### Doubts about teapot supremacy: my reply to Richard Borcherds

Tuesday, April 20th, 2021Richard Borcherds is a British mathematician at Berkeley, who won the 1998 Fields Medal for the proof of the monstrous moonshine conjecture among many other contributions. A couple months ago, Borcherds posted on YouTube a **self-described “rant” about quantum computing**, which was recently making the rounds on Facebook and which I found highly entertaining.

Borcherds points out that the term “quantum supremacy” means only that quantum computers can outperform existing classical computers on *some* benchmark, which can be chosen to show maximum advantage for the quantum computer. He allows that BosonSampling could have some value, for example in calibrating quantum computers or in comparing one quantum computer to another, but he decries the popular conflation of quantum supremacy with the actual construction of a scalable quantum computer able (for example) to run Shor’s algorithm to break RSA.

Borcherds also proposes a “teapot test,” according to which any claim about quantum computers can be dismissed if an analogous claim would hold for a teapot (which he brandishes for the camera). For example, there are many claims to solve practical optimization and machine learning problems by “quantum/classical hybrid algorithms,” wherein a classical computer does most of the work but a quantum computer is somehow involved. Borcherds points out that, at least as things stand in early 2021, in most or all such cases, the classical computer could’ve probably done as well entirely on its own. So then if you put a teapot on top of your classical computer while it ran, you could equally say you used a “classical/teapot hybrid approach.”

Needless to say, Borcherds is correct about all of this. I’ve made similar points on this blog for 15 years, although less Britishly. I’m delighted to have such serious new firepower on the scoffing-at-QC-hype team.

I do, however, have one substantive disagreement. At one point, Borcherds argues that sampling-based quantum supremacy itself fails his teapot test. For consider the computational problem of predicting how many pieces a teapot will break into if it’s dropped on the ground. Clearly, he says, the teapot itself will outperform any simulation running on any existing classical computer at that task, and will therefore achieve “teapot supremacy.” But who cares??

I’m glad that Borcherds has set out, rather crisply, an objection that’s been put to me many times over the past decade. The response is simple: *I don’t believe the teapot really does achieve teapot supremacy on the stated task! At the least, I’d need to be shown why. You can’t just assert it without serious argument.*

If we want to mirror the existing quantum supremacy experiments, then the teapot computational problem, properly formulated, should be: given as input a description of a teapot’s construction, the height from which it’s dropped, etc., *output a sample from the probability distribution* over the number of shards that the teapot will break into when it hits the floor.

If so, though, then clearly a classical computer can easily sample from the same distribution! Why? Because presumably we agree that there’s a negligible probability of more than (say) 1000 shards. So the distribution is characterized by a list of at most 1000 probabilities, which can be estimated empirically (at the cost of a small warehouse of smashed teapots) and thereafter used to generate samples. In the plausible event that the distribution is (say) a Gaussian, it’s even easier: just estimate the mean and variance.

A couple days ago, I was curious what the distribution looked like, so I decided to order some teapots from Amazon and check. Unfortunately, real porcelain teapots are *expensive*, and it seemed vaguely horrific to order dozens (as would be needed to get reasonable data) for the sole purpose of smashing them on my driveway. So I hit on what seemed like a perfect solution: I ordered *toy* teapots, which were much smaller and cheaper. Alas, when my toy “porcelain” teapots arrived yesterday, they turned out (unsurprisingly in retrospect for a children’s toy) to be some sort of plastic or composite material, meaning that they *didn’t* break unless one propelled them downward forcefully. So, while I can report that they tended to break into one or two large pieces along with two or three smaller shards, I found it impossible to get better data. (There’s a reason why I became a *theoretical* computer scientist…)

The good news is that my 4-year-old son had an absolute *blast* smashing toy teapots with me on our driveway, while my 8-year-old daughter was thrilled to take the remaining, unbroken teapots for her dollhouse. I apologize if this fails to defy gender stereotypes.

Anyway, it might be retorted that it’s not good enough to sample from a probability distribution: what’s wanted, rather, is to calculate how many pieces this *specific* teapot will break into, given all the microscopic details of it and its environment. Aha, this brings us to a crucial conceptual point: in order for something to count as an “input” to a computer, *you need to be able to set it freely*. Certainly, at the least, you need to be able to measure and record the input in its entirety, so that someone trying to reproduce your computation on a standard silicon computer would know exactly which computation to do. You don’t get to claim computational supremacy based on a problem with secret inputs: that’s like failing someone on a math test without having fully told them the problems.

Ability to set and know the inputs is *the* key property that’s satisfied by Google’s quantum supremacy experiment, and to a lesser extent by the USTC BosonSampling experiment, but that’s not satisfied at all by the “smash a teapot on the floor” experiment. Or perhaps it’s better to say: influences on a computation that vary uncontrollably and chaotically, like gusts of air hitting the teapot as it falls to the floor, shouldn’t be called “inputs” at all; they’re simply *noise sources*. And what one does with noise sources is to try to estimate their distribution and average over them—but in that case, as I said, there’s no teapot supremacy.

A Facebook friend said to me: that’s well and good, but surely we could change Borcherds’s teapot experiment to address this worry? For example: add a computer-controlled lathe (or even a 3D printer), with which you can build a teapot in an arbitrary shape of your choice. Then consider the problem of sampling from the probability distribution over how many pieces *that* teapot will smash into, when it’s dropped from some standard height onto some standard surface. I replied that this is indeed more interesting—in fact, it already seems more like what engineers do in practice (still, sometimes!) when building wind tunnels, than like a silly reductio ad absurdum of quantum supremacy experiments. On the other hand, *if* you believe the Extended Church-Turing Thesis, then as long as your analog computer is governed by classical physics, it’s presumably inherently limited to an Avogadro’s number type speedup over a standard digital computer, whereas with a quantum computer, you’re limited only by the exponential dimensionality of Hilbert space, which seems more interesting.

Or maybe I’m wrong—in which case, I look forward to the first practical demonstration of teapot supremacy! Just like with quantum supremacy, though, it’s not enough to *assert* it; you need to … put the tea where your mouth is.

**Update:** On the suggestion of Ernest Davis, who I can now reveal as the Facebook friend mentioned above, I just ordered some terra cotta flower pots, which look cheap, easily smashable, and environmentally friendly, and which will hopefully be acceptable substitutes for porcelain teapots in a new experiment. (Not that my main arguments in this post hinge on the results of such an experiment! That’s the power of theory.)

**Another Update:** Some of you might enjoy John Horgan’s *Scientific American* column on reality vs. hype in quantum computing, based on conversations with me and with Terry Rudolph of PsiQuantum.