force density, which is

The force density comes three parts:

• The force of the magnetic field on the current, as described by the Lorentz force law:

• The force caused by the gradient of pressure, pointing toward regions of lower pressure:

• the force caused by viscosity, where faster bits of fluid try to speed up their slower neighbors, and vice versa:

Here is called the coefficient of viscosity.

Thus, our equation of motion is this:

Here we aren’t using just the ordinary time derivative of : we want to keep track of how is changing for a bit of fluid that’s moving along with the flow of the fluid, so we need to add in the derivative of in the direction. For this we use the material derivative:

which also has many other names like ‘convective derivative’ or ‘substantial derivative’.

So, those are the equations of magnetohydrodynamics! Let’s see them in one place. I’ll use the equation of state to eliminate the pressure by writing it as a function of density:

Notice that in our simplified Maxwell equations, two terms involving the electric field are gone. That’s why these are called the equations of magnetohydrodynamics. You can even eliminate the current from these equations, replacing it with The magnetic field reigns supreme!

Magnetic diffusion

It feels unsatisfying to quit right after I show you the equations of magnetohydrodynamics. Having gotten this far, I can’t resist showing you a couple of cool things we can do with these equations!

First, we can use Ohm’s law to see how the magnetic field tends to ‘diffuse’, like heat spreads out through a medium.

We start with Ohm’s law:

We take the curl of both sides:

We can get every term here to involve if we use two of our simplified Maxwell equations:

We get this:

Then we can use this vector identity:

Since another of the Maxwell equations says we get

and thus

and finally we get

Except for the last term, this is the heat equation—but not for temperature, which is a scalar field, but for the magnetic field, which is a vector field! The constant says how fast the magnetic field spreads out, so it’s called the magnetic diffusivity.

The last term makes things more complicated and interesting.

Magnetic pressure and tension

Second, and finally, I want to give you more intuition for how the magnetic field exerts a force on the conductive fluid in magnetohydrodynamics. We’ll see that the magnetic field has both pressure and tension.

Remember, the magnetic field exerts a force

More precisely this is the force per volume: the magnetic force density. We can express this solely in terms of the magnetic field using one of our simplified Maxwell equations

We get

Next we can use a fiendish vector calculus identity: for any vector fields in 3d space we have

When this gives

or what we actually need now:

where

This identity gives a nice formula for

The point of all these manipulations was not merely to revive our flagging memories of vector calculus—though that was good too. The real point is that they reveal that the magnetic force density consists of two parts, each with a nice physical intepretation.

Of course the force acts on the fluid, not on the magnetic field lines themselves. But as you dig deeper into magnetohydrodynamics, you’ll see that sometimes the magnetic field lines get ‘frozen in’ to the fluid—that is, they get carried along by the fluid flow, like bits of rope in a raging river. Then the first term above tends to straighten out these field lines, while the second term tends to push them apart!

The second term is minus the gradient of

Since minus the gradient of ordinary pressure creates a force density on a fluid, we call the quantity above the magnetic pressure. Yes, the square of the magnitude of the magnetic field creates a kind of pressure! It pushes the fluid like this:

The other more subtle term is called magnetic tension:

It points like this:

Its magnitude is where is the radius of curvature of the circle that best fits the magnetic field lines at the given point. And it points toward the center of that circle!

I’m getting tired of proving formulas, so I’ll leave the proofs of these facts as puzzles. If you get stuck and want a hint, go here.

There’s a lot more to say about this, and you can find a lot of it here:

• Gordon I. Ogilvie, Lecture notes: astrophysical fluid dynamics.

I might or might not say more—but if I don’t, these notes will satisfy your curiosity about exactly when magnetic field lines get ‘frozen in’ to the motion of an electrically conductive fluid, and how magnetic pressure then tends to push these field lines apart, while magnetic tension tends to straighten them out!

All of this is very cool, and I think this subject is a place where all the subtler formulas of vector calculus really get put to good use.

Happy New Year! 2025 is the centenary of some very important events in the development of quantum physics — the birth of new insights, of new mathematics, and of great misconceptions. For this reason, I’ve decided that this year I’ll devote more of this blog to quantum fundamentals, and take on some of the tricky issues that I carefully avoided in my recent book.

My focus will be on very basic questions, such as: How does quantum physics work, to the extent we humans understand it? Which of the widely-held and widely-promulgated ideas about quantum weirdness are true? And for those that aren’t, what is the right way to think about them?

I’ll frame some of this discussion in the context of the quantum two-slit experiment, because

• it’s famous,
• it’s often poorly explained
• it’s often poorly understood,
• it highlights (when properly understood) an extraordinarily strange aspect of quantum physics.

Not that I’ll cover this subject all in one post… far from it! It’s going to take quite some time.

The Visualization Problem

We humans often prefer to understand things visually. The problem with explaining quantum physics, aside from the fact that no one understands it 100%, is that all but the ultra-simplest problems are impossible to depict in an image or animation. This forces us to use words instead. Unfortunately, words are inherently misleading. Even when partial visual depictions are possible, they too are almost always misleading. (Math is helpful, but not as much as you’d think; it’s usually subtle and complicated, too.) So communication and clear thinking are big challenges throughout quantum physics.

These difficulties lead to many widespread misconceptions (some of which I myself suffered from when I was a student first learning the subject.) For instance, one of the most prevalent and problematic, common among undergraduates taking courses in chemistry or atomic physics, is the wrong idea that each elementary particle has its own wavefunction — a function which tells us the probability of where it might currently be located. This confusion arises, as much as anything else, from a visualization challenge.

Consider the quantum physics of the three electrons in a lithium atom. If you’ve read anything about quantum physics, you may have been led to believe that that each of the three electrons has a wave function, describing its behavior in three-dimensional space. In other words,

• naively the system would be described by three wave functions, each in three dimensions; each electron’s wave function tells us the probability that it is located at this point or that one;
• but in fact the three electrons are described by one wave function in nine dimensions, telling us simultaneously the overall probability that the first electron is t be found at this point, the second at that point, and the third at some other point.

Unfortunately, drawing something that exists in nine dimensions is impossible! Three wave functions in three dimensions is much easier to draw, and so, as a compromise/approximation that has some merits but is very confusing, that method of depiction is widely used in images of multiple electrons. Here, for instance, two of the lithium atom’s electrons are depicted as though they have wave functions sharing the yellow region (the “inner shell”), while the third is drawn as though it has a wave function occuping the [somewhat overlapping] blue region (the “next shell”). [The atomic nucleus is shown in red, but far larger than it actually is.] Something similar is done in this image of the electrons in oxygen from a chemistry class.)

Yet the meat of the quantum lithium atom lies in the fact that there’s actually only one wave function for the entire system, not three. Most notably, the Pauli exclusion principle, which is responsible for keeping the electrons from all doing the same things and leads to the shell-like structure, makes sense only because there’s only one wave function for the system. And so, the usual visual depictions of the three electrons in the atom are all inherently misleading.

Yet there’s no visual image that can replace them that is both correct and practical. And that’s a real problem.

That said, it is possible to use visual images for two objects traveling in one dimension, as I did in a recent article that explains what it means for a system of two particles to have only one wave function. But for today, we can set this particular issue aside.

What We Can’t Draw Can Hurt Our Brains

Like most interesting experiments, the underyling quantum physics of the quantum double slit experiment cannot be properly drawn. But depicting it somehow, or at least parts of it, will be crucial in understanding how it works. Most existing images that are made to try to explain it leave out important conceptual points. The challenge for me — not yet solved — is to find a better one.

In this post, I’ll start the process, opening a conversation with readers about what people do and don’t understand about this experiment, about what’s often said about it that is misleading or even wrong, and about why it’s so hard to draw anything that properly represents it. Over the year I expect to come back to the subject occasionally. With luck, I’ll find a way to describe this experiment to my satisfaction, and maybe yours, before the end of the year. I don’t know if I’ll succeed. Even if I do, the end product won’t be short, sweet and simple.

But let’s start at the beginning, with the conventional story of the quantum double-slit experiment. The goal here is not so much to explain the experiment — there are many descriptions of it on the internet — but rather to focus on exactly what we say and think about it. So I encourage you to read slowly and pay very close attention; in this business, every word can matter.

Observing the Two Slits and the Screen

We begin by throwing an ultra-microscopic object — perhaps a photon, or an electron, or a neutrino — toward a wall with two narrow, closely spaced slits cut in it. (The details of how we do this are not very important, although we do need to choose the slits and the distance to the screen with some care.) If the object manages to pass through the wall, then on the other side it continues onward until it hits a phosphorescent screen. Where it strikes the screen, the screen lights up. This is illustrated in Fig. 1, where several such objects are showing being sent outward from the left; a few pass through the slits and cause the screen to light up where they arrive.

If we do this many times and watch the screen, we’ll see flashes randomly around the screen, something like what is shown in Fig. 2:

But now let’s keep a record of where the flashes on the screen appear; that’s shown in Fig. 3, where new flashes are shown in orange and past flashes are shown in blue. When we do this, we’ll see a strange pattern emerge, seen not in each individual flash but over many flashes, growing clearer as the number of flashes increases. This pattern is not simply a copy of the shape of the two slits.

After a couple of thousand flashes, we’ll recognize that the pattern is characteristic of something known as interference (discussed further in Figs. 6-7 below):

By the way, there’s nothing hypothetical about this. Performing this experiment is not easy, because both the source of the objects and the screen are delicate and expensive. But I’ve seen it done, and I can confirm that what I’ve told you is exactly what one observes.

Trying to Interpret the Observations

The question is: given what is observed, what is actually happening as these microscopic objects proceed from source through slits to screen? and what can we infer about their basic properties?

We can conclude right away that the objects are not like bullets — not like “particles” in the traditional sense of a localized object that travels upon a definite path. If we fired bullets or threw tiny balls at the slitted wall, the bullets or balls would pass through the two slits and leave two slit-shaped images on the screen behind them, as in Fig. 5.

Nor are these objects ripples, meaning “waves” of some sort. Caution! Here I mean what scientists and recording engineers mean by “wave”: not a single wave crest such as you’d surf at a beach, but rather something that is typically a series of wave crests and troughs. (Sometimes we call this a “wave set” in ordinary English.)

If each object were like a wave, we’d see no dot-like flashes. Instead each object would leave the interference pattern seen in Fig. 4. This is illustrated in Fig. 6 and explained in Fig. 7. A wave (consisting of multiple crests and troughs) approaches the slits from the left in Fig. 6. After it passes through the slits, a striking pattern appears on the screen, with roughly equally spaced bright and dark regions, the brightest one in the center.

Where does the interference pattern come from? This is clearest if we look at the system from above, as in Fig. 7. The wave is coming in from the left, as a linear set of ripples, with crests in blue-green and troughs in red. The wall (represented in yellow) has two slits, from which emerge two sets of circular ripples. These ripples add and subtract from one another, making a complex, beautiful “interference” pattern. When this pattern reaches the screen at the opposite wall, it creates a pattern on the screen similar to that sketched in Fig. 6, with some areas that actively flicker separated by areas that are always dark.

It’s important to notice that the center of the phosphorescent screen is dark in Fig. 5 and bright in Fig. 6. The difference between particle-like bullets and wave-like ripples is stark.

And yet, whatever objects we’re dealing with in Figs. 2-4, they are clearly neither like the balls of Fig. 5 nor the waves of Fig. 6. Their arrival is marked with individual flashes, and the interference pattern builds up flash by flash; one object alone does not reveal the pattern. Strangely, each object seems to “know” about the pattern. After all, each one, independently, manages to avoid the dark zones and to aim for one of the bright zones.

How can these objects do this? What are they?

What Are These Objects?!

According to the conventional wisdom, Fig. 2 proves that the objects are somewhat like particles. When each object hits the wall, it instantaneously causes a single, tiny, localized flash on the screen, showing that it is itself a single, tiny, point-like object. It’s like a bullet leaving a bullet-hole: localized, sudden, and individual.

According to the conventional wisdom, Figs. 3-4 prove that the objects are somewhat like waves. They leave the same pattern that we would see if ocean swell were passing through two gaps in a harbor’s breakwater, as in Fig. 7. Interference patterns are characteristic only of waves. And because the interference pattern builds up over many independent flashes, occurring at different times, each object seems to “know,” independent of the others, what the interference pattern is. The logical conclusion is that each object interferes with itself, just as the waves of Figs. 6-7 do; otherwise how could each object “know” anything about the pattern? Interfering with oneself is something a wave can do, but a bullet or ball or anything else particle-like certainly cannot.

To review:

• A set of particles going through two slits wouldn’t leave an interference pattern; it would leave the pattern we’d expect of a set of bullets, as in Fig. 5.
• But waves going through two slits wouldn’t leave individual flashes on a screen; each wave would interfere with itself and leave activity all over the screen, with stronger and weaker effects in a predictable interference pattern, as in Figs. 6-7.

It’s as though the object is a wave when it goes through and past the slits, and turns into a particle before it hits the screen. (Note my careful use of the words “as though”; I did not say that’s what actually happens.)

And thus, according to the conventional wisdom, each object going through the slits is… well… depending on who you talk to or read…

• both a wave and a particle, or
• sometimes a wave and sometimes a particle, or
• equally wave and particle, or
• a thing with wave-like properties and with particle-like properties, or
• a particle-like thing described by a probability-wave (also known as a “wave function”), or
• a wave-like thing that can only be absorbed like a particle, or…
• ???

So… which is it?

Or is it any of the above?

Looking More Closely

We could try to explore this further. For instance, we could try to look more closely at what is going on, by asking whether our object is a particle that goes through one slit or is a wave that goes through both.

But the very process of looking at the object to see what slit it went through changes the interference pattern of Figs. 4 and 6 into the pattern in Fig. 5, shown in Fig. 9, that we’d expect for particles. We find two blobs, one for each slit, and no noticeable interference. It’s as though, by looking at an ocean wave, we turned it into a bullet, whereas when we don’t look at the ocean wave, it remains an ocean wave as it goes through the gaps, and only somehow coalesces into a bullet before it hits (or as it hits) the screen.

Said another way: it seems we cannot passively look at the objects. Looking at them is an active process, and it changes how they behave.

So this really doesn’t clarify anything. If anything, it muddies the waters further.

What sense can we make of this?

Before we even begin to try to make a coherent understanding out of this diverse set of observations, we’d better double-check that the logic of the conventional wisdom is accurate in the first place. To do that, each of us should read very carefully and think very hard about what has been observed and what has been written about it. For instance, in the list of possible interpretations given above, do the words “particle” and “wave” always mean what we think they do? They have multiple meanings even in English, so are we all thinking and meaning the same thing when we describe something as, say, “sometimes a wave and sometimes a particle”?

If we are very careful about what is observed and what is inferred from what is observed, as well as the details of language used to communicate that information, we may well worry about secret and perhaps unjustified assumptions lurking in the conventional wisdom.

For instance, does the object’s behavior at the screen, as in Fig. 2, really resemble a bullet hitting a wall? Is its interaction with the screen really instantaneous and tiny? Are its effects really localized and sudden?

Exactly how localized and sudden are they?

All we saw at the screen is a flash that is fast by human standards, and localized by human standards. But why would we apply human standards to something that might be smaller than an atom? Should we instead be judging speed and size using atomic standards? Perhaps even the standards of tiny atomic nuclei?

If our objects are among those things usually called “elementary particles” — such as photons, electrons, or neutrinos — then the very naming of these objects as “elementary particles” seems to imply that they are smaller than an atom, and even than an atom’s nucleus. But do the observations shown in Fig. 2 actually give some evidence that this is true? And if not… well, what do they show?

What do we precisely mean by “particle”? By “elementary particle”? By “subatomic particle”?

What actually happened at the slits? at the screen? between them? Can we even say, or know?

These are among the serious questions that face us. Something strange is going on, that’s for sure. But if we can’t first get our language, our logic, and our thinking straight — and as a writer, if I don’t choose and place every single word with great care — we haven’t a hope of collectively making sense of quantum physics. And that’s why this on-and-off discussion will take us all of 2025, at a minimum. Maybe it will take the rest of the decade. This is a challenge for the human mind, both for novices and for experts.

A while back, I was surprised when I saw the writer of a well-researched webcomic assume that academics are paid for their articles. I ended up writing a post explaining how academic publishing actually works.

Now that I’m out of academia, I’m noticing some confusion on the other side. I’m doing freelance journalism, and the academics I talk to tend to have some common misunderstandings. So academics, this post is for you: a FAQ of questions I’ve been asked about freelance journalism. Freelance journalism is more varied than academia, and I’ve only been doing it a little while, so all of my answers will be limited to my experience.

Q: What happens first? Do they ask you to write something? Do you write an article and send it to them?

Academics are used to writing an article, then sending it to a journal, which sends it out to reviewers to decide whether to accept it. In freelance journalism in my experience, you almost never write an article before it’s accepted. (I can think of one exception I’ve run into, and that was for an opinion piece.)

Sometimes, an editor reaches out to a freelancer and asks them to take on an assignment to write a particular sort of article. This happens more freelancers that have been working with particular editors for a long time. I’m new to this, so the majority of the time I have to “pitch”. That means I email an editor describing the kind of piece I want to write. I give a short description of the topic and why it’s interesting. If the editor is interested, they’ll ask some follow-up questions, then tell me what they want me to focus on, how long the piece should be, and how much they’ll pay me. (The last two are related, many places pay by the word.) After that, I can write a draft.

Q: Wait, you’re paid by the word? Then why not make your articles super long, like Victor Hugo?

I’m paid per word assigned, not per word in the finished piece. The piece doesn’t have to strictly stick to the word limit, but it should be roughly the right size, and I work with the editor to try to get it there. In practice, places seem to have a few standard size ranges and internal terminology for what they are (“blog”, “essay”, “short news”, “feature”). These aren’t always the same as the categories readers see online. Some places have a web page listing these categories for prospective freelancers, but many don’t, so you have to either infer them from the lengths of articles online or learn them over time from the editors.

Q: Why didn’t you mention this important person or idea?

Because pieces pay more by the word, it’s easier as a freelancer to sell shorter pieces than longer ones. For science news, favoring shorter pieces also makes some pedagogical sense. People usually take away only a few key messages from a piece, if you try to pack in too much you run a serious risk of losing people. After I’ve submitted a draft, I work with the editor to polish it, and usually that means cutting off side-stories and “by-the-ways” to make the key points as vivid as possible.

Q: Do you do those cool illustrations?

Academia has a big focus on individual merit. The expectation is that when you write something, you do almost all of the work yourself, to the extent that more programming-heavy fields like physics and math do their own typesetting.

Industry, including journalism, is more comfortable delegating. Places will generally have someone on-staff to handle illustrations. I suggest diagrams that could be helpful to the piece and do a sketch of what they could look like, but it’s someone else’s job to turn that into nice readable graphic design.

Q: Why is the title like that? Why doesn’t that sound like you?

Editors in journalistic outlets are much more involved than in academic journals. Editors won’t just suggest edits, they’ll change wording directly and even input full sentences of their own. The title and subtitle of a piece in particular can change a lot (in part because they impact SEO), and in some places these can be changed by the editor quite late in the process. I’ve had a few pieces whose title changed after I’d signed off on them, or even after they first appeared.

Q: Are your pieces peer-reviewed?

The news doesn’t have peer review, no. Some places, like Quanta Magazine, do fact-checking. Quanta pays independent fact-checkers for longer pieces, while for shorter pieces it’s the writer’s job to verify key facts, confirming dates and the accuracy of quotes.

Q: Can you show me the piece before it’s published, so I can check it?

That’s almost never an option. Journalists tend to have strict rules about showing a piece before it’s published, related to more political areas where they want to preserve the ability to surprise wrongdoers and the independence to find their own opinions. Science news seems like it shouldn’t require this kind of thing as much, it’s not like we normally write hit pieces. But we’re not publicists either.

In a few cases, I’ve had people who were worried about something being conveyed incorrectly, or misleadingly. For those, I offer to do more in the fact-checking stage. I can sometimes show you quotes or paraphrase how I’m describing something, to check whether I’m getting something wrong. But under no circumstances can I show you the full text.

Q: What can I do to make it more likely I’ll get quoted?

Pieces are short, and written for a general, if educated, audience. Long quotes are harder to use because they eat into word count, and quotes with technical terms are harder to use because we try to limit the number of terms we ask the reader to remember. Quotes that mention a lot of concepts can be harder to find a place for, too: concepts are introduced gradually over the piece, so a quote that mentions almost everything that comes up will only make sense to the reader at the very end.

In a science news piece, quotes can serve a couple different roles. They can give authority, an expert’s judgement confirming that something is important or real. They can convey excitement, letting the reader see a scientist’s emotions. And sometimes, they can give an explanation. This last only happens when the explanation is very efficient and clear. If the journalist can give a better explanation, they’re likely to use that instead.

So if you want to be quoted, keep that in mind. Try to say things that are short and don’t use a lot of technical jargon or bring in too many concepts at once. Convey judgement, which things are important and why, and convey passion, what drives you and excited you about a topic. I am allowed to edit quotes down, so I can take a piece of a longer quote that’s cleaner or cut a long list of examples from an otherwise compelling statement. I can correct grammar and get rid of filler words and obvious mistakes. But I can’t put words in your mouth, I have to work with what you actually said, and if you don’t say anything I can use then you won’t get quoted.

Wow! Biologists seem to have discovered an entirely new kind of life form. They’re called ‘obelisks’, and you probably have some in you.

They were discovered in 2024—not by somebody actually seeing one, but by analyzing huge amounts of genetic data from the human gut. This search found 29,959 new RNA sequences, similar to each other, but very different from any previously known. Thus, we don’t know where these things fit into the tree of life!

Biologists found them when they were trying to solve a puzzle. Even smaller than viruses, there exist ‘viroids’ that are just loops of RNA that cleverly manage to reproduce using the machinery of the cell they infect. Viruses have a protein coat. Viroids are just bare RNA—it doesn’t even code for any proteins!

But all known viroids only infect plants. The first one found causes a disease in potatoes; another causes a disease in avocados, and so on. This raised the puzzle: why aren’t there viroids that infect bacteria, or animals?

Now perhaps we’ve found them! But not quite: while obelisks may work in a similar way, they seem genetically unrelated. Also, their RNA seems to code for two proteins.

Given how little we know about this stuff, I think some caution is in order. Still, this is really cool. Do any of you biologists out there know any research going on now to learn more?

The original paper is free to read on the bioRxiv:

I see just one other paper, about an automated system for detecting obelisks:

There’s also a budding Wikipedia article on obelisks:

Thanks to Mike Stay for pointing this out!

I visited an old estate today
Whose gardens, much acclaimed throughout the world,
Spread out beyond the gated entranceway
In scenic splendors gradually unfurled.
Bright potted blooms sprung beaming by the drive,
While further off, large topiary yews
Rose stoutly in the air. The site, alive
With summer, traded sunned and shaded views.
This composition—classical, restrained—
Bespoke the glories of a golden age:
An equilibrium perchance ordained
By god-directors on an earthly stage.
I set off lightly down a stone-laid path
With boxwood gathered cloudlike at each side;
Here, no hint of nature’s chastening wrath
Impinged upon man’s flourishes of pride.
Proceeding, I pressed forward on a walk
Of gravel, through an arch of climbing vines
And ramblers—where a long, inquiring stalk
Of fragrant musk rose, with its fuzzy spines,
Pressed toward my face with luscious pale-pink blooms.
Aha! At once, I almost could distill
A wayward slant within these outdoor rooms!
Though order reigned, I glimpsed some riot spill
From bush and bending bough. Each bordered sward
Spoke elegance, encasing gemlike pools—
Appointed, each, with cherubs keeping guard—
Yet moss had gathered round the sculpted stools.
Still rapt by these core plats, I shortly passed
Into the grounds beyond—deep green and vast.
Mincing, I issued from that inner fold
To find ahead a trilling rivulet
With flowers on either side; yet how controlled
Was even this—a cautious, cool vignette!
Nonetheless, as in some scattered spots
I’d seen before, there crept a shaggy clump
Of unmown grass—a few forgotten blots
Upon this bloom-besprinkled sphere; a bump
Of wild daylilies, mowed along with lawn;
And, further yet afield, a sprawling mire
Spread forth, less fettered still. Here berries’ brawn
Arose obscenely through the bulbs—each briar
Announcing more desuetude in this place—
Until, at once, stopped short all tended space.
There meadows shot up—shaggy, coarse, and plain;
Wiry weeds and scabbed grass, and the mad buzz
Of insects’ millions; here the splattering rain
Had mothered slug and bug and mugworts’ fuzz.
Beyond, thick woodlands, reckless and abrupt,
Loomed, calling, “Ho, enough of tended stuff!
We are what’s real!”—threatening to erupt—
Ah, dizzy, blowsy trees—nature’s high rough
Abandon!
And so I mused upon the human mind:
Its own mild garths and cool Augustan plots
Were laid for promenades—sedate, refined—
A genteel garden park, it seems, of thoughts.
Or so it might appear—yet gather close,
O marveling guest, round something slantwise spied:
An errant feature free of plan or pose—
A rankling thing you’d wish you hadn’t eyed!
Here, stark, the prankster stands—perhaps a spire
Of malice rising prideful in the air;
Perhaps a wild confusion of desire;
Perhaps a raw delusion, none too rare.
Unchecked, untrimmed, they hint at countless more
Uncomely details sprung at every edge
Of reason’s fair cross-axis, past the door
Of harmony’s last stand; truth’s final hedge.
Observe: my own best traits were raised by force
In soil hauled in from some more fertile strand.
My consciousness, when nature takes its course,
Still bristles, as if tended by no hand.
Stripped were the grounds from which my grace was carved;
Spaded, seeded, hoed its gaping womb—
Beaten and blazed its weeds; its vermin starved
To press my brain toward paths and gorgeous bloom.
Friend, pass no further from my watered spheres,
Well-groomed to please men’s civil hands and eyes!
For past these bounds, a roiling madness rears;
The way of chaos rules and wilds arise!

This was written by my sister Alexandra Baez, who died on Saturday, September 21st, 2024. She had cancer of the tongue, which she left untreated for too long, and it metastatized: in the end, despite radiation therapy and chemotherapy, she was asphixiated by a tumor in her throat.

She had her own landscape maintenance business, which fit perfectly with her deep love and knowledge of plants. She put a lot of energy into formal poetry—that is, poetry with a strict meter and rhyme scheme. This poem is longer than most of hers, and I think this length helps bring the reader into the world of the formal garden, and then out of it into wild world of nature: the real.

We’re brought up to say that the dual concept of injection is surjection, and of course there’s a perfectly good reason for this. The monics in the category of sets are the injections, the epics are the surjections, and monics and epics are dual concepts in the usual categorical sense.

But there’s another way of looking at things, which gives a different answer to the question “what is the dual concept of injection?”

This different viewpoint comes from the observation that in the category of sets, the injections are precisely the coproduct inclusions (coprojections)

$$A \to A + B.$$

Certainly every coproduct inclusion in the category of sets is injective. And conversely, any injection is a coproduct inclusion, because every subset of a set has a complement.

So, the injections between sets are the specialization to $\mathbf{Set}$ of the general categorical concept of coproduct coprojection. The dual of that general categorical concept is product projection. Hence one can reasonably say that the dual concept of injection is that of product projection

$$A \times B \to A,$$

where $A$ and $B$ are sets.

Which maps of sets are product projections? It’s not hard to show that they’re exactly the functions $f: X \to Y$ whose fibres are all isomorphic:

$$f^{-1}(y) \cong f^{-1}(y')$$

for all $y, y' \in Y$. You could reasonably call these “uniform” maps, or “even coverings”, or maybe there’s some other good name, but in this post I’ll just call them projections (with the slightly guilty feeling that this is already an overworked term).

Projections are always surjections, except in the trivial case where the fibres are all empty; this happens when our map is of the form $\varnothing \to Y$. On the other hand, most surjections aren’t projections. A surjection is a function whose fibres are nonempty, but there’s no guarantee that all the fibres will be isomorphic, and usually they’re not.

A projection appears in a little puzzle I heard somewhere. Suppose someone hands you a tangle of string, a great knotted mass with who knows how many bits of string all jumbled together. How do you count the number of pieces of string?

The slow way is to untangle everything, separate the strands, then count them. But the fast way is to simply count the ends of the pieces of string, then divide by two. The point is that there’s a projection — specifically, a two-to-one map — from the set of ends to the set of pieces of string.

String theory aside, does this alternative viewpoint on the dual concept of injection have any importance? I think it does, at least a little. Let me give two illustrations.

Factorization, graphs and cographs

It’s a standard fact that any function between sets can be factorized as a surjection followed by an injection, and the same is true in many other familiar categories. But here are two other methods of factorization, dual to one another, that also work in many familiar categories. One involves injections, and the other projections.

• In any category with finite coproducts, any map $f: X \to Y$ factors as

  $$X \to X + Y \stackrel{\binom{f}{1}}{\to} Y,$$

  where the first map is the coproduct coprojection. This is a canonical factorization of $f$ as a coproduct coprojection followed by a (canonically) split epic. In $\mathbf{Set}$, it’s a canonical factorization of a function $f$ as an injection followed by a surjection (or “split surjection”, if you don’t want to assume the axiom of choice).

  Unlike the usual factorization involving a surjection followed by an injection, this one isn’t unique: there are many ways to factor $f$ as an injection followed a surjection. But it is canonical.

• In any category with finite products, any map $f: X \to Y$ factors as

  $$X \stackrel{(1, f)}{\to} X \times Y \to Y,$$

  where the second map is the product projection. This is a canonical factorization of $f$ as a split monic followed by a projection. In $\mathbf{Set}$, “split monic” just means a function with a retraction, or concretely, an injection with the property that if the domain is empty then so is the codomain.

  (I understand that in some circles, this or a similar statement is called the “fundamental theorem of reversible computing”. This seems like quite a grand name, but I don’t know the context.)

  Again, even in $\mathbf{Set}$, this factorization is not unique: most functions can be factored as a split monic followed by a projection in multiple ways. But again, it is canonical.

These factorizations have a role in basic set theory. Let’s consider the second one first.

Take a function $f: X \to Y$. The resulting function

$$(1, f): X \to X \times Y$$

is injective, which means it represents a subset of $X \times Y$, called the graph of $f$. A subset of $X \times Y$ is also called a relation between $X$ and $Y$; so the graph of $f$ is a relation between $X$ and $Y$.

The second factorization shows that $f$ can be recovered from its graph, by composing with the projection $X \times Y \to Y$. Thus, the set of functions from $X$ to $Y$ embeds into the set of relations between $X$ and $Y$. Which relations between $X$ and $Y$ correspond to functions? The answer is well-known: it’s the set of relations that are “functional”, meaning that each element of $X$ is related to exactly one element of $Y$.

This correspondence between functions and their graphs is very important for many reasons, which I won’t go into here. But it has a much less well known dual, which involves the first factorization.

Here I have to take a bit of a run-up. The injections into a set $S$, taken up to isomorphism over $S$, correspond to the subsets of $S$. Dually, the surjections out of $S$, taken up to isomorphism under $S$, correspond to the equivalence relations on $S$. (This is more or less the first isomorphism theorem for sets.) And just as we define a relation between sets $X$ and $Y$ to be a subset of $X \times Y$, it’s good to define a corelation between $X$ and $Y$ to be an equivalence relation on $X + Y$.

Now, given a function $X \to Y$, the resulting function

$$\binom{f}{1}: X + Y \to Y$$

is surjective, and so represents an equivalence relation on $X + Y$. This equivalence relation on $X + Y$ is called the cograph of $f$, and is a corelation between $X$ and $Y$.

The first of the two factorizations above shows that $f$ can be recovered from its cograph, by composing with the injection $X \to X + Y$. Thus, the set of functions from $X$ to $Y$ embeds into the set of corelations between $X$ and $Y$. Which corelations between $X$ and $Y$ correspond to functions? This isn’t so well known, but — recalling that a corelation between $X$ and $Y$ is an equivalence relation on $X + Y$ — you can show that it’s the corelations with the property that every equivalence class contains exactly one element of $Y$.

I’ve digressed enough already, but I can’t resist adding that the dual graph/cograph approaches correspond to the two standard types of picture that we draw when talking about functions: graphs on the left, and cographs on the right.

For example, cographs are discussed in Lawvere and Rosebrugh’s book Sets for Mathematics.

Back to the injection-projection duality! I said I’d give two illustrations of the role it plays. That was the first one: the canonical factorization of a map as an injection followed by a split epic or a split monic followed by a projection. Here’s the second.

Free and cofree presheaves

Let $A$ be a small category. If you’re given a set $P(a)$ for each object $a$ of $A$ then this does not, by itself, constitute a functor $A \to \mathbf{Set}$. A functor has to be defined on maps too. But $P$ does generate a functor $A \to \mathbf{Set}$. In fact, it generates two of them, in two dual universal ways.

More exactly, we’re starting with a family $(P(a))_{a \in A}$ of sets, which is in object of the category $\mathbf{Set}^{ob\ A}$. There’s a forgetful functor

$$\mathbf{Set}^A \to \mathbf{Set}^{ob\ A},$$

and the general lore of Kan extensions tells us that it has both a left adjoint $L$ and a right adjoint $R$. Better still, it provides explicit formulas: the functors $L(P), R(P): A \to \mathbf{Set}$ are given by

$$(L(P))(a) = \sum_b A(b, a) \times P(b)$$

(where $\sum$ means coproduct or disjoint union), and

$$(R(P))(a) = \prod_b P(b)^{A(a, b)}.$$

I’ll call $L(P)$ the free functor on $P$, and $R(P)$ the cofree functor on $P$.

Example   Let $G$ be a group, seen as a one-object category. Then we’re looking at the forgetful functor $\mathbf{Set}^G \to \mathbf{Set}$ from $G$-sets to sets, which has both adjoints.

The left adjoint $L$ sends a set $P$ to $G \times P$ with the obvious $G$-action, which is indeed usually called the free $G$-set on $P$.

The right adjoint $R$ sends a set $P$ to the set $P^G$ with its canonical action: acting on a family $(p_g)_{g \in G} \in P^G$ by an element $u \in G$ produces the family $(p_{g u})_{g \in G}$. This is a cofree $G$-set.

Cofree group actions are important in some parts of the theory of dynamical systems, where $G$ is often the additive group $\mathbb{Z}$. A $\mathbb{Z}$-set is a set equipped with an automorphism. The cofree $\mathbb{Z}$-set on a set $P$ is the set $R(P) = P^{\mathbb{Z}}$ of double sequences of elements of the “alphabet” $P$, where the automorphism shifts a sequence along by one. This is called the full shift on $P$, and it’s a fundamental object in symbolic dynamics.

What’s this got to do with the injection-projection duality? The next example gives a strong clue.

Example   Let $A$ be the category $(0 \to 1)$ consisting of a single nontrivial map. Then an object of $\mathbf{Set}^{ob\ A}$ is a pair $(P_0, P_1)$ of sets, and an object of $\mathbf{Set}^A$ is a function $X_0 \stackrel{f}{\to} X_1$ between a pair of sets.

The free functor $A \to \mathbf{Set}$ on $P$ is the injection

$$A \to A + B.$$

The cofree functor $A \to \mathbf{Set}$ on $P$ is the projection

$$A \times B \to B.$$

More generally, it’s a pleasant exercise to prove:

• Any free functor $X: A \to \mathbf{Set}$ preserves monics. In other words, if $u$ is a monic in $A$ then $X(u)$ is an injection.

• Any cofree functor $X: A \to \mathbf{Set}$ sends epics to projections. In other words, if $u$ is an epic in $A$ then $X(u)$ is a projection.

These necessary conditions for being a free or cofree functor aren’t sufficient. (If you want a counterexample, think about group actions.) But they give you some feel for what free and cofree functors are like.

For instance, if $A$ is the free category on a directed graph, then every map in $A$ is both monic and epic. So when $X: A \to \mathbf{Set}$ is free, the functions $X(u)$ are all injections, for every map $u$ in $A$. When $X$ is cofree, they’re all projections. This imposes a severe restriction on which functors can be free or cofree.

(Question: do quiver representation people ever look at cofree representations? Here we’d replace $\mathbf{Set}$ by $\mathbf{Vect}$.)

In another post, I’ll explain why I’ve been thinking about this.

We recently got back from a family trip to Mexico, a country I’d almost never been to (a couple of childhood day trips to Nogales, a walk across the border into Juárez in 1999, a day and a half in Cabo San Lucas giving a lecture.) I’m a fan! Some surprises:

1. Drinking pulque under a highway bridge with our family (part of the food tour!) I heard this incredible banger playing on the radio:

“Reaktorn läck i Barsebäck” doesn’t sound like the name of a Mexican song, and indeed, this is a 1980 album track by Sweidish raggare rocker Eddie Meduza, which, for reasons that seem to be completely unknown, is extremely popular in Mexico, where it is (for reasons that, etc etc) known as “Himno a la Banda.”

2. Mexico had secession movements. Yucatán was an independent state in the 1840s. (One of our guides, who moved there from Mexico City, told us it still feels like a different country.) And a Maya revolt in the peninsula created a de facto independent state for decades in the second half of the 19th century. Apparently the perceived weakness of the national government was one cause of the Pastry War.

3. Partly as a result of the above, sisal plantation owners in Yucatán were short of indentured workers, so they imported 1,000 desperate Korean workers in 1905. By the time their contracts ended, independent Korea had been overrun by Japan. So they stayed in Mexico and their descendants still live there today.

4. It is customary in Mexico, or at least in Mexico City, to put a coat rack right next to the table in a restaurant. I guess that makes sense! Why wouldn’t you want your coat right there, and isn’t it nicer for it to be on a rack than hung over the back of your chair?

5. The torta we usually see in America, on a big round roll, is a Central Mexico torta. In Yucatán, a torta is served on French bread, because the filling is usually cocinita pibil, which is so juicy it would make a soft roll soggy. A crusty French roll can soak up the juices without losing its structural integrity. The principle is essentially the same as that of an Italian beef.

6. The Mesoamerican ballgame is not only still played throughout the region, there’s a World Cup of it.

After living in Japan for about four months, we left in mid-December. We miss it already.

One of the pleasures we discovered is the onsen, or hot spring. Originally referring to the natural volcanic springs themselves, and the villages around them, there are now onsens all over Japan. Many hotels have an onsen, and most towns will have several. Some people still use them as their primary bath and shower for keeping clean. (Outside of actual volcanic locations, these are technically sento rather than onsen.) You don’t actually wash yourself in the hot baths themselves; they are just for soaking, and there are often several, at different temperatures, mineral content, indoor and outdoor locations, whirlpools and even “electric baths” with muscle-stimulating currents. For actual cleaning, there is a bank of hand showers, usually with soap and shampoo. Some can be very basic, some much more like a posh spa, with massages, saunas, and a restaurant.

Our favourite, about 25 minutes away by bicycle, was Kirari Onsen Tsukuba. When not traveling, we tried to go every weekend, spending a day soaking in the hot water, eating the good food, staring at the gardens, snacking on Hokkaido soft cream — possibly the best soft-serve ice cream in the world (sorry, Carvel!), and just enjoying the quiet and peace. Even our seven- and nine-year old girls have found the onsen spirit, calming and quieting themselves down for at least a few hours.

Living in Tsukuba, lovely but not a common tourist destination, although with plenty of foreigners due to the constellation of laboratories and universities, we were often one of only one or two western families in our local onsen. It sometimes takes Americans (and those from other buttoned-up cultures) some time to get used to their sex-segregated but fully-naked policies of the baths themselves. The communal areas, however, are mixed, and fully-clothed. In fact, many hotels and fancier onsen facilities supply a jinbei, a short-sleeve pyjama set in which you can softly pad around the premises during your stay. (I enjoyed wearing jinbei so much that I purchased a lightweight cotton set for home, and am also trying to get my hands on samue, a somewhat heavier style of traditional Japanese clothing.)

And my newfound love for the onsen is another reason not to get a tattoo beyond the sagging flesh and embarrassment of my future self: in Japan, tattoos are often a symbol of the yakuza, and are strictly forbidden in the onsen, even for foreigners.

Later in our sabbatical, we will be living in the Netherlands, which also has a good public bath culture, but it will be hard to match the calm of the Japanese onsen.

Update (Jan. 13): Readers might enjoy the Bankless Podcast, in which I and Justin Drake of the Ethereum engineering team discuss quantum computing and its impact on cryptocurrency. I learned something interesting from Justin—namely that Satoshi has about $90 billion worth of bitcoin that’s never been touched since the cryptocurrency’s earliest days, much of which (added: the early stuff, the stuff not additionally protected by a hash function) would be stealable by anyone who could break elliptic curve cryptography—for example, by using a scalable quantum computer. At what point in time, if any, would this stash acquire the moral or even legal status of (say) gold doubloons just lying on the bottom of the ocean? Arrr, ’tis avast Hilbert space!

Apparently Jensen Huang, the CEO of NVIDIA, opined on an analyst call this week that quantum computing was plausibly still twenty years away from being practical. As a direct result, a bunch of publicly-traded quantum computing companies (including IonQ, Rigetti, and D-Wave) fell 40% or more in value, and even Google/Alphabet stock fell on the news.

So then friends and family attuned to the financial markets started sending me messages asking for my reaction, as the world’s semi-unwilling Quantum Computing Opiner-in-Chief.

My reaction? Mostly just that it felt really weird for all those billions of dollars to change hands, or evaporate, based on what a microchip CEO offhandedly opined about my tiny little field, while I (like much of that field) could’ve remained entirely oblivious to it, were it not for all of their messages!

But was Jensen Huang right in his time estimate? And, relatedly, what is the “correct” valuation of quantum computing companies? Alas, however much more I know about quantum computing than Jensen Huang does, the knowledge does not enable me to answer to either question.

I can, of course, pontificate about the questions, as I can pontificate about anything.

To start with the question of timelines: yes, there’s a lot still to be done, and twenty years might well be correct. But as I’ve pointed out before, within the past year we’ve seen 2-qubit gates with ~99.9% fidelity, which is very near the threshold for practical fault-tolerance. And of course, Google has now demonstrated fault-tolerance that becomes more and more of a win with increasing code size. So no, I can’t confidently rule out commercially useful quantum simulations within the next decade. Like, it sounds fanciful, but then I remember how fanciful it would’ve seemed in 2012 that we’d have conversational AI by 2022. I was alive in 2012! And speaking of which, if you really believe (as many people now do) AI will match or exceed human capabilities in most fields in the next decade, then that will scramble all the other timelines too. And presumably Jensen Huang understands these points as well as anyone.

Now for the valuation question. On the one hand, Shtetl-Optimized readers will know that there’s been plenty of obfuscation and even outright lying, to journalists, the public, and investors, about what quantum computing will be good for and how soon. To whatever extent the previous valuations were based on that lying, a brutal correction was of course in order, regardless of what triggered it.

On the other hand, I can’t say with certainty that high valuations are wrong! After all, even if there’s only a 10% chance that something will produce $100B in value, that would still justify a $10B valuation. It’s a completely different way of thinking than what we’re used to in academia.

For whatever it’s worth, my own family’s money is just sitting in index funds and CDs. I have no quantum computing investments of any kind. I do sometimes accept consulting fees to talk to quantum computing startups and report back my thoughts. When I do, my highest recommendation is: “these people are smart and honest, everything they say about quantum algorithms is correct insofar as I can judge, and I hope they succeed. I wouldn’t invest my own money, but I’m very happy if you or anyone else does.” Meanwhile, my lowest recommendation is: “these people are hypesters and charlatans, and I hope they fail. But even then, I can’t say with confidence that their valuation won’t temporarily skyrocket, in which case investing in them would presumably have been the right call.”

So basically: it’s good that I became an academic rather than an investor.

Having returned from family vacation, I hope to get back to a more regular blogging schedule … let’s see how it goes!

Tonight (January 13th) offers a wonderful opportunity for all of us who love the night sky, and also for science teachers. For those living within the shaded region of Fig. 1, the planet Mars will disappear behind the Moon, somewhere between 9 and 10 pm Eastern (6 and 7 pm Pacific), before reappearing an hour later. Most easily enjoyed with binoculars. (And, umm, without clouds, which will be my own limitation, I believe…)

For everyone else, look up anyway! Mars and the Moon will appear very close together, a lovely pair.

Why is this Cool?

“Occultations”, in which a planet or star disappears behind our Moon, are always cool. Normally, even though we know that the planets and the Moon move across the sky, we don’t get to actually see the motion. But here we can really watch the Moon close in on Mars — a way to visually experience the Moon’s motion around the Earth. You can see this minute by minute with the naked eye until Mars gets so close that the Moon’s brightness overwhelms it. Binoculars will allow you to see much more. With a small telescope, where you’ll see Mars as a small red disk, you can actually watch it gradually disappear as the Moon crosses in front of it. This takes less than a minute.

A particularly cool thing about this particular occultation is that it is happening at full Moon. Occultations like this can happen at any time of year or month, but when they happen at full Moon, it represents a very special geometry in the sky. In particular, it means that the Sun, Earth, Moon and Mars lie in almost a straight line, as shown (not to scale!!!) in Fig. 2.

• The Moon is full because it is fully lit from our perspective, which means that it must lie almost directly behind the Earth relative to the Sun. [If it were precisely behind it, then it would be in Earth’s shadow, leading to a lunar eclipse; instead it is slightly offset, as it is at most full Moons.]
• And when the Moon covers Mars from our perspective, that must mean Mars lies almost directly behind the Moon relative to the Earth.

So all four objects must lie nearly in a line, a relatively rare coincidence.

What Does This Occultation Teach Us?

Aside from the two things I’ve already mentioned — that an occultation is an opportunity to see the Moon’s motion, and that an occultation at full Moon implies the geometry of Fig. 2 — what else can we learn from this event, considered both on its own and in the context of others like it?

Distances and Sizes

Let’s start with one very simple thing: Mars is obviously farther from Earth than is the Moon, since it passes behind it. In fact, the Moon has occultations with all the planets, and all of them disappear behind the Moon instead of passing in front of it. This is why it has been understood for millennia that the Moon is closer to Earth than any of the planets.

Less obvious is that the map in Fig. 1 teaches us the size of the Moon. That’s because the width of the band where the Moon-Mars meeting is visible is approximately the diameter of the Moon. Why is that? Simple geometry. I’ve explained this here.

“Oppositions” and Orbital Periods

The moment when Mars is closest to Earth and brightest in the sky is approximately when the Sun, Earth and Mars lie in a straight line, known as “opposition”. Fig. 2 implies that an occultation of a planet at full Moon can only occur at or around that planet’s opposition. And indeed, while today’s occultation occurs on January 13th, Mars’ opposition occurs on January 15th.

Oppositions are very interesting for another reason; you can use them to learn a planet’s year. Mars’ most recent oppositions (and the next ones) are given in Fig. 3. You notice they occur about 25-26 months apart — just a bit more than two years.

This, in turn, implies something interesting, but not instantly obvious: the time between Martian oppositions tells us that a Martian year is slightly less than two Earth years. Why?

Fig. 4 shows what would happen if (a) a Martian year (the time Mars takes to orbit the Sun) were exactly twice as long as an Earth year, and (b) both orbits were perfect circles around the Sun. Then the time between oppositions would be exactly two Earth years.

But neither (a) nor (b) is exactly true. In fact a Martian year is 687 days, slightly less than two Earth years, whereas the time between oppositions is slightly more than two Earth years. Why? It takes a bit of thought, and is explained in detail here (for solar conjuctions rather than oppositions, but the argument is identical.)

The Planets, Sun and Moon are In a Line — Always!

And finally, one more thing about occultations of planets by the Moon: they happen for all the planets, and they actually happen pretty often, though some are much harder to observe than others. Here is a partial list, showing occultations of all planets [except Neptune is not listed for some unknown reason], as well as occultations of a few bright stars, in our current period. Why are these events so common?

Well (although the news media seems not to be aware of it!) the Moon and the planets are always laid out roughly in a (curved) line across the sky, though not all are visible at the same time. Since the Moon crosses the whole sky once a month, the chance of it passing in front of a planet is not particularly small!

Why are they roughly in a line? This is because the Sun and its planets lie roughly in a disk, with the Earth-Moon system also oriented in roughly the same disk. A disk, seen from someone sitting inside it, look like a line that goes across the sky… or rather, a huge circle that goes round the Earth.

To get a sense of how this works, look at Fig. 5. It shows a flat disk, seen from three perspectives (left to right): first head on, then obliquely (where it appears as an ellipse), and finally from the side (where it appears as a line segment.) The closer we come to the disk, the larger it will appear — and thus the longer the line segment will appear in side view. If we actually enter the disk from the side, the line segment will appear to wrap all the way around us, as a circle that we sit within.

Specifically for the planets, this means the following. Most planetary systems with a single star have the star at the near-center and planets orbiting in near-circles, with all the orbits roughly in a disk around the star. This is shown in Fig. 6. Just as in Fig. 5, when the star and planets are viewed obliquely, their orbits form an ellipse; and when they are viewed from the side, their orbits form a line segment, as a result of which the planets lie in a line. When we enter the planetary disk, so that some planets sit farther from the Sun than we do, then this line becomes a circle that wraps around us. That circle is the ecliptic, and all the planets and the Sun always lie close to it.

Reversing the logic, the fact that we observe that the planets and Sun lie on a curved line across the sky teaches us that the planetary orbits lie in a disk. This, too, has been known for millennia, long before humans understood that the planets orbit the Sun, not the Earth.

(This is also true of our galaxy, the Milky Way, in which the Sun is just one of nearly a trillion stars. The fact that the Milky Way always forms a cloudy band across the sky provides evidence that our galaxy is in the shape of a disk, probably somewhat like this one.)

The Mysteries of the Moon

But why does the Moon also lie on the ecliptic? That is, since the Moon orbits the Earth and not the Sun, why does its orbit have to lie in the same disk as the planets all do?

This isn’t obvious at all! (Indeed it was once seen as evidence that the planets and Sun must, like the Moon, all orbit the Earth.) But today we know this orientation of the Moon’s orbit is not inevitable. The moons of the planet Uranus, for instance, don’t follow this pattern; they and Uranus’ rings orbit in the plane of Uranus’ equator, tipped almost perpendicular to the plane of planetary orbits.

Well, the fact that the Moon’s orbit is almost in the same plane as the planets’ orbits — and that of Earth’s equator — is telling us something important about Earth’s history and about how the Moon came to be. The current leading explanation for the Moon’s origin is that the current Earth and Moon were born from the collision of two planets. Those planets would have been traveling in the same plane as all the others, and if they suffered a glancing blow within that plane, then the debris from the collision would also have been mostly in that plane. As the debris coalesced to form the Earth and Moon we know, they would have ended up orbiting each other, and spinning around their axes, in roughly this very same plane. (Note: This is a consequence of the conservation of angular momentum.)

This story potentially explains the orientation of the Moon’s orbit, as well as many other strange things about the Earth-Moon system. But evidence in favor of this explanation is still not overwhelmingly strong, and so we should consider this as an important question that astronomy has yet to fully settle.

So occultations, oppositions, and their near-simultaneous occurrence have a great deal to teach us and our students. Let’s not miss the opportunity!

I was just in Seattle for the Joint Meetings and had dinner with an old friend. I asked her 15-year-old daughter, who’s lived in Seattle all her life, if she knew who Kurt Cobain was, and she looked at me with a slight tinge of recognition and said, “Did…. he play basketball?”

I do really love Seattle. I love the rocks coming out of the water, I love the pointy trees. Breath of happiness whenever I show up there.

Thanks to everyone who made all those kind remarks in various places last month after my mother died. I've not responded individually (I did not have the strength) but I did read them all and they were deeply appreciated. Yesterday would’ve been mum‘s 93rd birthday. A little side-note occurred to me the other day: Since she left us a month ago, she was just short of having seen two perfect square years. (This year and 1936.) Anyway, still on the theme of playing with numbers, my siblings and I agreed that as a tribute to her on the day, we would all do some kind of outdoor activity for 93 minutes. Over in London, my brother and sister did a joint (probably chilly) walk together in Regents Park and surrounds. I decided to take out a piece of the afternoon at low tide and run along the beach. It went pretty well, [...] Click to continue reading this post →

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People sometimes say there is a crisis of trust in science. In controversial subjects, from ecology to health, increasingly many people are rejecting not only mainstream ideas, but the scientists behind them.

I think part of the problem is media literacy, but not in the way you’d think. When we teach media literacy, we talk about biased sources. If a study on cigarettes is funded by the tobacco industry or a study on climate change is funded by an oil company, we tell students to take a step back and consider that the scientists might be biased.

That’s a worthwhile lesson, as far as it goes. But it naturally leads to another idea. Most scientific studies aren’t funded by companies, most studies are funded by the government. If you think the government is biased, does that mean the studies are too?

I’m going to argue here that government science funding is a very different thing than corporations funding individual studies. Governments do have an influence on scientists, and a powerful one, but that influence is diffuse and long-term. They don’t have control over the specific conclusions scientists reach.

If you picture a stereotypical corrupt scientist, you might imagine all sorts of perks. They might get extra pay from corporate consulting fees. Maybe they get invited to fancy dinners, go to corporate-sponsored conferences in exotic locations, and get gifts from the company.

Grants can’t offer any of that, because grants are filtered through a university. When a grant pays a scientist’s salary, the university pays less to compensate, instead reducing their teaching responsibilities or giving them a slightly better chance at future raises. Any dinners or conferences have to obey not only rules from the grant agency (a surprising number of grants these days can’t pay for alcohol) but from the university as well, which can set a maximum on the price of a dinner or require people to travel economy using a specific travel agency. They also have to be applied for: scientists have to write their planned travel and conference budget, and the committee evaluating grants will often ask if that budget is really necessary.

Actual corruption isn’t the only thing we teach news readers to watch out for. By funding research, companies can choose to support people who tend to reach conclusions they agree with, keep in contact through the project, then publicize the result with a team of dedicated communications staff.

Governments can’t follow up on that level of detail. Scientific work is unpredictable, and governments try to fund a wide breadth of scientific work, so they have to accept that studies will not usually go as advertised. Scientists pivot, finding new directions and reaching new opinions, and government grant agencies don’t have the interest or the staff to police them for it. They also can’t select very precisely, with committees that often only know bits and pieces about the work they’re evaluating because they have to cover so many different lines of research. And with the huge number of studies funded, the number that can be meaningfully promoted by their comparatively small communications staff is only a tiny fraction.

In practice, then, governments can’t choose what conclusions scientists come to. If a government grant agency funds a study, that doesn’t tell you very much about whether the conclusion of the study is biased.

Instead, governments have an enormous influence on the general type of research that gets done. This doesn’t work on the level of conclusions, but on the level of topics, as that’s about the most granular that grant committees can get. Grants work in a direct way, giving scientists more equipment and time to do work of a general type that the grant committees are interested in. It works in terms of incentives, not because researchers get paid more but because they get to do more, hiring more students and temporary researchers if they can brand their work in terms of the more favored type of research. And it works by influencing the future: by creating students and sustaining young researchers who don’t yet have temporary positions, and by encouraging universities to hire people more likely to get grants for their few permanent positions.

So if you’re suspicious the government is biasing science, try to zoom out a bit. Think about the tools they have at their disposal, about how they distribute funding and check up on how it’s used. The way things are set up currently, most governments don’t have detailed control over what gets done. They have to filter that control through grant committees of opinionated scientists, who have to evaluate proposals well outside of their expertise. Any control you suspect they’re using has to survive that.

When it comes to the weak nuclear force and why it is weak, there’s a strange story which floats around. It starts with a true but somewhat misleading statement:

• The weak nuclear force (which is weak because its effects only extend over a short range) has its short range because the particles which mediate the force, the W and Z bosons, have mass [specifically, they have “rest mass”.] This is in contrast to electromagnetic forces which can reach out over great distances; that’s because photons, the particles of light which mediate that force, have no rest mass.

This is misleading because fields mediate forces, not particles; it’s the W and Z fields that are the mediators for the weak nuclear force, just as the electromagnetic field is the mediator for the electromagnetic force. (When people speak of forces as due to exchange of “virtual particles” — which aren’t particles — they’re using fancy math language for a simple idea from first-year undergraduate physics.)

Then things get worse, because it is stated that

• The connection between the W and Z bosons’ rest mass and the short range of the weak nuclear force is that
  • the force is created by the exchange of virtual W and Z bosons, and
  • due to the quantum uncertainty principle, these virtual particles with mass can’t live as long and/or travel as far as virtual photons can, shortening their range.

This is completely off-base. In fact, quantum physics plays no role in why the weak nuclear force is weak and short-range. (It plays a big role in why the strong nuclear force is strong and short-range, but that’s a tale for another day.)

I’ve explained the real story in a new webpage that I’ve added to my site; it has a non-technical explanation, and then some first-year college math for those who want to see it. It’s gotten some preliminary comments that have helped me improve it, but I’m sure it could be even better, and I’d be happy to get your comments, suggestions, questions and critiques if you have any.

[P.S. — if you try but are unable to leave a comment on that page, please leave one here and tell me what went wrong; and if you try but are unable to leave a comment here too for some reason, please send me a message to let me know.]

As a French citizen I should probably disavow the following post and remind myself that I have access to some of the best food in the world. Yet it's impossible to forget the tastes of your childhood. And indeed there are lots of British things that are difficult or very expensive to get hold of in France. Some of them (Marmite, Branston pickle ...) I can import via occasional trips across the channel, or in the luggage of visiting relatives. However, since Brexit this no longer works for fresh food like bacon and sausages. This is probably a good thing for my health, but every now and then I get a hankering for a fry-up or a bacon butty, and as a result of their rarity these are amongst the favourite breakfasts of my kids too. So I've learnt how to make bacon and sausages (it turns out that boudin noir is excellent with a fry-up and I even prefer it to black pudding). 

Sausages are fairly labour-intensive, but after about an hour or so's work it's possible to make one or two kilos worth. Back bacon, on the other hand, takes three weeks to make one batch, and I thought I'd share the process here.

1. Cut of meat

The first thing is to get the right piece of pork, since animals are divided up differently in different countries. I've made bacon several times now and keep forgetting which instructions I previously gave to the butcher at my local Grand Frais ... Now I have settled on asking for a carré de porc, and when they (nearly always) tell me that they don't have that in I ask for côtes de porc première in one whole piece, and try to get them to give me a couple of kilos. As you can find on wikipedia, I need the same piece of meat used to make pork chops. I then ask them to remove the spine, but it should still have the ribs. So I start with this:

2. Cure

Next the meat has to be cured for 10 days (I essentially follow the River Cottage recipe). I mix up a 50-50 batch of PDV salt and brown sugar (1 kg in total here), and add some pepper, juniper berries and bay leaves:

Notice that this doesn't include any nitrites or nitrates. I have found that nitrates/nitrites are essential for the flavour in sausages, but in bacon the only thing that they will do (other than be a carcinogen) as far as I can tell is make the meat stay pink when you cook it. I can live without that. This cure makes delicious bacon as far as I'm concerned. 

The curing process involves applying 1/10th of the mixture each day for ten days and draining off the liquid produced at each step. After the first coating it looks like this:

The salt and sugar remove water from the meat, and penetrate into it, preserving it. Each day I get liquid at the bottom, which I drain off and apply the next cure. After one day it looks like this:

This time I still had liquid after 10 days:

3. Drying

After ten days, I wash/wipe off the cure and pat it down with some vinegar. If you leave cure on the meat it will be much too salty (and, to be honest, this cure always gives quite salty bacon). So at this point it looks like this:

4. Smoking

After the curing/drying, the bacon is ready to eat and should in principle keep almost indefinitely. However, I prefer smoked bacon, so I cold smoke it. This involves sticking it in a smoker (essentially just a box where you can suspend the meat above some smouldering sawdust) for several hours:

The sawdust is beech wood and slowly burns round in the little spiral device you can see above. Of course, I close the smoker up and usually put it in the shed to protect against the elements:

5. All done!

And then that's it! Delicious back bacon that really doesn't take very long to eat:

As I mentioned above, it's usually still a bit salty, so when I slice it to cook I put the pieces in water for a few minutes before grilling/frying:

Three papers caught my eye the other day on the arXiv at the start of the new year:

arXiv:2501.00098 - J. Yu et al., "Quantum geometry in quantum materials" - I hope to write up something about quantum geometry soon, but I wanted to point out this nice review even if I haven't done my legwork yet.  The ultrabrief point:  The single-particle electronic states in crystalline solids may be written as Bloch waves, of the form \(u_{n \mathbf{k}}(\mathbf{r}) \exp(i \mathbf{k} \cdot \mathbf{r})\), where the (crystal) momentum is given by \(\hbar \mathbf{k}\) and \(u_{n \mathbf{k}}\) is a function with the real-space periodicity of the crystal lattice and contains an implicit \(\mathbf{k}\) dependence.  You can get very far in understanding solid-state physics without worrying about this, but it turns out that there are a number of very important phenomena that originate from the oft-neglected \(\mathbf{k}\) dependence of \(u_{n \mathbf{k}}\).  These include the anomalous Hall effect, the (intrinsic) spin Hall effect, the orbital Hall effect, etc.  Basically the \(\mathbf{k}\) dependence of \(u_{n \mathbf{k}}\) in the form of derivatives defines an internal "quantum" geometry of the electronic structure.  This review is a look at the consequences of quantum geometry on things like superconductivity, magnetic excitations, excitons, Chern insulators, etc. in quantum materials.

Fig. 1 from arXiv:2501.01253

arXiv:2501.00536 - G. Bartolucci et al., "Phase behavior of Cacio and Pepe sauce" - Cacio e pepe is a wonderful Italian pasta dish with a sauce made from pecorino cheese, pepper, and hot pasta cooking water that contains dissolved starch.  When prepared well, it's incredibly creamy, smooth, and satisfying.  The authors here perform a systematic study of the sauce properties as a function of temperature and starch concentration relative to cheese content, finding the part of parameter space to avoid if you don't want the sauce to "break" (condensing out clumps of cheese-rich material and ruining the sauce texture).  That's cool, but what is impressive is that they are actually able to model the phase stability mathematically and come up with a scientifically justified version of the recipe.  Very fun.

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(Why mention Greenland? It’s a movie reference.)

I figured I’d give an update on my personal life.

A year ago, I resigned from my position in France and moved back to Denmark. I had planned to spend a few months as a visiting researcher in my old haunts at the Niels Bohr Institute, courtesy of the spare funding of a generous friend. There turned out to be more funding than expected, and what was planned as just a few months was extended to almost a year.

I spent that year learning something new. It was still an amplitudes project, trying to make particle physics predictions more efficient. But this time I used Python. I looked into reinforcement learning and PyTorch, played with using a locally hosted Large Language Model to generate random code, and ended up getting good results from a classic genetic programming approach. Along the way I set up a SQL database, configured Docker containers, and puzzled out interactions with CUDA. I’ve got a paper in the works, I’ll post about it when it’s out.

All the while, on the side, I’ve been seeking out stories. I’ve not just been a writer, but a journalist, tracking down leads and interviewing experts. I had three pieces in Quanta Magazine and one in Ars Technica.

Based on that, I know I can make money doing science journalism. What I don’t know yet is whether I can make a living doing it. This year, I’ll figure that out. With the project at the Niels Bohr Institute over, I’ll have more time to seek out leads and pitch to more outlets. I’ll see whether I can turn a skill into a career.

So if you’re a scientist with a story to tell, if you’ve discovered something or accomplished something or just know something that the public doesn’t, and that you want to share: do reach out. There’s a lot that can be of interest, passion that can be shared.

At the same time, I don’t know yet whether I can make a living as a freelancer. Many people try and don’t succeed. So I’m keeping my CV polished and my eyes open. I have more experience now with Data Science tools, and I’ve got a few side projects cooking that should give me a bit more. I have a few directions in mind, but ultimately, I’m flexible. I like being part of a team, and with enthusiastic and competent colleagues I can get excited about pretty much anything. So if you’re hiring in Copenhagen, if you’re open to someone with ten years of STEM experience who’s just starting to see what industry has to offer, then let’s chat. Even if we’re not a good fit, I bet you’ve got a good story to tell.

In Kenneth Grahame’s 1908 novel The Wind in the Willows, a Mole meets a Water Rat who lives on a River. The Rat explains how the River permeates his life: “It’s brother and sister to me, and aunts, and company, and food and drink, and (naturally) washing.” As the River plays many roles in the Rat’s life, so does Carnot’s theorem play many roles in a thermodynamicist’s.

Nicolas Léonard Sadi Carnot lived in France during the turn of the 19th century. His father named him Sadi after the 13th-century Persian poet Saadi Shirazi. Said father led a colorful life himself,¹ working as a mathematician, engineer, and military commander for and before the Napoleonic Empire. Sadi Carnot studied in Paris at the École Polytechnique, whose members populate a “Who’s Who” list of science and engineering. 

As Carnot grew up, the Industrial Revolution was humming. Steam engines were producing reliable energy on vast scales; factories were booming; and economies were transforming. France’s old enemy Britain enjoyed two advantages. One consisted of inventors: Englishmen Thomas Savery and Thomas Newcomen invented the steam engine. Scotsman James Watt then improved upon Newcomen’s design until rendering it practical. Second, northern Britain contained loads of coal that industrialists could mine to power her engines. France had less coal. So if you were a French engineer during Carnot’s lifetime, you should have cared about engines’ efficiencies—how effectively engines used fuel.²

Carnot proved a fundamental limitation on engines’ efficiencies. His theorem governs engines that draw energy from heat—rather than from, say, the motional energy of water cascading down a waterfall. In Carnot’s argument, a heat engine interacts with a cold environment and a hot environment. (Many car engines fall into this category: the hot environment is burning gasoline. The cold environment is the surrounding air into which the car dumps exhaust.) Heat flows from the hot environment to the cold. The engine siphons off some heat and converts it into work. Work is coordinated, well-organized energy that one can directly harness to perform a useful task, such as turning a turbine. In contrast, heat is the disordered energy of particles shuffling about randomly. Heat engines transform random heat into coordinated work.

An engine’s efficiency is the bang we get for our buck—the upshot we gain, compared to the cost we spend. Running an engine costs the heat that flows between the environments: the more heat flows, the more the hot environment cools, so the less effectively it can serve as a hot environment in the future. An analogous statement concerns the cold environment. So a heat engine’s efficiency is the work produced, divided by the heat spent.

Carnot upper-bounded the efficiency achievable by every heat engine of the sort described above. Let denote the cold environment’s temperature; and , the hot environment’s. The efficiency can’t exceed . What a simple formula for such an extensive class of objects! Carnot’s theorem governs not only many car engines (Otto engines), but also the Stirling engine that competed with the steam engine, its cousin the Ericsson engine, and more.

In addition to generality and simplicity, Carnot’s bound boasts practical and fundamental significances. Capping engine efficiencies caps the output one can expect of a machine, factory, or economy. The cap also prevents engineers from wasting their time on daydreaming about more-efficient engines. 

More fundamentally than these applications, Carnot’s theorem encapsulates the second law of thermodynamics. The second law helps us understand why time flows in only one direction. And what’s deeper or more foundational than time’s arrow? People often cast the second law in terms of entropy, but many equivalent formulations express the law’s contents. The formulations share a flavor often synopsized with “You can’t win.” Just as we can’t grow younger, we can’t beat Carnot’s bound on engines. 

One might expect no engine to achieve the greatest efficiency imaginable: , called the Carnot efficiency. This expectation is incorrect in one way and correct in another. Carnot did design an engine that could operate at his eponymous efficiency: an eponymous engine. A Carnot engine can manifest as the thermodynamicist’s favorite physical system: a gas in a box topped by a movable piston. The gas undergoes four strokes, or steps, to perform work. The strokes form a closed cycle, returning the gas to its initial conditions.³ 

Steampunk artist Todd Cahill beautifully illustrated the Carnot cycle for my book. The gas performs useful work because a weight sits atop the piston. Pushing the piston upward, the gas lifts the weight.

The gas expands during stroke 1, pushing the piston and so outputting work. Maintaining contact with the hot environment, the gas remains at the temperature . The gas then disconnects from the hot environment. Yet the gas continues to expand throughout stroke 2, lifting the weight further. Forfeiting energy, the gas cools. It ends stroke 2 at the temperature .

The gas contacts the cold environment throughout stroke 3. The piston pushes on the gas, compressing it. At the end of the stroke, the gas disconnects from the cold environment. The piston continues compressing the gas throughout stroke 4, performing more work on the gas. This work warms the gas back up to .

In summary, Carnot’s engine begins hot, performs work, cools down, has work performed on it, and warms back up. The gas performs more work on the piston than the piston performs on it.

At what cost, if the engine operates at the Carnot efficiency? The engine mustn’t waste heat. One wastes heat by roiling up the gas unnecessarily—by expanding or compressing it too quickly. The gas must stay in equilibrium, a calm, quiescent state. One can keep the gas quiescent only by running the cycle infinitely slowly. The cycle will take an infinitely long time, outputting zero power (work per unit time). So one can achieve the perfect efficiency only in principle, not in practice, and only by sacrificing power. Again, you can’t win.

Carnot’s theorem may sound like the Eeyore of physics, all negativity and depression. But I view it as a companion and backdrop as rich, for thermodynamicists, as the River is for the Water Rat. Carnot’s theorem curbs diverse technologies in practical settings. It captures the second law, a foundational principle. The Carnot cycle provides intuition, serving as a simple example on which thermodynamicists try out new ideas, such as quantum engines. Carnot’s theorem also provides what physicists call a sanity check: whenever a researcher devises a new (for example, quantum) heat engine, they can confirm that the engine obeys Carnot’s theorem, to help confirm their proposal’s accuracy. Carnot’s theorem also serves as a school exercise and a historical tipping point: the theorem initiated the development of thermodynamics, which continues to this day. 

So Carnot’s theorem is practical and fundamental, pedagogical and cutting-edge—brother and sister, and aunts, and company, and food and drink. I just wouldn’t recommend trying to wash your socks in Carnot’s theorem.

¹To a theoretical physicist, working as a mathematician and an engineer amounts to leading a colorful life.

²People other than Industrial Revolution–era French engineers should care, too.

³A cycle doesn’t return the hot and cold environments to their initial conditions, as explained above.

As we head into 2025, and the prospects for increased (US) government investment in science, engineering, and STEM education seem very limited, I wanted to revisit a topic that I wrote about over a decade ago (!!!), the role of philanthropy and foundations in these areas.  

Personally I think the case for government support of scientific research and education is overwhelmingly clear; while companies depend on having an educated technical workforce (at least for now) and continually improving technology, they are under great short-term financial pressures and genuinely long-term investment in research is rare.  Foundations are not a substitute for nation-state levels of support, but they are a critical component of the research and education landscape.  

Annual citations of the EPR paper from Web of Science, a case study in the long-term impact of some "pure" scientific research, and giving hope to practicing scientists that surely our groundbreaking work will be appreciated sixty years  after publication.

The nature of long-term research is that it often takes a long time for its true impact (I don't mean just citation counts, but those are an indicator of activity) to be felt.  One (admittedly extreme) example is shown here, the citations-per-year (from Web of Science) of the 1935 Einstein/Podolsky/Rosen paper about entanglement.  (Side note:  You have to love the provocative title, "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?", which from the point of view of the authors at the time satisfies Betteridge's Law of Headlines.)  There are few companies that would be willing to invest in supporting research that won't have its true heyday for five or six decades.

One additional tricky bit is that grants are usually given to people and organizations who are already active.  It's often not simple to point to some clear result or change in output that absolutely would not have happened without foundation support.  This is exacerbated by the fact that grants in science and engineering are often given to people and organizations who are not just active but already very well supported - betting on an odds-on favorite is a low risk strategy. 

Many foundations do think very carefully about what areas to support, because they want to "move the needle".  For example, some scientific foundations are consciously reluctant to support closer-to-clinical-stage cancer research, since the total annual investment by governments and pharmaceutical companies in that area numbers in the many billions of dollars, and a modest foundation contribution would be a tiny delta on top of that.  

Here is a list of the wealthiest charitable foundations (only a few of which support scientific research and/or education) and their endowments.  Nearly all of the science-related ones are also plugged in here.  A rough estimate of annual expenditures from endowed entities is about 5% of their holdings.  Recently I've come to think about private universities as one crude comparator for impact.  If a foundation has the same size endowment as a prestigious research university, I think it's worth thinking about the relative downstream impacts of those entities.  (Novo Nordisk Foundation has an endowment three times the size of Harvard's endowment.)  

Another comparator would be the annual research expenditures of a relevant funding agency.  The US NSF put forward $234M into major research instrumentation and facilities in FY2024.  A foundation with a $5B endowment could in principle support all of that from endowment returns.  This lets me make my semiregular pitch about foundational or corporate support for research infrastructure and user facilities around the US.  The entire annual budget for the NSF's NNCI, which supports shared nanofabrication and characterization facilities around the US, is about $16M.   That's a niche where comparatively modest foundation (or corporate) support could have serious impact for interdisciplinary research and education across the country.  I'm sure there are other similar areas out there, and I hope someone is thinking about this.  

Anyway, thanks to my readers - this is now the 20th year of this blog's existence (!!! again), and I hope to be able to keep it up well in the new year.

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A farmer with a wolf, a goat, and a cabbage must cross a river by boat. The boat can carry only the farmer and a single item. If left unattended together, the wolf would eat the goat, or the goat would eat the cabbage. How can they cross the river without anything being eaten?

You probably know this puzzle. There are two efficient solutions, related by a symmetry that switches the wolf and the cabbage.

But what you might not know is that this puzzle goes back to a book written around 800 AD, sometimes attributed to Charlemagne’s advisor Alcuin! Charlemagne brought this monk from York to help set up the educational system of his empire. But Alcuin had a great fondness for logic. Nobody is sure if he wrote this book — but it’s fascinating nonetheless.

It has either 53 or 56 logic puzzles, depending on the version. It’s called Propositiones ad Acuendos Juvenes, or Problems to Sharpen the Young. If the wolf, goat and cabbage problem is too easy for you, you might like this one:

Three men, each with a sister, must cross a boat which can carry only two people, so that a woman whose brother is not present is never left in the company of another man.

There are also trick puzzles, like this:

A man has 300 pigs. He ordered all of them slaughtered in 3 days, but with an odd number killed each day. What number were to be killed each day?

Wikipedia says this was given to punish unruly students — presumably students who didn’t know that the sum of three odd numbers is always odd.

It’s fascinating to think that while some Franks were fighting Saxons and Lombards, students in more peaceful parts of the empire were solving these puzzles!

The book also has some of the first recorded packing problems, like this:

There is a triangular city which has one side of 100 feet, another side of 100 feet, and a third of 90 feet. Inside of this, I want to build rectangular houses in such a way that each house is 20 feet in length, 10 feet in width. How many houses can I fit in the city?

This is hard! There’s a nice paper about all the packing problems in this book:

• Nikolai Yu. Zolotykh, Alcuin’s Propositiones de Civitatibus: the earliest packing problems.

He shows two solutions to the above problem, both with 16 houses. In solution here there’s a tiny space between houses 7 and 15.

However, Alcuin — or whoever wrote the book — didn’t really solve the problem. They just used an approximate Egyptian formula for the area of a triangle in terms of its side lengths, and divided that by the area of the houses! This is consistent with the generally crappy knowledge of math in Charlemagne’s empire.

There’s an even harder packing problem in this book, which again isn’t solved correctly.

For more on this book of puzzles, check out this:

• Wikipedia, Propositiones ad Acuendos Juvenes.

You can get a translation into English here:

• John Hadley and David Singmaster,
Problems to Sharpen the Young, The Mathematical Gazette 76 (1992), 102–126.

Finally, here is a nice paper on the question of whether Alcuin wrote the book, and various bit of evidence for Alcuin’s interest in mathematics, numerology, and related fun activities:

• Michael N. Brennan, Alcuin, mathematics and the rational mind, in Insular Iconographies: Essays in Honor of Jane Hawkes, eds. Meg Boulton and Michael D. J. Bintley, Boydell Press, 2019, pp. 203–216.

It’s so fascinating that I’ll quote a bunch of it. It starts this way:

A medieval mathematical manuscript

The medieval cleric Alcuin (AD 735–804) is credited in surviving manuscripts with being the originator of a collection of fifty-three mathematical and logical puzzles, the Propositiones ad acuendos iuvenes (“Problems for Sharpening the Young”). There is no direct evidence to connect the collection with Alcuin either as compiler or creator, but even modern commentators continue to associate his name with the puzzles. There are at least fourteen extant or partial copies of the Propositiones, which date from the ninth to the fifteenth century, suggesting that the collection was in popular use at least from the time of Alcuin onwards. Michael Gorman confidently listed the Propositiones among ninety spurious prose works of Alcuin for two reasons: firstly, because Alcuin always attached a dedicatory letter to his works, and there is none here, and secondly, because the work falls, Gorman thinks, among those documents which Alcuin would have had neither the time nor the energy to write in the period AD 782-800. Alcuin himself admitted to having only “stolen hours” at night in which to write a life of Willibrord. Despite Gorman”s view that the work is pseudo-Alcuin, it is reasonable to ask if there is internal evidence in the puzzles which would support or oppose the assigning of the collection to Alcuin himself, committed as he was to the promotion of educational subjects, including mathematics, in the course of the Carolingian “renaissance”.

The majority of the problem types in the Propositiones are known from earlier Chinese, Indian, Egyptian, Byzantine, Greek, and Roman sources, whilst others appear in works by Boethius, Metrodorus, and Isidore of Seville. Among the puzzles, however, there are a small number of important types that are not as yet known from earlier sources. These include the so-called “river-crossing problems”, “strange family problems”, a transportation (or “desert-crossing”) problem, and a problem that relies on the summation of an arithmetical series. The case for the mathematical creativity of the author, if there was only one author, rests on these. One puzzle has outstripped all the others in having popular appeal down to the present day. This puzzle, passing presumably from later medieval reproductions of the Propositiones into oral tradition, concerns a farmer who needs to ferry three items across a river in a boat.

He then goes on to discuss the problem of the wolf, the goat and the cabbage — though in this paper, presumably more historically accurate, it’s a wolf, a goat and a bag of oats. Then he returns to the big question:

Did Alcuin compose the Propositiones?

In a letter to Charlemagne in AD 799, three years after Alcuin had moved from the Palace School to an abbacy at Tours, Alcuin wrote that he was sending, along with some examples of grammar and correct expression, quidquid figuras arithmeticas laetitiae causa (“certain arithmetical curiosities for [your] pleasure”). He added that he would write these on an empty sheet that Charlemagne sent to him and suggested that “our friend and helper Beselel will […] be able to look up the problems in an arithmetic book”. Beselel was Alcuin”s nickname for the technically skilled Einhard, later Charlemagne”s biographer. It would be convenient if the reference in Alcuin”s letter were to the Propositiones, but for many reasons it is almost certainly not. For one thing, fifty-three propositions and their solutions would not fit on the blank side of a folio, given that they occupy ten folio sides in most of the manuscripts in which they are currently found. Secondly, since Alcuin”s letter to Charlemagne dealt primarily with the importance of exactness of language, a grammarian and self-conscious Latinist like Alcuin would not describe the Propositiones as figurae arithmeticae, or refer to an “arithmetic” book in a case where the solutions required both geometry and logic methods. Thirdly, the idea of Einhard looking up the problems in an arithmetic book is an odd one, given that the Propositiones is usually found with answers, even if in most cases it does not show how these answers were arrived at.

Apart from the reasons given by Gorman for Alcuin not having been the author of the Propositiones — the effort and time involved and the absence of a dedication — there are deeper, internal reasons for concluding that Alcuin was not the author. The river-crossing puzzles, along with the “transportation problem”, and a puzzle about the number of birds on a100-step ladder took a considerable amount of time and mathematical sophistication to compose. Accompanying them are two types of puzzles that appear repeatedly in the collection and that demand far lesssophistication: area problems, often fanciful (such as problem 29: “how many rectangular houses will fit into a town which has a circular wall?”); and division problems where a “lord of the manor” wishes to divide grain among his household (problems 32-5, and others). Repetition of problems suggests that the Propositiones was intended for use as a practice book, but this supposed pedagogical purpose stumbles (even if it does not fall) on two counts. Firstly, the solutions to most of the mensuration problems, like the one involving a circular town, rely on late Roman methods for approximating the areas of common figures (circles, triangles, etc.) and these methods can be quite wrong. Secondly, the lack of worked solutions in the Propositiones deprived the student of the opportunity to learn the method required to solve another question of the same type. Solution methods might have been lost in transcription, but their almost total absence makes this unlikely. Mathematical methods (algebra in particular) that post-dated the Carolingian era would have been necessary to provide elegant and instructive solutions to many of the problems, and one cannot escape the suspicion that trial-and-error was the method used by whoever supplied answers (without worked solutions) to the Propositiones: a guess was made, refined, and then a new guess made. Such an approach is difficult to systemise, and even more difficult to describe in writing for the benefit of students. We are left with a mixture of “complex” mathematical problems, such as those cited earlier, and simpler questions whose answers are mostly not justified.

This lack of uniformity suggests that it was a compiler rather than a composer who produced the first edition of the Propositiones. If Alcuin was involved, he was no more than a medium through which the fifty-three puzzles were assembled. No one person was the author of the Propositiones problems, because no mathematically sophisticated person would have authored the weaker ones, and nobody who was not mathematically sophisticated could have authored the others. Furthermore, with the more sophisticated problems, there is a noticeable absence of the kind of refinement and repetition that is common in modern textbooks.

He then goes on to discuss Alcuin’s fascination with numerology, acrostics, and making up nicknames for his friends. It’s too bad we’ll never know for sure what, if anything, Alcuin had to do with this book of puzzles!

Today, December 24th 2024, the Parker Solar Probe got 7 times closer to the Sun than any spacecraft ever has, going faster than any spacecraft ever has—690,000 kilometers per hour. WHEEEEEE!!!!!!!

But the newspapers are barely talking about the really cool part: what it’s like down there. The Sun doesn’t have a surface like the Earth does, since it’s all just hot ionized gas, called ‘plasma‘. But the Sun has an ‘Alfvén surface’—and the probe has penetrated that.

What’s the Alfvén surface? In simple terms, it’s where the solar wind—the hot ionized gas emitted by the Sun—breaks free of the Sun and shoots out into space. But to understand how cool this is, we need to dig a bit deeper.

After all, how can we say where the solar wind “breaks free of the Sun”?

Hot gas shoots up from the Sun, faster and faster due to its pressure, even though it’s pulled down by gravity. At some point it goes faster than the speed of sound! This is the Alfvén surface. Above this surface, the solar wind becomes supersonic, so no disturbances in its flow can affect the Sun below.

It’s sort of like the reverse of a black hole! Light emitted from within the event horizon of a black hole can’t get out. Sound emitted from outside the Alfvén surface of the Sun can’t get in.

Or, it’s like the edge of a waterfall, where the water starts flowing so fast that waves can’t make it back upstream.

That’s pretty cool. But it’s even cooler than this, because ‘sound’ in the solar wind is very different from sound on Earth. Here we have air. The Sun has ions—atoms of gas so hot that electrons have been ripped off—interacting with powerful magnetic fields. You can visualize these fields as tight rubber bands, with the ions stuck to them. They vibrate back and forth together!

You could call these vibrations ‘sound’, but the technical term is Alfvén waves. Alfvén was the one who figured out how fast these waves move. Parker studied the surface where the solar wind’s speed exceeds the speed of the Alfvén waves.

And now we’ve gone deep below that surface!

This realm is a strange one, and the more we study it, the more complex it seems to get.

You’ve probably heard the joke that ends “consider a spherical cow”. Parker’s original model of the solar wind was spherically symmetric, so he imagined the solar wind shooting straight out of the Sun in all directions. In this model, the Alfvén surface is the sphere where the wind becomes faster the Alfvén waves. There are some nice simple formulas for all this.

But in fact the Sun’s surface is roiling and dynamic, with sunspots making solar flares, and all sorts of bizarre structures made of plasma and magnetic fields, like spicules, ‘coronal streamers’ and ‘pseudostreamers’… aargh, too complicated for me to understand. This is an entire branch of science!

So, the Alfvén surface is not a mere sphere: it’s frothy and randomly changing. The Parker Solar Probe will help us learn how it works—along with many other things.

Finally, here’s something mindblowing. There’s a red dwarf star 41 light years away from us, called TRAPPIST-1, which may have six planets beneath its Alfvén surface! This means these planets can create Alfvén waves in the star’s atmosphere. Truly the music of the spheres!

For more, check out these articles:

• Wikipedia, Alfvén wave.

• Wikipedia, Alfvén surface.

and this open-access article:

• Steven R. Cranmer, Rohit Chhiber, Chris R. Gilly, Iver H. Cairns, Robin C. Colaninno, David J. McComas, Nour E. Raouafi, Arcadi V. Usmanov, Sarah E. Gibson and Craig E. DeForest, The Sun’s Alfvén surface: recent insights and prospects for the Polarimeter to Unify the Corona and Heliosphere (PUNCH), Solar Physics 298 (2023).

A quote:

Combined with recent perihelia of Parker Solar Probe, these studies seem to indicate that the Alfvén surface spends most of its time at heliocentric distances between about 10 and 20 solar radii. It is becoming apparent that this region of the heliosphere is sufficiently turbulent that there often exist multiple (stochastic and time-dependent) crossings of the Alfvén surface along any radial ray.

This week, people all over the world celebrated the birth of someone whose universally attractive ideas spread around the globe. I’m talking, of course about Isaac Newton.

For Newtonmas this year, I’ve been pondering another aspect of Newton’s life. There’s a story you might have heard that physicists can do basically anything, with many people going from a career in physics to a job in a variety of other industries. It’s something I’ve been trying to make happen for myself. In a sense, this story goes back to the very beginning, when Newton quit his academic job to work at the Royal Mint.

On the surface, there are a lot of parallels. At the Mint, a big part of Newton’s job was to combat counterfeiting and “clipping”, where people would carve small bits of silver off of coins. This is absolutely a type of job ex-physicists do today, at least in broad strokes. Working as Data Scientists for financial institutions, people look for patterns in transactions that give evidence of fraud.

Digging deeper, though, the analogy falls apart a bit. Newton didn’t apply any cunning statistical techniques to hunt down counterfeiters. Instead, the stories that get told about his work there are basically detective stories. He hung out in bars to catch counterfeiter gossip and interviewed counterfeiters in prison, not exactly the kind of thing you’d hire a physicist to do these days. The rest of the role was administrative: setting up new mint locations and getting people to work overtime to replace the country’s currency. Newton’s role at the mint was less like an ex-physicist going into Data Science and more like Steven Chu as Secretary of Energy: someone with a prestigious academic career appointed to a prestigious government role.

If you’re looking for a patron saint of physicists who went to industry, Newton’s contemporary Robert Hooke may be a better bet. Unlike many other scientists of the era, Hooke wasn’t independently wealthy, and for a while he was kept quite busy working for the Royal Society. But a bit later he had another, larger source of income: working as a surveyor and architect, where he designed several of London’s iconic buildings. While Newton’s work at the Mint drew on his experience as a person of power and influence, working as an architect drew much more on skills directly linked to Hooke’s work as a scientist: understanding the interplay of forces in quantitative detail.

While Newton and Hooke’s time was an era of polymaths, in some sense the breadth of skills imparted by a physics education has grown. Physicists learn statistics (which barely existed in Newton’s time) programming (which did not exist at all) and a wider range of mathematical and physical models. Having a physics background isn’t the ideal way to go into industry (that would be having an industry background). But for those of us making the jump, it’s still a Newtonmas gift to be grateful for.

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Hamilton’s quaternion number system is a non-commutative extension of the complex numbers, consisting of numbers of the form where are real numbers, and are anti-commuting square roots of with , , . While they are non-commutative, they do keep many other properties of the complex numbers:

• Being non-commutative, the quaternions do not form a field. However, they are still a skew field (or division ring): multiplication is associative, and every non-zero quaternion has a unique multiplicative inverse.
• Like the complex numbers, the quaternions have a conjugation

  

  although this is now an antihomomorphism rather than a homomorphism: . One can then split up a quaternion into its real part and imaginary part by the familiar formulae

  

  (though we now leave the imaginary part purely imaginary, as opposed to dividing by in the complex case).
• The inner product

  

  is symmetric and positive definite (with forming an orthonormal basis). Also, for any , is real, hence equal to . Thus we have a norm

  

  Since the real numbers commute with all quaternions, we have the multiplicative property . In particular, the unit quaternions (also known as , , or ) form a compact group.
• We have the cyclic trace property

  

  which allows one to take adjoints of left and right multiplication:

  

• As are square roots of , we have the usual Euler formulae

  

  for real , together with other familiar formulae such as , , , etc.

The unit quaternions act on the imaginary quaternions by conjugation:

For instance, for any real , conjugation by is a rotation by around :

Given how quaternions relate to three-dimensional rotations, it is not surprising that one can also be used to recover the basic laws of spherical trigonometry – the study of spherical triangles on the unit sphere. This is fairly well known, but it took a little effort for me to locate the required arguments, so I am recording the calculations here.

The first observation is that every unit quaternion induces a unit tangent vector on the unit sphere , located at ; the third unit vector is then another tangent vector orthogonal to the first two (and oriented to the left of the original tangent vector), and can be viewed as the cross product of and . Right multplication of this quaternion then corresponds to various natural operations on this unit tangent vector:

• Right multiplying by does not affect the location of the tangent vector, but rotates the tangent vector anticlockwise by in the direction of the orthogonal tangent vector , as it replaces by .
• Right multiplying by advances the tangent vector by geodesic flow by angle , as it replaces by , and replaces by .

Now suppose one has a spherical triangle with vertices , with the spherical arcs subtending angles respectively, and the vertices subtending angles respectively; suppose also that is oriented in an anti-clockwise direction for sake of discussion. Observe that if one starts at with a tangent vector oriented towards , advances that vector by , and then rotates by , the tangent vector now at and pointing towards . If one advances by and rotates by , one is now at pointing towards ; and if one then advances by and rotates by , one is back at pointing towards . This gives the fundamental relation

Remark 1 The law (4) has an evident symmetry , which corresponds to the operation of replacing a spherical triangle with its dual triangle. Also, there is nothing particularly special about the choice of imaginaries in (4); one can conjugate (4) by various quaternions and replace here by any other orthogonal pair of unit quaternions.

Remark 2 If we work in the small scale regime, replacing by for some small , then we expect spherical triangles to behave like Euclidean triangles. Indeed, (4) to zeroth order becomes

which reflects the classical fact that the sum of angles of a Euclidean triangle is equal to . To first order, one obtains

which reflects the evident fact that the vector sum of the sides of a Euclidean triangle sum to zero. (Geometrically, this correspondence reflects the fact that the action of the (projective) quaternion group on the unit sphere converges to the action of the special Euclidean group on the plane, in a suitable asymptotic limit.)

The identity (4) is an identity of two unit quaternions; as the unit quaternion group is three-dimensional, this thus imposes three independent constraints on the six real parameters of the spherical triangle. One can manipulate this constraint in various ways to obtain various trigonometric identities involving some subsets of these six parameters. For instance, one can rearrange (4) to get

In a similar fashion, from (5) we see that the quantity

As a variant of the above analysis, we have from (5) again that

Example 3 One application of Napier’s rule (6) is to determine the sunrise equation for when the sun rises and sets at a given location on the Earth, and a given time of year. For sake of argument let us work in summer, in which the declination of the Sun is positive (due to axial tilt, it reaches a maximum of at the summer solstice). Then the Sun subtends an angle of from the pole star (Polaris in the northern hemisphere, Sigma Octantis in the southern hemisphere), and appears to rotate around that pole star once every hours. On the other hand, if one is at a latitude , then the pole star an elevation of above the horizon. At extremely high latitudes , the sun will never set (a phenomenon known as “midnight sun“); but in all other cases, at sunrise or sunset, the sun, pole star, and horizon point below the pole star will form a right-angled spherical triangle, with hypotenuse subtending an angle and vertical side subtending an angle . The angle subtended by the pole star in this triangle is , where is the solar hour angle – the angle that the sun deviates from its noon position. Equation (6) then gives the sunrise equation

or equivalently

A similar rule determines the time of sunset. In particular, the number of daylight hours in summer (assuming one is not in the midnight sun scenario ) is given by

The situation in winter is similar, except that is now negative, and polar night (no sunrise) occurs when .

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In a previous post, I showed you that the Standard Model, armed with its special angle θ_w of approximately 30 degrees, does a pretty good job of predicting a whole host of processes in the Standard Model. I focused attention on the decays of the Z boson, but there were many more processes mentioned in the bonus section of that post.

But the predictions aren’t perfect. They’re not enough to convince a scientist that the Standard Model might be the whole story. So today let’s bring these predictions into better focus.

There are two major issues that we have to correct in order to make more precise predictions using the Standard Model:

• In contrast to what I assumed in the last post, θ_w isn’t exactly 30 degrees (i.e. sin θ_w isn’t 1/2)
• Although I ignored them so far, the strong nuclear force makes small but important effects

But before we deal with these, we have to fix something with the experimental measurements themselves.

Knowledge and Uncertainty: At the Center of Science

No one complained — but everyone should have — that when I presented the experimental results in my previous post, I expressed them without the corresponding uncertainties. I did that to keep things simple. But it wasn’t professional. As every well-trained scientist knows, when you are comparing an experimental result to a theoretical prediction, the uncertainties, both experimental and theoretical, are absolutely essential in deciding whether your prediction works or not. So we have to discuss this glaring omission.

Here’s how to read typical experimental uncertainties (see Figure 1). Suppose a particle physicist says that a quantity is measured to be x ± y — for instance, that the top quark mass is measured to be 172.57± 0.29 GeV/c². Usually (unless explicitly noted) that means that the true value has a 68% chance of lying between x-y and x+y — “within one standard deviation” — and a 95% chance of lying between x-2y and x+2y — “within two standard deviations.” (See Figure 1, where x and y are called and .) The chance of the true value being more than two standard deviations away from x is about 5% — about 1/20. That’s not rare! It will happen several times if you make a hundred different measurements.

But the chance of being more than three standard deviations away from x is a small fraction of a percent — as long as the cause is purely a statistical fluke — and that is indeed rare. (That said, one has to remember that big differences between prediction and measurement can also be due to an unforseen measurement problem or feature. That won’t be an issue today.)

W Boson Decays, More Precisely

Let’s first look at W decays, where we don’t have the complication of θ_w , and see what happens when we account for the effect of the strong nuclear force and the impact of experimental uncertainies.

The strong nuclear force slightly increases the rate for the W boson to decay to any quark/anti-quark pair, by about 3%. This is due to the same effect discussed in the “Understanding the Remaining Discrepancy” and “Strength of a Force” sections of this post… though the effect here is a little smaller (as it decreases at shorter distances and higher energies.) This slightly increases the percentages for quarks and, to compensate, slightly reduces the percentages for the electron, muon and tau (the “leptons”).

In Figure 2 are shown predictions of the Standard Model for the probabilities of the W- boson’s various decays:

• At left are the predictions made in the previous post.
• At center are better predictions that account for the strong nuclear force.

(To do this properly, uncertainties on these predictions should also be provided. But I don’t think that doing so would add anything to this post, other than complications.) These predictions are then compared with the experimental measurements of several quantities, shown at right: certain combinations of these decays that are a little easier to measure are also shown. (The measurements and uncertainties are published by the Particle Data Group here.)

The predictions and measurements do not perfectly agree. But that’s fine; because of the uncertainties in the measurements, they shouldn’t perfectly agree! All of the differences are less than two standard deviations, except for the probability for decay of a W^– to a tau and its anti-neutrino. That deviation is less than three standard deviations — and as I noted, if you have enough measurements, you’ll occasionally get one that differs by more than two standard deviations. We still might wonder if something funny is up with the tau, but we don’t have enough evidence of that yet. Let’s see what the Z boson teaches us later.

In any case, to a physicist’s eye, there is no sign here of any notable disgreement between theory and experiment in these results. Within current uncertainties, the Standard Model correctly predicts the data.

Z Boson Decays, More Precisely

Now let’s do the same for the Z boson, but here we have three steps:

• first, the predictions when we take sin θ_w = 1/2, as we did in the previous post;
• second, the predictions when we take sin θ_w = 0.48;
• third, the better predictions when we also include the effect of the strong nuclear force.

And again Figure 3 compares predictions with the data.

You notice that some of the experimental measurements have extremely small uncertainties! This is especially true of the decays to electrons, to muons, to taus, and (collectively) to the three types of neutrinos. Let’s look at them closely.

If you look at the predictions with sin θ_w = 1/2 for the electrons, muons and taus, they are in disagreement with the measurements by a lot. For example, in Z decay to muons, the initial prediction differs from the data by 19 standard deviations!! Not even close. For sin θ_w = 0.48 but without accounting for the strong nuclear force, the disagreement drops to 11 standard deviations; still terrible. But once we account also for the strong nuclear force, the predictions agree with data to within 1 to 2 standard deviations for all three types of particles.

As for the decays to neutrinos, the three predictions differ by 16 standard deviations, 9 standard deviations, and… below 2 standard deviations.

My reaction, when this data came in in the 1990s, was “Wow.” I hope yours is similar. Such close matching of the Standard Model’s predictions with highly precise measurements is a truly stunning sucesss.

Notice that the successful prediction requires three of the Standard Model’s forces: the mixture of the electromagnetic and weak nuclear forces given by the magic angle, with a small effect from the strong nuclear force. Said another way, all of the Standard Model’s particles except the Higgs boson and top quark play a role in Figs. 2 and 3. (The Higgs field, meanwhile, is secretly in the background, giving the W and Z bosons their masses and affecting the Z boson’s interactions with the other particles; and the top quark is hiding in the background too, since it can’t be removed without changing how the Z boson interacts with bottom quarks.) You can’t take any part of the Standard Model out without messing up these predictions completely.

Oh, and by the way, remember how the probability for W decay to a tau and a neutrino in Fig. 2 was off the prediction by more than two standard deviations? Well there’s nothing weird about the tau or the neutrinos in Fig. 3 — predictions and measurements agree just fine — and indeed, no numbers in Z decay differ from predictions by more than two standard deviations. As I said earlier, the expectation is that about one in every twenty measurements should differ from its true value by more than two standard deviations. Since we have over a dozen measurements in Figs. 2 and 3, it’s no surprise that one of them might be two standard deviations off… and so we can’t use that single disagreement as evidence that the Standard Model doesn’t work.

Asymmetries, Precisely

Let’s do one more case: one of the asymmetries that I mentioned in the bonus section of the previous post. Consider a forward-backward asymmetry shown in Fig. 4. Take all collisions in which an electron strikes a positron (the anti-particle of an electron) and turns into a muon and an anti-muon. Now compare the probability that the muon goes “forward” (roughly the direction that the electron is heading) to the probability that it goes “backward” (roughly the direction that the positron is heading.) If the two probabilities are equal, then the asymmetry would be zero; if the muon always goes to the left, then the asymmetry would be 100%; if always to the right, the asymmetry would be -100%.

Asymmetries are special because the effect of the strong nuclear force cancels out of them completely, and so they only depend on sin θ_w. And this particular “leptonic forward-backward” asymmetry is an example with a special feature: if sin θ_w were exactly 1/2, this asymmetry for lepton production would be predicted to be exactly zero.

But the measured value of this asymmetry, while quite small (less than 2%), is definitely not zero, and so this is another confirmation that sin θ_w is not exactly 1/2. So let’s instead compare the prediction for this asymmetry using sin θ_w = 0.48, the choice that worked so well for the Z boson’s decays in Fig. 3, with the data.

In Figure 5, the horizontal axis shows the lepton forward-backward asymmetry. The prediction of 1.8% that one obtains for sin θ_w = 0.48, widened slightly to cover 1.65% to 2.0%, which is what obtains for sin θ_w between 0.479 and 0.481, is shown in pink. The four open circles represent four measurements of the asymmetry by the four experiments that were located at the LEP collider; the dashes through the circles show the standard deviations on their measurements. The dark circle shows what one gets when one combines the four experiments’ data together, obtaining an even better statistical estimate: 1.71 ± 0.10%, the uncertainty being indicated both as the dash going through the solid circle and as the yellow band. Since the yellow band extends to just above 1.8%, we see that the data differs from the sin θ_w = 0.480 prediction (the center of the pink band) by less than one standard deviation… giving precise agreement of the Standard Model with this very small but well-measured asymmetry.

Predictions of other asymmetries show similar success, as do numerous other measurements.

The Big Picture

Successful predictions like these, especially ones in which both theory and experiment are highly precise, explain why particle physicists have such confidence in the Standard Model, despite its clear limitations.

What limitations of the Standard Model am I referring too? They are many, but one of them is simply that the Standard Model does not predict θ_w . No one can say why θ_w takes the value that it has, or whether the fact that it is close to 30 degrees is a clue to its origin or a mere coincidence. Instead, of the many measurements, we use a single one (such as one of the asymmetries) to extract its value, and then can predict many other quantities.

One thing I’ve neglected to do is to convey the complexity of the calculations that are needed to compare the Standard Model predictions to data. To carry out these computations much more carefully than I did in Figs. 2, 3 and 5, in order to make them as precise as the measurements, demands specialized knowledge and experience. (As an example of how tricky these computations can be: even defining what one means by sin θ_w can be ambiguous in precise enough calculations, and so one needs considerable expertise [which I do not have] to define it correctly and use that definition consistently.) So there are actually still more layers of precision that I could go into…!

But I think perhaps I’ve done enough to convince you that the Standard Model is a fortress. Sure, it’s not a finished construction. Yet neither will it be easily overthrown.

You might have heard of the conundrum “What do you give the man who has everything?” I discovered a variation on it last October: how do you celebrate the man who studied (nearly) everything? Physicist Edwin Thompson Jaynes impacted disciplines from quantum information theory to biomedical imaging. I almost wrote “theoretical physicist,” instead of “physicist,” but a colleague insisted that Jaynes had a knack for electronics and helped design experiments, too. Jaynes worked at Washington University in St. Louis (WashU) from 1960 to 1992. I’d last visited the university in 2018, as a newly minted postdoc collaborating with WashU experimentalist Kater Murch. I’d scoured the campus for traces of Jaynes like a pilgrim seeking a saint’s forelock or humerus. The blog post “Chasing Ed Jaynes’s ghost” documents that hunt.

I found his ghost this October.

Kater and colleagues hosted the Jaynes Centennial Symposium on a brilliant autumn day when the campus’s trees were still contemplating shedding their leaves. The agenda featured researchers from across the sciences and engineering. We described how Jaynes’s legacy has informed 21st-century developments in quantum information theory, thermodynamics, biophysics, sensing, and computation. I spoke about quantum thermodynamics and information theory—specifically, incompatible conserved quantities, about which my research-group members and I have blogged many times.

Irfan Siddiqi spoke about quantum technologies. An experimentalist at the University of California, Berkeley, Irfan featured on Quantum Frontiers seven years ago. His lab specializes in superconducting qubits, tiny circuits in which current can flow forever, without dissipating. How can we measure a superconducting qubit? We stick the qubit in a box. Light bounces back and forth across the box. The light interacts with the qubit while traversing it, in accordance with the Jaynes–Cummings model. We can’t seal any box perfectly, so some light will leak out. That light carries off information about the qubit. We can capture the light using a photodetector to infer about the qubit’s state.

Bill Bialek, too, spoke about inference. But Bill is a Princeton biophysicist, so fruit flies preoccupy him more than qubits do. A fruit fly metamorphoses from a maggot that hatches from an egg. As the maggot develops, its cells differentiate: some form a head, some form a tail, and so on. Yet all the cells contain the same genetic information. How can a head ever emerge, to differ from a tail? 

A fruit-fly mother, Bill revealed, injects molecules into an egg at certain locations. These molecules diffuse across the egg, triggering the synthesis of more molecules. The knock-on molecules’ concentrations can vary strongly across the egg: a maggot’s head cells contain molecules at certain concentrations, and the tail cells contain the same molecules at other concentrations.

At this point in Bill’s story, I was ready to take my hat off to biophysicists for answering the question above, which I’ll rephrase here: if we find that a certain cell belongs to a maggot’s tail, why does the cell belong to the tail? But I enjoyed even more how Bill turned the question on its head (pun perhaps intended): imagine that you’re a maggot cell. How can you tell where in the maggot you are, to ascertain how to differentiate? Nature asks this question (loosely speaking), whereas human observers ask Bill’s first question.

To answer the second question, Bill recalled which information a cell accesses. Suppose you know four molecules’ concentrations: , , , and . How accurately can you predict the cell’s location? That is, what probability does the cell have of sitting at some particular site, conditioned on the s? That probability is large only at one site, biophysicists have found empirically. So a cell can accurately infer its position from its molecules’ concentrations.

I’m no biophysicist (despite minor evidence to the contrary), but I enjoyed Bill’s story as I enjoyed Irfan’s. Probabilities, information, and inference are abstract notions; yet they impact physical reality, from insects to quantum science. This tension between abstraction and concreteness arrested me when I first encountered entropy, in a ninth-grade biology lecture. The tension drew me into information theory and thermodynamics. These toolkits permeate biophysics as they permeate my disciplines. So, throughout the symposium, I spoke with engineers, medical-school researchers, biophysicists, thermodynamicists, and quantum scientists. They all struck me as my kind of people, despite our distribution across the intellectual landscape. Jaynes reasoned about distributions—probability distributions—and I expect he’d have approved of this one. The man who studied nearly everything deserves a celebration that illuminates nearly everything.

On December 11, I gave a keynote address at the Q2B 2024 Conference in Silicon Valley. This is a transcript of my remarks. The slides I presented are here.

NISQ and beyond

I’m honored to be back at Q2B for the 8^th year in a row.

The Q2B conference theme is “The Roadmap to Quantum Value,” so I’ll begin by showing a slide from last year’s talk. As best we currently understand, the path to economic impact is the road through fault-tolerant quantum computing. And that poses a daunting challenge for our field and for the quantum industry.

We are in the NISQ era. And NISQ technology already has noteworthy scientific value. But as of now there is no proposed application of NISQ computing with commercial value for which quantum advantage has been demonstrated when compared to the best classical hardware running the best algorithms for solving the same problems. Furthermore, currently there are no persuasive theoretical arguments indicating that commercially viable applications will be found that do not use quantum error-correcting codes and fault-tolerant quantum computing.

NISQ, meaning Noisy Intermediate-Scale Quantum, is a deliberately vague term. By design, it has no precise quantitative meaning, but it is intended to convey an idea: We now have quantum machines such that brute force simulation of what the quantum machine does is well beyond the reach of our most powerful existing conventional computers. But these machines are not error-corrected, and noise severely limits their computational power.

In the future we can envision FASQ* machines, Fault-Tolerant Application-Scale Quantum computers that can run a wide variety of useful applications, but that is still a rather distant goal. What term captures the path along the road from NISQ to FASQ? Various terms retaining the ISQ format of NISQ have been proposed [here, here, here], but I would prefer to leave ISQ behind as we move forward, so I’ll speak instead of a megaquop or gigaquop machine and so on meaning one capable of executing a million or a billion quantum operations, but with the understanding that mega means not precisely a million but somewhere in the vicinity of a million.

Naively, a megaquop machine would have an error rate per logical gate of order 10^{-6}, which we don’t expect to achieve anytime soon without using error correction and fault-tolerant operation. Or maybe the logical error rate could be somewhat larger, as we expect to be able to boost the simulable circuit volume using various error mitigation techniques in the megaquop era just as we do in the NISQ era. Importantly, the megaquop machine would be capable of achieving some tasks beyond the reach of classical, NISQ, or analog quantum devices, for example by executing circuits with of order 100 logical qubits and circuit depth of order 10,000.

What resources are needed to operate it? That depends on many things, but a rough guess is that tens of thousands of high-quality physical qubits could suffice. When will we have it? I don’t know, but if it happens in just a few years a likely modality is Rydberg atoms in optical tweezers, assuming they continue to advance in both scale and performance.

What will we do with it? I don’t know, but as a scientist I expect we can learn valuable lessons by simulating the dynamics of many-qubit systems on megaquop machines. Will there be applications that are commercially viable as well as scientifically instructive? That I can’t promise you.

The road to fault tolerance

To proceed along the road to fault tolerance, what must we achieve? We would like to see many successive rounds of accurate error syndrome measurement such that when the syndromes are decoded the error rate per measurement cycle drops sharply as the code increases in size. Furthermore, we want to decode rapidly, as will be needed to execute universal gates on protected quantum information. Indeed, we will want the logical gates to have much higher fidelity than physical gates, and for the logical gate fidelities to improve sharply as codes increase in size. We want to do all this at an acceptable overhead cost in both the number of physical qubits and the number of physical gates. And speed matters — the time on the wall clock for executing a logical gate should be as short as possible.

A snapshot of the state of the art comes from the Google Quantum AI team. Their recently introduced Willow superconducting processor has improved transmon lifetimes, measurement errors, and leakage correction compared to its predecessor Sycamore. With it they can perform millions of rounds of surface-code error syndrome measurement with good stability, each round lasting about a microsecond. Most notably, they find that the logical error rate per measurement round improves by a factor of 2 (a factor they call Lambda) when the code distance increases from 3 to 5 and again from 5 to 7, indicating that further improvements should be achievable by scaling the device further. They performed accurate real-time decoding for the distance 3 and 5 codes. To further explore the performance of the device they also studied the repetition code, which corrects only bit flips, out to a much larger code distance. As the hardware continues to advance we hope to see larger values of Lambda for the surface code, larger codes achieving much lower error rates, and eventually not just quantum memory but also logical two-qubit gates with much improved fidelity compared to the fidelity of physical gates.

Last year I expressed concern about the potential vulnerability of superconducting quantum processors to ionizing radiation such as cosmic ray muons. In these events, errors occur in many qubits at once, too many errors for the error-correcting code to fend off. I speculated that we might want to operate a superconducting processor deep underground to suppress the muon flux, or to use less efficient codes that protect against such error bursts.

The good news is that the Google team has demonstrated that so-called gap engineering of the qubits can reduce the frequency of such error bursts by orders of magnitude. In their studies of the repetition code they found that, in the gap-engineered Willow processor, error bursts occurred about once per hour, as opposed to once every ten seconds in their earlier hardware.  Whether suppression of error bursts via gap engineering will suffice for running deep quantum circuits in the future is not certain, but this progress is encouraging. And by the way, the origin of the error bursts seen every hour or so is not yet clearly understood, which reminds us that not only in superconducting processors but in other modalities as well we are likely to encounter mysterious and highly deleterious rare events that will need to be understood and mitigated.

Real-time decoding

Fast real-time decoding of error syndromes is important because when performing universal error-corrected computation we must frequently measure encoded blocks and then perform subsequent operations conditioned on the measurement outcomes. If it takes too long to decode the measurement outcomes, that will slow down the logical clock speed. That may be a more serious problem for superconducting circuits than for other hardware modalities where gates can be orders of magnitude slower.

For distance 5, Google achieves a latency, meaning the time from when data from the final round of syndrome measurement is received by the decoder until the decoder returns its result, of about 63 microseconds on average. In addition, it takes about another 10 microseconds for the data to be transmitted via Ethernet from the measurement device to the decoding workstation. That’s not bad, but considering that each round of syndrome measurement takes only a microsecond, faster would be preferable, and the decoding task becomes harder as the code grows in size.

Riverlane and Rigetti have demonstrated in small experiments that the decoding latency can be reduced by running the decoding algorithm on FPGAs rather than CPUs, and by integrating the decoder into the control stack to reduce communication time. Adopting such methods may become increasingly important as we scale further. Google DeepMind has shown that a decoder trained by reinforcement learning can achieve a lower logical error rate than a decoder constructed by humans, but it’s unclear whether that will work at scale because the cost of training rises steeply with code distance. Also, the Harvard / QuEra team has emphasized that performing correlated decoding across multiple code blocks can reduce the depth of fault-tolerant constructions, but this also increases the complexity of decoding, raising concern about whether such a scheme will be scalable.

Trading simplicity for performance

The Google processors use transmon qubits, as do superconducting processors from IBM and various other companies and research groups. Transmons are the simplest superconducting qubits and their quality has improved steadily; we can expect further improvement with advances in materials and fabrication. But a logical qubit with very low error rate surely will be a complicated object due to the hefty overhead cost of quantum error correction. Perhaps it is worthwhile to fashion a more complicated physical qubit if the resulting gain in performance might actually simplify the operation of a fault-tolerant quantum computer in the megaquop regime or well beyond. Several versions of this strategy are being pursued.

One approach uses cat qubits, in which the encoded 0 and 1 are coherent states of a microwave resonator, well separated in phase space, such that the noise afflicting the qubit is highly biased. Bit flips are exponentially suppressed as the mean photon number of the resonator increases, while the error rate for phase flips induced by loss from the resonator increases only linearly with the photon number. This year the AWS team built a repetition code to correct phase errors for cat qubits that are passively protected against bit flips, and showed that increasing the distance of the repetition code from 3 to 5 slightly improves the logical error rate. (See also here.)

Another helpful insight is that error correction can be more effective if we know when and where the errors occur in a quantum circuit. We can apply this idea using a dual rail encoding of the qubits. With two microwave resonators, for example, we can encode a qubit by placing a single photon in either the first resonator (the 10) state, or the second resonator (the 01 state). The dominant error is loss of a photon, causing either the 01 or 10 state to decay to 00. One can check whether the state is 00, detecting whether the error occurred without disturbing a coherent superposition of 01 and 10. In a device built by the Yale / QCI team, loss errors are detected over 99% of the time and all undetected errors are relatively rare. Similar results were reported by the AWS team, encoding a dual-rail qubit in a pair of transmons instead of resonators.

Another idea is encoding a finite-dimensional quantum system in a state of a resonator that is highly squeezed in two complementary quadratures, a so-called GKP encoding. This year the Yale group used this scheme to encode 3-dimensional and 4-dimensional systems with decay rate better by a factor of 1.8 than the rate of photon loss from the resonator. (See also here.)

A fluxonium qubit is more complicated than a transmon in that it requires a large inductance which is achieved with an array of Josephson junctions, but it has the advantage of larger anharmonicity, which has enabled two-qubit gates with better than three 9s of fidelity, as the MIT team has shown.

Whether this trading of simplicity for performance in superconducting qubits will ultimately be advantageous for scaling to large systems is still unclear. But it’s appropriate to explore such alternatives which might pay off in the long run.

Error correction with atomic qubits

We have also seen progress on error correction this year with atomic qubits, both in ion traps and optical tweezer arrays. In these platforms qubits are movable, making it possible to apply two-qubit gates to any pair of qubits in the device. This opens the opportunity to use more efficient coding schemes, and in fact logical circuits are now being executed on these platforms. The Harvard / MIT / QuEra team sampled circuits with 48 logical qubits on a 280-qubit device –- that big news broke during last year’s Q2B conference. Atom computing and Microsoft ran an algorithm with 28 logical qubits on a 256-qubit device. Quantinuum and Microsoft prepared entangled states of 12 logical qubits on a 56-qubit device.

However, so far in these devices it has not been possible to perform more than a few rounds of error syndrome measurement, and the results rely on error detection and postselection. That is, circuit runs are discarded when errors are detected, a scheme that won’t scale to large circuits. Efforts to address these drawbacks are in progress. Another concern is that the atomic movement slows the logical cycle time. If all-to-all coupling enabled by atomic movement is to be used in much deeper circuits, it will be important to speed up the movement quite a lot.

Toward the megaquop machine

How can we reach the megaquop regime? More efficient quantum codes like those recently discovered by the IBM team might help. These require geometrically nonlocal connectivity and are therefore better suited for Rydberg optical tweezer arrays than superconducting processors, at least for now. Error mitigation strategies tailored for logical circuits, like those pursued by Qedma, might help by boosting the circuit volume that can be simulated beyond what one would naively expect based on the logical error rate. Recent advances from the Google team, which reduce the overhead cost of logical gates, might also be helpful.

What about applications? Impactful applications to chemistry typically require rather deep circuits so are likely to be out of reach for a while yet, but applications to materials science provide a more tempting target in the near term. Taking advantage of symmetries and various circuit optimizations like the ones Phasecraft has achieved, we might start seeing informative results in the megaquop regime or only slightly beyond.

As a scientist, I’m intrigued by what we might conceivably learn about quantum dynamics far from equilibrium by doing simulations on megaquop machines, particularly in two dimensions. But when seeking quantum advantage in that arena we should bear in mind that classical methods for such simulations are also advancing impressively, including in the past year (for example, here and here).

To summarize, advances in hardware, control, algorithms, error correction, error mitigation, etc. are bringing us closer to megaquop machines, raising a compelling question for our community: What are the potential uses for these machines? Progress will require innovation at all levels of the stack.  The capabilities of early fault-tolerant quantum processors will guide application development, and our vision of potential applications will guide technological progress. Advances in both basic science and systems engineering are needed. These are still the early days of quantum computing technology, but our experience with megaquop machines will guide the way to gigaquops, teraquops, and beyond and hence to widely impactful quantum value that benefits the world.

I thank Dorit Aharonov, Sergio Boixo, Earl Campbell, Roland Farrell, Ashley Montanaro, Mike Newman, Will Oliver, Chris Pattison, Rob Schoelkopf, and Qian Xu for helpful comments.

*The acronym FASQ was suggested to me by Andrew Landahl.

I want to go back over something from Part 11, but in a more systematic and self-contained way.

Namely, I want to prove a wonderful known fact about random permutations, the Cycle Length Lemma, using a bit of category theory. The idea here is that the number of $k$-cycles in a random permutation of $n$ things is a random variable. Then comes a surprise: in the limit as $n \to \infty$, this random variable approaches a Poisson distribution with mean $1/k$. And even better, for different choices of $k$ these random variables become independent in the $n \to \infty$ limit.

I’m stating these facts roughly now, to not get bogged down. But I’ll state them precisely, prove them, and categorify them. That is, I’ll state equations involving random variables — but I’ll prove that these equations come from equivalences of groupoids!

First I’ll state the Cycle Length Lemma, which summarizes a lot of interesting facts about random permutations. Then I’ll state and prove a categorified version of the Cycle Length Lemma, which asserts an equivalence of groupoids. Then I’ll derive the original version of the lemma from this categorified version by taking the cardinalities of these groupoids. The categorified version contains more information, so it’s not just a trick for proving the original lemma.

What do groupoids have to do with random permutations? You’ll see, but it’s an example of ‘principle of indifference’, especially in its modern guise, called the ‘principle of transformation groups’: the idea that outcomes related by a symmetry should have the same probability. This sets up a connection between groupoids and probability theory — and as we’ll see, we can “go down” from groupoids to probabilities using the theory of groupoid cardinalities.

The Cycle Length Lemma

In the theory of random permutations, we treat the symmetric group $S_n$ as a probability measure space where each element has the same measure, namely $1/n!$. Functions $f \colon S_n \to \mathbb{R}$ then become random variables, and we can study their expected values:

$$E(f) = \frac{1}{n!} \sum_{\sigma \in S_n} f(\sigma).$$

An important example is the function

$$C_k \colon S_n \to \mathbb{N}$$

that counts, for any permutation $\sigma \in S_n$, its number of cycles of length $k$, also called $k$-cycles. A well-known but striking fact about random permutations is that whenever $k \le n$, the expected number of $k$-cycles is $1/k$:

$$E(C_k) = \frac{1}{k}$$

For example, a random permutation of any finite set has, on average, one fixed point!

Another striking fact is that whenever $j \ne k$ and $j + k \le n$, so that it’s possible for a permutation $\sigma \in S_n$ to have both a $j$-cycle and a $k$-cycle, the random variables $C_j$ and $C_k$ are uncorrelated in the following sense:

$$E(C_j C_k) = E(C_j) E(C_k) .$$

You might at first think that having lots of $j$-cycles for some large $j$ would tend to inhibit the presence of $k$-cycles for some other large value of $k$, but that’s not true unless $j + k \gt n$, when it suddenly becomes impossible to have both a $j$-cycle and a $k$-cycle!

These two facts are special cases of the Cycle Length Lemma. To state this lemma in full generality, recall that the number of ordered $p$-tuples of distinct elements of an $n$-element set is the falling power

$$n^{\underline{p}} = n(n-1)(n-2) \, \cdots \, (n-p+1).$$

It follows that the function

$$C_k^{\underline{p}} \colon S_n \to \mathbb{N}$$

counts, for any permutation in $S_n$, its ordered $p$-tuples of distinct $k$-cycles. We can also replace the word ‘distinct’ here by ‘disjoint’, without changing the meaning, since distinct cycles must be disjoint.

The two striking facts mentioned above generalize as follows:

1) First, whenever $p k \le n$, so that it is possible for a permutation in $S_n$ to have $p$ distinct $k$-cycles, then

$$E(C_k^{\underline{p}}) = \frac{1}{k^p}.$$

If you know about the moments of a Poisson distribution here’s a nice equivalent way to state this equation: when $p k \le n$, the $p$th moment of the random variable $C_k$ equals that of a Poisson distribution with mean $1/k$.

2) Second, the random variables $C_k$ are better and better approximated by independent Poisson distributions. To state this precisely we need a bit of notation. Let $\vec{p}$ denote an $n$-tuple $(p_1 , \dots, p_n)$ of natural numbers, and let

$$|\vec{p}| = p_1 + 2p_2 + \cdots + n p_n.$$

If $|\vec{p}| \le n$, it is possible for a permutation $\sigma \in S_n$ to have a collection of distinct cycles, with $p_1$ cycles of length 1, $p_2$ cycles of length 2, and so on up to $p_n$ cycles of length $n$. If $|\vec{p}| \gt n$, this is impossible. In the former case, where $|\vec{p}| \le n$, we always have

$$E\left( \prod_{k=1}^n C_k^{\underline{p}_k} \right) = \prod_{k=1}^n E( C_k^{\underline{p}_k}) .$$

Taken together, 1) and 2) are equivalent to the Cycle Length Lemma, which may be stated in a unified way as follows:

Cycle Length Lemma. Suppose $p_1 , \dots, p_n \in \mathbb{N}$. Then

$$E\left( \prod_{k=1}^n C_k^{\underline{p}_k} \right) = \left\{ \begin{array}{ccc} \displaystyle{ \prod_{k=1}^n \frac{1}{k^{p_k}} } & & \mathrm{if} \; |\vec{p}| \le n \\ \\ 0 & & \mathrm{if} \; |\vec{p}| \gt n \end{array} \right.$$

This appears, for example, in Ford’s course notes on random permutations and the statistical properties of prime numbers [Lemma 1.1, F]. The most famous special case is when $|\vec{p}| = n$. Apparently this goes back to Cauchy, but I don’t know where he proved it. I believe he would have phrased it in terms of counting permutations, not probabilities.

I won’t get into details of precisely the sense in which random variables $C_k$ approach independent Poisson distributions. For that, see Arratia and Tavaré [AT].

The Categorified Cycle Length Lemma

To categorify the Cycle Length Lemma, the key is to treat a permutation as an extra structure that we can put on a set, and then consider the groupoid of $n$-element sets equipped with this extra structure:

Definition. Let $\mathsf{Perm}(n)$ be the groupoid in which

• an object is an $n$-element set equipped with a permutation $\sigma \colon X \to X$

and

• a morphism from $\sigma \colon X \to X$ to $\sigma' \colon X' \to X'$ is a bijection $f \colon X \to X'$ that is permutation-preserving in the following sense: