Shtetl-Optimized

This morning Xavier Waintal, coauthor of the new arXiv preprint “””refuting””” Grover’s algorithm, which I dismantled here yesterday, emailed me a two-paragraph response. He remarked that the “classy” thing for me to do would be to post the response on my blog, but: “I would totally understand if you did not want to be contradicted in your own zone of influence.”

Here is Waintal’s response, exactly as sent to me:

The elephant in the (quantum computing) room: opening the Pandora box of the quantum oracle

One of the problem we face in the field of quantum computing is a vast diversity of cultures between, say, complexity theorists on one hand and physicists on the other hand. The former define mathematical objects and consider any mathematical problem as legitimate. The hypothesis are never questioned, by definition. Physicists on the other hand spend their life questioning the hypothesis, wondering if they do apply to the real world. This dichotomy is particularly acute in the context of the emblematic Grover search algorithm, one of the cornerstone of quantum computing. Grover’s algorithm uses the concept of “oracle”, a black box function that one can call, but of which one is forbidden to see the source code. There are well known complexity theorems that show that in this context a quantum computer can solve the “search problem” faster than a classical computer.

But because we closed our eyes and decided not to look at the source code does not mean it does not exist. In https://arxiv.org/pdf/2303.11317.pdf, Miles Stoudenmire and I deconstruct the concept of oracle and show that as soon as we give the same input to both quantum and classical computers (the quantum circuit used to program the oracle on the actual quantum hardware) then the *generic* quantum advantage disappears. The charge of the proof is reversed: one must prove certain properties of the quantum circuit in order to possibly have a theoretical quantum advantage. More importantly – for the physicist that I am – our classical algorithm is very fast and we show that we can solve large instances of any search problem. This means that even for problems where *asymptotically* quantum computers are faster than classical ones, the crossing point where they become so is for astronomically large computing time, in the tens of millions of years. Who is willing to wait that long for the answer to a single question, even if the answer is 42?

The above explicitly confirms something that I realized immediately on reading the preprint, and that fully explains the tone of my response. Namely, Stoudenmire and Waintal’s beef isn’t merely with Grover’s algorithm, or even with the black-box model; it’s with the entire field of complexity theory. If they were right that complexity theorists never “questioned hypotheses” or wondered what did or didn’t apply to the real world, then complexity theory shouldn’t exist in CS departments at all—at most it should exist in pure math departments.

But a converse claim is also true. Namely, suppose it turned out that complexity theorists had already fully understood, for decades, all the elementary points Stoudenmire and Waintal were making about oracles versus explicit circuits. Suppose complexity theorists hadn’t actually been confused, at all, about under what sorts of circumstances the square-root speedup of Grover’s algorithm was (1) provable, (2) plausible but unproven, or (3) nonexistent. Suppose they’d also been intimately familiar with the phenomenon of asymptotically faster algorithms that get swamped in practice by unfavorable constants, and with the overhead of quantum error-correction. Suppose, indeed, that complexity theorists hadn’t merely understood all this stuff, but expressed it clearly and accurately where Stoudenmire and Waintal’s presentation was garbled and mixed with absurdities (e.g., the Grover problem “being classically solvable with a linear number of queries,” the Grover speedup not being “generic,” their being able to “solve large instances of any search problem” … does that include, for example, CircuitSAT? do they still not get the point about CircuitSAT?). Then Stoudenmire and Waintal’s whole objection would collapse.

Anyway, we don’t have to suppose! In the SciRate discussion of the preprint, a commenter named Bibek Pokharel helpfully digs up some undergraduate lecture notes from 2017 that are perfectly clear about what Stoudenmire and Waintal treat as revelations (though one could even go 20+ years earlier). The notes are focused here on Simon’s algorithm, but the discussion generalizes to any quantum black-box algorithm, including Grover’s:

The difficulty in claiming that we’re getting a quantum speedup [via Simon’s algorithm] is that, once we pin down the details of how we’re computing [the oracle function] f—so, for example, the actual matrix A [such that f(x)=Ax]—we then need to compare against classical algorithms that know those details as well. And as soon as we reveal the innards of the black box, the odds of an efficient classical solution become much higher! So for example, if we knew the matrix A, then we could solve Simon’s problem in classical polynomial time just by calculating A‘s nullspace. More generally, it’s not clear whether anyone to this day has found a straightforward “application” of Simon’s algorithm, in the sense of a class of efficiently computable functions f that satisfy the Simon promise, and for which any classical algorithm plausibly needs exponential time to solve Simon’s problem, even if the algorithm is given the code for f.

In the same lecture notes, one can find the following discussion of Grover’s algorithm, and how its unconditional square-root speedup becomes conditional (albeit, still extremely plausible in many cases) as soon as the black box is instantiated by an explicit circuit:

For an NP-complete problem like CircuitSAT, we can be pretty confident that the Grover speedup is real, because no one has found any classical algorithm that’s even slightly better than brute force. On the other hand, for more “structured” NP-complete problems, we do know exponential-time algorithms that are faster than brute force. For example, 3SAT is solvable classically in about O(1.3ⁿ) time. So then, the question becomes a subtle one of whether Grover’s algorithm can be combined with the best classical tricks that we know to achieve a polynomial speedup even compared to a classical algorithm that uses the same tricks. For many NP-complete problems the answer seems to be yes, but it need not be yes for all of them.

The notes in question were written by some random complexity theorist named Scot Aronsen (sp?). But if you don’t want to take it from that guy, then take it from (for example) the Google quantum chemist Ryan Babbush, again on the SciRate page:

It is well understood that applying Grover’s algorithm to 3-SAT in the standard way would not give a quadratic speedup over the best classical algorithm for 3-SAT in the worst case (and especially not on average). But there are problems for which Grover is expected to give a quadratic speedup over any classical algorithm in the worst case. For example, the problem “Circuit SAT” starts by me handing you a specification of a poly-size classical circuit with AND/OR/NOT gates, so it’s all explicit. Then you need to solve SAT on this circuit. Classically we strongly believe it will take time 2^n (this is even the basis of many conjectures in complexity theory, like the exponential time hypothesis), and quantumly we know it can be done with 2^{n/2}*poly(n) quantum gates using Grover and the explicitly given classical circuit. So while I think there are some very nice insights in this paper, the statement in the title “Grover’s Algorithm Offers No Quantum Advantage” seems untrue in a general theoretical sense. Of course, this is putting aside issues with the overheads of error-correction for quadratic speedups (a well understood practical matter that is resolved by going to large problem sizes that wouldn’t be available to the first fault-tolerant quantum computers). What am I missing?

More generally, over the past few days, as far as I can tell, every actual expert in quantum algorithms who’s looked at Stoudenmire and Waintal’s preprint—every one, whether complexity theorist or physicist or chemist—has reached essentially the same conclusions about it that I did. The one big difference is that many of the experts, who are undoubtedly better people than I am, extended a level of charity to Stoudenmire and Waintal (“well, this of course seems untrue, but here’s what it could have meant”) that Stoudenmire and Waintal themselves very conspicuously failed to extend to complexity theory.

Unrelated Update: Huge congratulations to Ethernet inventor Bob Metcalfe, for winning UT Austin’s third Turing Award after Dijkstra and Emerson!

And also to mathematician Luis Caffarelli, for winning UT Austin’s third Abel Prize!

---

I was really, really hoping that I’d be able to avoid blogging about this new arXiv preprint, by E. M. Stoudenmire and Xavier Waintal:

Grover’s Algorithm Offers No Quantum Advantage

Grover’s algorithm is one of the primary algorithms offered as evidence that quantum computers can provide an advantage over classical computers. It involves an “oracle” (external quantum subroutine) which must be specified for a given application and whose internal structure is not part of the formal scaling of the quantum speedup guaranteed by the algorithm. Grover’s algorithm also requires exponentially many steps to succeed, raising the question of its implementation on near-term, non-error-corrected hardware and indeed even on error-corrected quantum computers. In this work, we construct a quantum inspired algorithm, executable on a classical computer, that performs Grover’s task in a linear number of call to the oracle – an exponentially smaller number than Grover’s algorithm – and demonstrate this algorithm explicitly for boolean satisfiability problems (3-SAT). Our finding implies that there is no a priori theoretical quantum speedup associated with Grover’s algorithm. We critically examine the possibility of a practical speedup, a possibility that depends on the nature of the quantum circuit associated with the oracle. We argue that the unfavorable scaling of the success probability of Grover’s algorithm, which in the presence of noise decays as the exponential of the exponential of the number of qubits, makes a practical speedup unrealistic even under extremely optimistic assumptions on both hardware quality and availability.

Alas, inquiries from journalists soon made it clear that silence on my part wasn’t an option.

So, desperately seeking an escape, this morning I asked GPT-4 to read the preprint and comment on it just like I would. Sadly, it turns out the technology isn’t quite ready to replace me at this blogging task. I suppose I should feel good: in every such instance, either I’m vindicated in all my recent screaming here about generative AI—what the naysayers call “glorified autocomplete”—being on the brink of remaking civilization, or else I still, for another few months at least, have a role to play on the Internet.

So, on to the preprint, as reviewed by the human Scott Aaronson. Yeah, it’s basically a tissue of confusions, a mishmash of the well-known and the mistaken. As they say, both novel and correct, but not in the same places.

The paper’s most eye-popping claim is that the Grover search problem—namely, finding an n-bit string x such that f(x)=1, given oracle access to a Boolean function f:{0,1}ⁿ→{0,1}—is solvable classically, using a number of calls that’s only linear in n, or in many cases only constant (!!). Since this claim contradicts a well-known, easily provable lower bound—namely, that Ω(2ⁿ) oracle calls are needed for classical brute-force searching—the authors must be using words in nonstandard ways, leaving only the question of how.

It turns out that, for their “quantum-inspired classical algorithm,” the authors assume you’re given, not merely an oracle for f, but the actual circuit to compute f. They then use that circuit in a non-oracular way to extract the marked item. In which case, I’d prefer to say that they’ve actually solved the Grover problem with zero queries—simply because they’ve entirely left the black-box setting where Grover’s algorithm is normally formulated!

What could possibly justify such a move? Well, the authors argue that sometimes one can use the actual circuit to do better classically than Grover’s algorithm would do quantumly, and therefore, they’ve shown that the Grover speedup is not “generic,” as the quantum algorithms people always say it is.

But this is pure wordplay around the meaning of “generic.” When we say that Grover’s algorithm achieves a “generic” square-root speedup, what we mean is that it solves the generic black-box search problem in O(2^n/2) queries, whereas any classical algorithm for that generic problem requires Ω(2ⁿ) queries. We don’t mean that for every f, Grover achieves a quadratic speedup for searching that f, compared to the best classical algorithm that could be tailored to that f. Of course we don’t; that would be trivially false!

Remarkably, later in the paper, the authors seem to realize that they haven’t delivered the knockout blow against Grover’s algorithm that they’d hoped for, because they then turn around and argue that, well, even for those f’s where Grover does provide a quadratic speedup over the best (or best-known) classical algorithm, noise and decoherence could negate the advantage in practice, and solving that problem would require a fault-tolerant quantum computer, but fault-tolerance could require an enormous overhead, pushing a practical Grover speedup far into the future.

The response here boils down to “no duh.” Yes, if Grover’s algorithm can yield any practical advantage in the medium term, it will either be because we’ve discovered much cheaper ways to do quantum fault-tolerance, or else because we’ve discovered “NISQy” ways to exploit the Grover speedup, which avoid the need for full fault-tolerance—for example, via quantum annealing. The prospects are actually better for a medium-term advantage from Shor’s factoring algorithm, because of its exponential speedup. Hopefully everyone in quantum computing theory has realized all this for a long time.

Anyway, as you can see, by this point we’ve already conceded the principle of Grover’s algorithm, and are just haggling over the practicalities! Which brings us back to the authors’ original claim to have a principled argument against the Grover speedup, which (as I said) rests on a confusion over words.

Some people dread the day when GPT will replace them. In my case, for this task, I can’t wait.

---

Thanks to students Yuxuan Zhang (UT) and Alex Meiburg (UCSB) for discussions of the Stoudenmire-Waintal preprint that informed this post. Of course, I take sole blame for anything anyone dislikes about the post!

---

For a much more technical response—one that explains how this preprint’s detailed attempt to simulate Grover classically fails, rather than merely proving that it must fail—check out this comment by Alex Meiburg.

Just days after we celebrated my wife’s 40th birthday, she came down with COVID, meaning she’s been isolating and I’ve been spending almost all my time dealing with our kids.

But if experience has taught me anything, it’s that the quantum hype train never slows down. In the past 24 hours, at least four people have emailed to ask me about a new paper entitled “Factoring integers with sublinear resources on a superconducting quantum processor.” Even the security expert Bruce Schneier, while skeptical, took the paper surprisingly seriously.

The paper claims … well, it’s hard to pin down what it claims, but it’s certainly given many people the impression that there’s been a decisive advance on how to factor huge integers, and thereby break the RSA cryptosystem, using a near-term quantum computer. Not by using Shor’s Algorithm, mind you, but by using the deceptively similarly named Schnorr’s Algorithm. The latter is a classical algorithm based on lattices, which the authors then “enhance” using the heuristic quantum optimization method called QAOA.

For those who don’t care to read further, here is my 3-word review:

No. Just No.

And here’s my slightly longer review:

Schnorr ≠ Shor. Yes, even when Schnorr’s algorithm is dubiously “enhanced” using QAOA—a quantum algorithm that, incredibly, for all the hundreds of papers written about it, has not yet been convincingly argued to yield any speedup for any problem whatsoever (besides, as it were, the problem of reproducing its own pattern of errors) (one possible recent exception from Sami Boulebnane and Ashley Montanaro).

In the new paper, the authors spend page after page saying-without-saying that it might soon become possible to break RSA-2048, using a NISQ (i.e., non-fault-tolerant) quantum computer. They do so via two time-tested strategems:

1. the detailed exploration of irrelevancies (mostly, optimization of the number of qubits, while ignoring the number of gates), and
2. complete silence about the one crucial point.

Then, finally, they come clean about the one crucial point in a single sentence of the Conclusion section:

It should be pointed out that the quantum speedup of the algorithm is unclear due to the ambiguous convergence of QAOA.

“Unclear” is an understatement here. It seems to me that a miracle would be required for the approach here to yield any benefit at all, compared to just running the classical Schnorr’s algorithm on your laptop. And if the latter were able to break RSA, it would’ve already done so.

All told, this is one of the most actively misleading quantum computing papers I’ve seen in 25 years, and I’ve seen … many. Having said that, this actually isn’t the first time I’ve encountered the strange idea that the exponential quantum speedup for factoring integers, which we know about from Shor’s algorithm, should somehow “rub off” onto quantum optimization heuristics that embody none of the actual insights of Shor’s algorithm, as if by sympathetic magic. Since this idea needs a name, I’d hereby like to propose Cargo Cult Quantum Factoring.

And with that, I feel I’ve adequately discharged my duties here to sanity and truth. If I’m slow to answer comments, it’ll be because I’m dealing with two screaming kids.

Update (Dec. 6): I’m having a blast at the Workshop on Spacetime and Quantum Information at the Institute for Advanced Study in Princeton. I’m learning a huge amount from the talks and discussions here—and also simply enjoying being back in Princeton, to see old friends and visit old haunts like the Bent Spoon. Tomorrow I’ll speak about my recent work with Jason Pollack on polynomial-time AdS bulk reconstruction. [New: click here for video of my talk!]

But there’s one thing, relevant to this post, that I can’t let pass without comment. Tonight, David Nirenberg, Director of the IAS and a medieval historian, gave an after-dinner speech to our workshop, centered around how auspicious it was that the workshop was being held a mere week after the momentous announcement of a holographic wormhole on a microchip (!!)—a feat that experts were calling the first-ever laboratory investigation of quantum gravity, and a new frontier for experimental physics itself. Nirenberg asked whether, a century from now, people might look back on the wormhole achievement as today we look back on Eddington’s 1919 eclipse observations providing the evidence for general relativity.

I confess: this was the first time I felt visceral anger, rather than mere bemusement, over this wormhole affair. Before, I had implicitly assumed: no one was actually hoodwinked by this. No one really, literally believed that this little 9-qubit simulation opened up a wormhole, or helped prove the holographic nature of the real universe, or anything like that. I was wrong.

To be clear, I don’t blame Professor Nirenberg at all. If I were a medieval historian, everything he said about the experiment’s historic significance might strike me as perfectly valid inferences from what I’d read in the press. I don’t blame the It from Qubit community—most of which, I can report, was grinding its teeth and turning red in the face right alongside me. I don’t even blame most of the authors of the wormhole paper, such as Daniel Jafferis, who gave a perfectly sober, reasonable, technical talk at the workshop about how he and others managed to compress a simulation of a variant of the SYK model into a mere 9 qubits—a talk that eschewed all claims of historic significance and of literal wormhole creation.

But it’s now clear to me that, between

(1) the It from Qubit community that likes to explore speculative ideas like holographic wormholes, and

(2) the lay news readers who are now under the impression that Google just did one of the greatest physics experiments of all time,

something went terribly wrong—something that risks damaging trust in the scientific process itself. And I think it’s worth reflecting on what we can do to prevent it from happening again.

---

This is going to be one of the many Shtetl-Optimized posts that I didn’t feel like writing, but was given no choice but to write.

News, social media, and my inbox have been abuzz with two claims about Google’s Sycamore quantum processor, the one that now has 72 superconducting qubits.

The first claim is that Sycamore created a wormhole (!)—a historic feat possible only with a quantum computer. See for example the New York Times and Quanta and Ars Technica and Nature (and of course, the actual paper), as well as Peter Woit’s blog and Chad Orzel’s blog.

The second claim is that Sycamore’s pretensions to quantum supremacy have been refuted. The latter claim is based on this recent preprint by Dorit Aharonov, Xun Gao, Zeph Landau, Yunchao Liu, and Umesh Vazirani. No one—least of all me!—doubts that these authors have proved a strong new technical result, solving a significant open problem in the theory of noisy random circuit sampling. On the other hand, it might be less obvious how to interpret their result and put it in context. See also a YouTube video of Yunchao speaking about the new result at this week’s Simons Institute Quantum Colloquium, and of a panel discussion afterwards, where Yunchao, Umesh Vazirani, Adam Bouland, Sergio Boixo, and your humble blogger discuss what it means.

On their face, the two claims about Sycamore might seem to be in tension. After all, if Sycamore can’t do anything beyond what a classical computer can do, then how exactly did it bend the topology of spacetime?

I submit that neither claim is true. On the one hand, Sycamore did not “create a wormhole.” On the other hand, it remains pretty hard to simulate with a classical computer, as far as anyone knows. To summarize, then, our knowledge of what Sycamore can and can’t do remains much the same as last week or last month!

---

Let’s start with the wormhole thing. I can’t really improve over how I put it in Dennis Overbye’s NYT piece:

“The most important thing I’d want New York Times readers to understand is this,” Scott Aaronson, a quantum computing expert at the University of Texas in Austin, wrote in an email. “If this experiment has brought a wormhole into actual physical existence, then a strong case could be made that you, too, bring a wormhole into actual physical existence every time you sketch one with pen and paper.”

More broadly, Overbye’s NYT piece explains with admirable clarity what this experiment did and didn’t do—leaving only the question “wait … if that’s all that’s going on here, then why is it being written up in the NYT??” This is a rare case where, in my opinion, the NYT did a much better job than Quanta, which unequivocally accepted and amplified the “QC creates a wormhole” framing.

Alright, but what’s the actual basis for the “QC creates a wormhole” claim, for those who don’t want to leave this blog to read about it? Well, the authors used 9 of Sycamore’s 72 qubits to do a crude simulation of something called the SYK (Sachdev-Ye-Kitaev) model. SYK has become popular as a toy model for quantum gravity. In particular, it has a holographic dual description, which can indeed involve a spacetime with one or more wormholes. So, they ran a quantum circuit that crudely modelled the SYK dual of a scenario with information sent through a wormhole. They then confirmed that the circuit did what it was supposed to do—i.e., what they’d already classically calculated that it would do.

So, the objection is obvious: if someone simulates a black hole on their classical computer, they don’t say they thereby “created a black hole.” Or if they do, journalists don’t uncritically repeat the claim. Why should the standards be different just because we’re talking about a quantum computer rather than a classical one?

Did we at least learn anything new about SYK wormholes from the simulation? Alas, not really, because 9 qubits take a mere 2⁹=512 complex numbers to specify their wavefunction, and are therefore trivial to simulate on a laptop. There’s some argument in the paper that, if the simulation were scaled up to (say) 100 qubits, then maybe we would learn something new about SYK. Even then, however, we’d mostly learn about certain corrections that arise because the simulation was being done with “only” n=100 qubits, rather than in the n→∞ limit where SYK is rigorously understood. But while those corrections, arising when n is “neither too large nor too small,” would surely be interesting to specialists, they’d have no obvious bearing on the prospects for creating real physical wormholes in our universe.

And yet, this is not a sensationalistic misunderstanding invented by journalists. Some prominent quantum gravity theorists themselves—including some of my close friends and collaborators—persist in talking about the simulated SYK wormhole as “actually being” a wormhole. What are they thinking?

Daniel Harlow explained the thinking to me as follows (he stresses that he’s explaining it, not necessarily endorsing it). If you had two entangled quantum computers, one on Earth and the other in the Andromeda galaxy, and if they were both simulating SYK, and if Alice on Earth and Bob in Andromeda both uploaded their own brains into their respective quantum simulations, then it seems possible that the simulated Alice and Bob could have the experience of jumping into a wormhole and meeting each other in the middle. Granted, they couldn’t get a message back out from the wormhole, at least not without “going the long way,” which could happen only at the speed of light—so only simulated-Alice and simulated-Bob themselves could ever test this prediction. Nevertheless, if true, I suppose some would treat it as grounds for regarding a quantum simulation of SYK as “more real” or “more wormholey” than a classical simulation.

Of course, this scenario depends on strong assumptions not merely about quantum gravity, but also about the metaphysics of consciousness! And I’d still prefer to call it a simulated wormhole for simulated people.

For completeness, here’s Harlow’s passage from the NYT article:

Daniel Harlow, a physicist at M.I.T. who was not involved in the experiment, noted that the experiment was based on a model of quantum gravity that was so simple, and unrealistic, that it could just as well have been studied using a pencil and paper.

“So I’d say that this doesn’t teach us anything about quantum gravity that we didn’t already know,” Dr. Harlow wrote in an email. “On the other hand, I think it is exciting as a technical achievement, because if we can’t even do this (and until now we couldn’t), then simulating more interesting quantum gravity theories would CERTAINLY be off the table.” Developing computers big enough to do so might take 10 or 15 years, he added.

---

Alright, let’s move on to the claim that quantum supremacy has been refuted. What Aharonov et al. actually show in their new work, building on earlier work by Gao and Duan, is that Random Circuit Sampling, with a constant rate of noise per gate and no error-correction, can’t provide a scalable approach to quantum supremacy. Or more precisely: as the number of qubits n goes to infinity, and assuming you’re in the “anti-concentration regime” (which in practice probably means: the depth of your quantum circuit is at least ~log(n)), there’s a classical algorithm to approximately sample the quantum circuit’s output distribution in poly(n) time (albeit, not yet a practical algorithm).

Here’s what’s crucial to understand: this is 100% consistent with what those of us working on quantum supremacy had assumed since at least 2016! We knew that if you tried to scale Random Circuit Sampling to 200 or 500 or 1000 qubits, while you also increased the circuit depth proportionately, the signal-to-noise ratio would become exponentially small, meaning that your quantum speedup would disappear. That’s why, from the very beginning, we targeted the “practical” regime of 50-100 qubits: a regime where

1. you can still see explicitly that you’re exploiting a 2⁵⁰– or 2¹⁰⁰-dimensional Hilbert space for computational advantage, thereby confirming one of the main predictions of quantum computing theory, but
2. you also have a signal that (as it turned out) is large enough to see with heroic effort.

To their credit, Aharonov et al. explain all this perfectly clearly in their abstract and introduction. I’m just worried that others aren’t reading their paper as carefully as they should be!

So then, what’s the new advance in the Aharonov et al. paper? Well, there had been some hope that circuit depth ~log(n) might be a sweet spot, where an exponential quantum speedup might both exist and survive constant noise, even in the asymptotic limit of n→∞ qubits. Nothing in Google’s or USTC’s actual Random Circuit Sampling experiments depended on that hope, but it would’ve been nice if it were true. What Aharonov et al. have now done is to kill that hope, using powerful techniques involving summing over Feynman paths in the Pauli basis.

Stepping back, what is the current status of quantum supremacy based on Random Circuit Sampling? I would say it’s still standing, but more precariously than I’d like—underscoring the need for new and better quantum supremacy experiments. In more detail, Pan, Chen, and Zhang have shown how to simulate Google’s 53-qubit Sycamore chip classically, using what I estimated to be 100-1000X the electricity cost of running the quantum computer itself (including the dilution refrigerator!). Approaching from the problem from a different angle, Gao et al. have given a polynomial-time classical algorithm for spoofing Google’s Linear Cross-Entropy Benchmark (LXEB)—but their algorithm can currently achieve only about 10% of the excess in LXEB that Google’s experiment found.

So, though it’s been under sustained attack from multiple directions these past few years, I’d say that the flag of quantum supremacy yet waves. The Extended Church-Turing Thesis is still on thin ice. The wormhole is still open. Wait … no … that’s not what I meant to write…

---

Note: With this post, as with future science posts, all off-topic comments will be ruthlessly left in moderation. Yes, even if the comments “create their own reality” full of anger and disappointment that I talked about what I talked about, instead of what the commenter wanted me to talk about. Even if merely refuting the comments would require me to give in and talk about their preferred topics after all. Please stop. This is a wormholes-‘n-supremacy post.

Happy Thanksgiving to my American readers! While I enjoy a family holiday-week vacation in exotic Dallas—and yes, I will follow up on my old JFK post by visiting Dealey Plaza—please enjoy the following Thanksgiving victuals:

I recently recorded a 3-hour (!) YouTube video with Timothy Nguyen, host of the Cartesian Cafe. Our episode is entitled Quantum Computing: Dismantling the Hype. In it, I teach a sort of extremely compressed version of my undergraduate Intro to Quantum Information Science course, unburdening myself about whatever Tim prompts me to explain: the basic rules of quantum information, quantum circuits, the quantum black-box model, the Deutsch-Jozsa algorithm, BQP and its relationship to classical complexity classes, and sampling-based quantum supremacy experiments. This is a lot more technical than an average podcast, a lot less technical than an actual course, and hopefully just right for some nonempty subset of readers.

Outside of his podcasting career, some of you might recognize Nguyen as the coauthor, with Theo Polya, of a rebuttal of “Geometric Unity.” This latter is the proposal by the financier, podcaster, and leading “Intellectual Dark Web” figure Eric Weinstein for a unified theory of particle physics. Now, I slightly know Weinstein, and have even found him fascinating, eloquent, and correct about various issues. So, in an addendum to the main video, Nguyen chats with me about his experience critiquing Weinstein’s theory, and also about something where my knowledge is far greater: namely, my 2002 rebuttal of some of the central claims in Stephen Wolfram’s A New Kind of Science, and whether there are any updates to that story twenty years later.

Enjoy!

These days, I often need to remind myself that, as an undergrad, grad student, postdoc, or professor, I’ve now been doing quantum computing research for a quarter-century—i.e., well over half of the subject’s existence. As a direct result, when I feel completely jaded about a new development in QC, it might actually be exciting. When I feel moderately excited, it might actually be the most exciting thing for years.

With that in mind:

---

(1) Last week National Public Radio’s Marketplace interviewed me, John Martinis, and others about the current state of quantum computing. While the piece wasn’t entirely hype-free, I’m pleased to report that my own views were represented accurately! To wit:

“There is a tsunami of hype about what quantum computers are going to revolutionize,” said Scott Aaronson, a professor of computer science at the University of Texas at Austin. “Quantum computing has turned into a word that venture capitalists or people seeking government funding will sprinkle on anything because it sounds good.”

Aaronson warned we can’t be certain that these computers will in fact revolutionize machine learning and finance and optimization problems.  “We can’t prove that there’s not a quantum algorithm that solves all these problems super fast, but we can’t even prove there’s not an algorithm for a conventional computer that does it,” he said. [In the recorded version, they replaced this by a simpler but also accurate thought: namely, that we can’t prove one way or the other whether there’s a useful quantum advantage for these tasks.]

---

(2) I don’t like to use this blog to toot my own research horn, but on Thursday my postdoc Jason Pollack and I released a paper, entitled Discrete Bulk Reconstruction. And to be honest, I’m pretty damned excited about it. It represents about 8 months of Jason—a cosmologist and string theorist who studied under Sean Carroll—helping me understand AdS/CFT in the language of the undergraduate CS curriculum, like min-cuts on undirected graphs, so that we could then look for polynomial-time algorithms to implement the holographic mapping from boundary quantum states to the spatial geometry in the bulk. We drew heavily on previous work in the same direction, especially the already-seminal 2015 holographic entropy cone paper by Ning Bao et al. But I’d like to think that, among other things, our work represents a new frontier in just how accessible AdS/CFT itself can be made to CS and discrete math types. Anyway, here’s the abstract if you’re interested:

According to the AdS/CFT correspondence, the geometries of certain spacetimes are fully determined by quantum states that live on their boundaries — indeed, by the von Neumann entropies of portions of those boundary states. This work investigates to what extent the geometries can be reconstructed from the entropies in polynomial time. Bouland, Fefferman, and Vazirani (2019) argued that the AdS/CFT map can be exponentially complex if one wants to reconstruct regions such as the interiors of black holes. Our main result provides a sort of converse: we show that, in the special case of a single 1D boundary, if the input data consists of a list of entropies of contiguous boundary regions, and if the entropies satisfy a single inequality called Strong Subadditivity, then we can construct a graph model for the bulk in linear time. Moreover, the bulk graph is planar, it has O(N²) vertices (the information-theoretic minimum), and it’s “universal,” with only the edge weights depending on the specific entropies in question. From a combinatorial perspective, our problem boils down to an “inverse” of the famous min-cut problem: rather than being given a graph and asked to find a min-cut, here we’re given the values of min-cuts separating various sets of vertices, and need to find a weighted undirected graph consistent with those values. Our solution to this problem relies on the notion of a “bulkless” graph, which might be of independent interest for AdS/CFT. We also make initial progress on the case of multiple 1D boundaries — where the boundaries could be connected via wormholes — including an upper bound of O(N⁴) vertices whenever a planar bulk graph exists (thus putting the problem into the complexity class NP).

---

(3) Anand Natarajan and Chinmay Nirkhe posted a preprint entitled A classical oracle separation between QMA and QCMA, which makes progress on a problem that’s been raised on this blog all the way back to its inception. A bit of context: QMA, Quantum Merlin-Arthur, captures what can be proven using a quantum state with poly(n) qubits as the proof, and a polynomial-time quantum algorithm as the verifier. QCMA, or Quantum Classical Merlin-Arthur, is the same as QMA except that now the proof has to be classical. A fundamental problem of quantum complexity theory, first raised by Aharonov and Naveh in 2002, is whether QMA=QCMA. In 2007, Greg Kuperberg and I introduced the concept of quantum oracle separation—that is, a unitary that can be applied in a black-box manner—in order to show that there’s a quantum oracle relative to which QCMA≠QMA. In 2015, Fefferman and Kimmel improved this, to show that there’s a “randomized in-place” oracle relative to which QCMA≠QMA. Natarajan and Nirkhe now remove the “in-place” part, meaning the only thing still “wrong” with their oracle is that it’s randomized. Derandomizing their construction would finally settle this 20-year-old open problem (except, of course, for the minor detail of whether QMA=QCMA in the “real,” unrelativized world!).

---

(4) Oh right, the Google group reports the use of their superconducting processor to simulate non-abelian anyons. Cool.

For anyone living under a rock with no access to nerd social media, Alain Aspect, John Clauser, and Anton Zeilinger have finally won the Nobel Prize in Physics, for their celebrated experiments that rubbed everyone’s faces in the reality of quantum entanglement (including Bell inequality violation and quantum teleportation). I don’t personally know Aspect or Clauser, but Zeilinger extremely graciously hosted me and my wife Dana when we visited Vienna in 2012, even bringing us to the symphony (he knows the director and has front-row seats), and somehow making me feel more cultured rather than less.

As usual, the recipe for winning the Nobel Prize in Physics is this:

(1) Do something where anyone who knows about it is like, “why haven’t they given the Nobel Prize in Physics for that yet?”

(2) Live long enough.

Huge congratulations to Aspect, Clauser, and Zeilinger!

---

Elham Kashefi, my quantum complexity theory colleague and treasured friend for more than 20 years, brought to my attention a Statement of Solidarity with Students in Iran from the International Academic Community. Of course I was happy to sign the statement, just like I was back in 2009 when brave Iranian students similarly risked their lives and freedom for women’s rights and other Enlightenment values against the theocracy. I urge you to sign the statement as well. If enough Shtetl-Optimized readers disapprove of their brutal repression, surely the mullahs will reconsider! More seriously though: if any readers can recommend a charity that’s actually making a difference in helping Iranians participate in the modern world, I’d be happy to do another of my matching donation drives.

Here they are [PDF].

They’re 155 pages of awesome—for a certain extremely specific definition of “awesome”—which I’m hereby offering to the world free of charge (for noncommercial use only, of course). They cover material that I taught, for the first time, in my Introduction to Quantum Information Science II undergrad course at UT Austin in Spring 2022.

The new notes pick up exactly where my older QIS I lecture notes left off, and they presuppose familiarity with the QIS I material. So, if you’re just beginning your quantum information journey, then please start with my QIS I notes, which presuppose only linear algebra and a bit of classical algorithms (e.g., recurrence relations and big-O notation), and which self-containedly explain all the rules of QM, moving on to (e.g.) quantum circuits, density matrices, entanglement entropy, Wiesner’s quantum money, QKD, quantum teleportation, the Bell inequality, interpretations of QM, the Shor 9-qubit code, and the algorithms of Deutsch-Jozsa, Bernstein-Vazirani, Simon, Shor, and Grover. Master all that, and you’ll be close to the quantum information research frontier of circa 1996.

My new QIS II notes cover a bunch of topics, but the main theme is “perspectives on quantum computing that go beyond the bare quantum circuit model, and that became increasingly central to the field from the late 1990s onwards.” Thus, it covers:

• Hamiltonians, ground states, the adiabatic algorithm, and the universality of adiabatic QC
• The stabilizer formalism, the 1996 Gottesman-Knill Theorem on efficient classical simulation of stabilizer QC, my and Gottesman’s 2004 elaborations, boosting up to universality via “magic states,” transversal codes, and the influential 2016 concept of stabilizer rank
• Bosons and fermions: the formalism of Fock space and of creation and annihilation operators, connection to the permanents and determinants of matrices, efficient classical simulation of free fermionic systems (Valiant’s 2002 “matchcircuits”), the 2001 Knill-Laflamme-Milburn (KLM) theorem on universal optical QC, BosonSampling and its computational complexity, and the current experimental status of BosonSampling
• Cluster states, Raussendorf and Briegel’s 2000 measurement-based quantum computation (MBQC), and Gottesman and Chuang’s 1999 “gate teleportation” trick
• Matrix product states, and Vidal’s 2003 efficient classical simulation of “slightly entangled” quantum computations

Extra bonus topics include:

• The 2007 Broadbent-Fitzsimons-Kashefi (BFK) protocol for blind and authenticated QC; brief discussion of later developments including Reichardt-Unger-Vazirani 2012 and Mahadev 2018
• Basic protocols for quantum state tomography
• My 2007 work on PAC-learnability of quantum states
• The “dessert course”: the black hole information problem, and the Harlow-Hayden argument on the computational hardness of decoding Hawking radiation

Master all this, and hopefully you’ll have the conceptual vocabulary to understand a large fraction of what people in quantum computing and information care about today, how they now think about building scalable QCs, and what they post to the quant-ph arXiv.

Note that my QIS II course is complementary to my graduate course on quantum complexity theory, for which the lecture notes are here. There’s very little overlap between the two (and even less overlap between QIS II and Quantum Computing Since Democritus).

The biggest, most important topic related to the QIS II theme that I didn’t cover was topological quantum computing. I’d wanted to, but it quickly became clear that topological QC begs for a whole course of its own, and that I had neither the time nor the expertise to do it justice. In retrospect, I do wish I’d at least covered the Kitaev surface code.

Crucially, these lecture notes don’t represent my effort alone. I worked from draft scribe notes prepared by the QIS II students, who did a far better job than I had any right to expect (including creating the beautiful figures). My wonderful course TA and PhD student Daniel Liang, along with students Ethan Tan, Samuel Ziegelbein, and Steven Han, then assembled everything, fixed numerous errors, and compiled the bibliography. I’m grateful to all of them. At the last minute, we had a LaTeX issue that none of us knew how to fix—but, in response to a plea, Shtetl-Optimized reader Pablo Cingolani generously volunteered to help, completed the work by the very next day (I’d imagined it taking a month!), and earned a fruit basket from me in gratitude.

Anyway, let me know of any mistakes you find! We’ll try to fix them.

Several people have asked me to comment about a Financial Times opinion piece entitled The Quantum Computing Bubble (subtitle: “The industry has yet to demonstrate any real utility, despite the fanfare, billions of VC dollars and three Spacs”) (archive link). The piece is purely deflationary—not a positive word in it—though it never goes so far as to suggest that QC is blocked by any Gil-Kalai-like fundamental principle, nor does it even evince curiosity about that question.

As it happens, the author, physicist Nikita Gourianov, had emailed me a few days ago with some nice words about my own skeptical efforts on Shtetl-Optimized, and a request for comment on his article. So, as a way to get back into blogging after a 2-week hiatus, I figured I’d share my respoinse.

---

Hi Nikita,

Thanks for the kind words about my blog, and for your piece, which I just read.  There’s a great deal of truth in what you write, but I also take issue with a few points.  You say:

A convincing strategy for overcoming these errors has not yet been demonstrated, making it unclear as to when — if ever — it will become possible to build a large-scale, fault-tolerant quantum computer.

In one sense this is tautologically true — the only fully convincing and clear demonstration that something is possible is to do it, as with the Wright brothers or the Trinity nuclear test.  In other sense, though, we’ve known the “strategy” since the 1990s.  It’s just that the fault-tolerance theorem called for gate fidelities 5-6 orders of magnitude better than anything achievable at the time.  In the 25 years since, about 3 of those orders of magnitude have been achieved, so it doesn’t take any great imagination to foresee that the remainder could be as well.  A layperson reading your piece might not understand this.

As for applications, my position has always been that if there were zero applications, it would still be at least as scientifically important to try to build QCs as it was to build the LHC, LIGO, or the James Webb telescope.  If there are real applications, such as simulating chemical dynamics, or certifiable randomness — and there very well might be — then those are icing on the cake.  This, of course, radically differs from the vision that now gets presented to investors and the press (hence all the railing on my blog!), but it also differs from what a reader of your piece would take away.

Anyway, thanks again for sharing!

Best,
Scott

Update: We’re now finalizing the lecture notes—basically, a textbook—for the brand-new Quantum Information Science II course that I taught this past spring! The notes will be freely shared on this blog. But the bibliographies for the various lectures need to be merged, and we don’t know how. Would any TeXpert like to help us, in exchange for a generous acknowledgment? A reader named Pablo Cingolani has now graciously taken care of this. Thanks so much, Pablo!

---

I returned last week from the NSF Workshop on Quantum Advantage and Next Steps at the University of Chicago. Thanks so much to Chicago CS professor (and my former summer student) Bill Fefferman and the other organizers for making this workshop a reality. Given how vividly I remember the situation 12 years ago, when the idea of sampling-based quantum supremacy was the weird obsession of me and a few others, it was particularly special to attend a workshop on the topic with well over a hundred participants, some in-person and some on Zoom, delayed by covid but still excited by the dramatic experimental progress of the past few years.

Of course there’s a lot still to do. Many of the talks drew an exclamation point on something I’ve been saying for the past couple years: that there’s an urgent need for better quantum supremacy experiments, which will require both theoretical and engineering advances. The experiments by Google and USTC and now Xanadu represent a big step forward for the field, but since they started being done, the classical spoofing attacks have also steadily improved, to the point that whether “quantum computational supremacy” still exists depends on exactly how you define it.

Briefly: if you measure by total operations, energy use, or CO2 footprint, then probably yes, quantum supremacy remains. But if you measure by number of seconds, then it doesn’t remain, not if you’re willing to shell out for enough cores on AWS or your favorite supercomputer. And even the quantum supremacy that does remain might eventually fall to, e.g., further improvements of the algorithm due to Gao et al. For more details, see, e.g., the now-published work of Pan, Chen, and Zhang, or this good popular summary by Adrian Cho for Science.

If the experimentalists care enough, they could easily regain the quantum lead, at least for a couple more years, by (say) repeating random circuit sampling with 72 qubits rather than 53-60, and hopefully circuit depth of 30-40 rather than just 20-25. But the point I made in my talk was that, as long as we remain in the paradigm of sampling experiments that take ~2ⁿ time to verify classically and also ~2ⁿ time to spoof classically (where n is the number of qubits), all quantum speedups that we can achieve will be fragile and contingent, however interesting and impressive. As soon as we go way beyond what classical computers can keep up with, we’ve also gone way beyond where classical computers can check what was done.

I argued that the only solution to this problem is to design new quantum supremacy experiments: ones where some secret has been inserted in the quantum circuit that lets a classical computer efficiently verify the results. The fundamental problem is that we need three properties—

1. implementability on near-term quantum computers,
2. efficient classical verifiability, and
3. confidence that quantum computers have a theoretical asymptotic advantage,

and right now we only know how to get any two out of the three. We can get 1 and 2 with QAOA and various other heuristic quantum algorithms, 1 and 3 with BosonSampling and Random Circuit Sampling, or 2 and 3 with Shor’s algorithm or Yamakawa-Zhandry or recent interactive protocols. To get all three, there are three obvious approaches:

1. Start with the heuristic algorithms and find a real advantage from them,
2. Start with BosonSampling or Random Circuit Sampling or the like and figure out how to make them efficiently verifiable classically, or
3. Start with protocols like Kahanamoku-Meyer et al.’s and figure out how to run them on near-term devices.

At the Chicago workshop, I’d say that the most popular approach speakers talked about was 3, although my own focus was more on 2.

Anyway, until a “best-of-all-worlds” quantum supremacy experiment is discovered, there’s plenty to do in the meantime: for example, better understand the classical hardness of spoofing Random Circuit Sampling with a constant or logarithmic number of gate layers. Understand the best that classical algorithms can do to spoof the Linear Cross-Entropy Benchmark for BosonSampling, and build on Grier et al. to understand the power of BosonSampling with a linear number of modes.

I’ll be flying back with my family to Austin today after seven weeks at the Jersey shore, but I’ll try to field any questions about the state of quantum supremacy in general or this workshop in particular in the comments!