Archive for the ‘Quantum’ Category

Two new talks and an interview

Thursday, December 2nd, 2021
  1. A talk to UT Austin’s undergraduate math club (handwritten PDF notes) about Hao Huang’s proof of the Sensitivity Conjecture, and its implications for quantum query complexity and more. I’m still not satisfied that I’ve presented Huang’s beautiful proof as clearly and self-containedly as I possibly can, which probably just means I need to lecture on it a few more times.
  2. A Zoom talk at the QPQIS conference in Beijing (PowerPoint slides), setting out my most recent thoughts about Google’s and USTC’s quantum supremacy experiments and the continuing efforts to spoof them classically.
  3. An interview with me in Communications of the ACM, mostly about BosonSampling and the quantum lower bound for the collision problem.

Enjoy y’all!

The Acrobatics of BQP

Friday, November 19th, 2021

Just in case anyone is depressed this afternoon and needs something to cheer them up, students William Kretschmer, DeVon Ingram, and I have finally put out a new paper:

The Acrobatics of BQP

Abstract: We show that, in the black-box setting, the behavior of quantum polynomial-time (BQP) can be remarkably decoupled from that of classical complexity classes like NP. Specifically:

– There exists an oracle relative to which NPBQP⊄BQPPH, resolving a 2005 problem of Fortnow. Interpreted another way, we show that AC0 circuits cannot perform useful homomorphic encryption on instances of the Forrelation problem. As a corollary, there exists an oracle relative to which P=NP but BQP≠QCMA.

– Conversely, there exists an oracle relative to which BQPNP⊄PHBQP.

– Relative to a random oracle, PP=PostBQP is not contained in the “QMA hierarchy” QMAQMA^QMA^…, and more generally PP⊄(MIP*)(MIP*)^(MIP*)^… (!), despite the fact that MIP*=RE in the unrelativized world. This result shows that there is no black-box quantum analogue of Stockmeyer’s approximate counting algorithm.

– Relative to a random oracle, Σk+1⊄BQPΣ_k for every k.

– There exists an oracle relative to which BQP=P#P and yet PH is infinite. (By contrast, if NP⊆BPP, then PH collapses relative to all oracles.)

– There exists an oracle relative to which P=NP≠BQP=P#P.

To achieve these results, we build on the 2018 achievement by Raz and Tal of an oracle relative to which BQP⊄PH, and associated results about the Forrelation problem. We also introduce new tools that might be of independent interest. These include a “quantum-aware” version of the random restriction method, a concentration theorem for the block sensitivity of AC0 circuits, and a (provable) analogue of the Aaronson-Ambainis Conjecture for sparse oracles.

Incidentally, particularly when I’ve worked on a project with students, I’m often tremendously excited and want to shout about it from the rooftops for the students’ sake … but then I also don’t want to use this blog to privilege my own papers “unfairly.” Can anyone suggest a principle that I should follow going forward?

Scott Aaronson, when reached for comment, said…

Tuesday, November 16th, 2021

About IBM’s new 127-qubit superconducting chip: As I told New Scientist, I look forward to seeing the actual details! As far as I could see, the marketing materials that IBM released yesterday take a lot of words to say absolutely nothing about what, to experts, is the single most important piece of information: namely, what are the gate fidelities? How deep of a quantum circuit can they apply? How have they benchmarked the chip? Right now, all I have to go on is a stats page for the new chip, which reports its average CNOT error as 0.9388—in other words, close to 1, or terrible! (But see also a tweet by James Wootton, which explains that such numbers are often highly misleading when a new chip is first rolled out.) Does anyone here have more information? Update (11/17): As of this morning, the average CNOT error has been updated to 2%. Thanks to multiple commenters for letting me know!

About the new simulation of Google’s 53-qubit Sycamore chip in 5 minutes on a Sunway supercomputer (see also here): This is an exciting step forward on the classical validation of quantum supremacy experiments, and—ironically, what currently amounts to almost the same thing—on the classical spoofing of those experiments. Congratulations to the team in China that achieved this! But there are two crucial things to understand. First, “5 minutes” refers to the time needed to calculate a single amplitude (or perhaps, several correlated amplitudes) using tensor network contraction. It doesn’t refer to the time needed to generate millions of independent noisy samples, which is what Google’s Sycamore chip does in 3 minutes. For the latter task, more like a week still seems to be needed on the supercomputer. (I’m grateful to Chu Guo, a coauthor of the new work who spoke in UT Austin’s weekly quantum Zoom meeting, for clarifying this point.) Second, the Sunway supercomputer has parallel processing power equivalent to approximately ten million of your laptop. Thus, even if we agreed that Google no longer had quantum supremacy as measured by time, it would still have quantum supremacy as measured by carbon footprint! (And this despite the fact that the quantum computer itself requires a noisy, closet-sized dilution fridge.) Even so, for me the new work underscores the point that quantum supremacy is not yet a done deal. Over the next few years, I hope that Google and USTC, as well as any new entrants to this race (IBM? IonQ? Harvard? Rigetti?), will push forward with more qubits and, even more importantly, better gate fidelities leading to higher Linear Cross-Entropy scores. Meanwhile, we theorists should try to do our part by inventing new and better protocols with which to demonstrate near-term quantum supremacy—especially protocols for which the classical verification is easier.

About the new anti-woke University of Austin (UATX): In general, I’m extremely happy for people to experiment with new and different institutions, and of course I’m happy for more intellectual activity in my adopted city of Austin. And, as Shtetl-Optimized readers will know, I’m probably more sympathetic than most to the reality of the problem that UATX is trying to solve—living, as we do, in an era when one academic after another has been cancelled for ideas that a mere decade ago would’ve been considered unexceptional, moderate, center-left. Having said all that, I wish I could feel more optimistic about UATX’s prospects. I found its website heavy on free-speech rhetoric but frustratingly light on what the new university is actually going to do: what courses it will offer, who will teach them, where the campus will be, etc. etc. Arguably this is all excusable for a university still in ramp-up mode, but had I been in their shoes, I might have held off on the public launch until I had at least some sample content to offer. Certainly, the fact that Steven Pinker has quit UATX’s advisory board is a discouraging sign. If UATX asks me to get involved—to lecture there, to give them advice about their CS program, etc.—I’ll consider it as I would any other request. So far, though, they haven’t.

About the Association for Mathematical Research: Last month, some colleagues invited me to join a brand-new society called the Association for Mathematical Research. Many of the other founders (Joel Hass, Abigail Thompson, Colin Adams, Richard Borcherds, Jeff Cheeger, Pavel Etingof, Tom Hales, Jeff Lagarias, Mark Lackenby, Cliff Taubes, …) were brilliant mathematicians who I admired, they seemed like they could use a bit of theoretical computer science representation, there was no time commitment, maybe they’d eventually do something good, so I figured why not? Alas, to say that AMR has proved unpopular on Twitter would be an understatement: it’s received the same contemptuous reception that UATX has. The argument seems to be: starting a new mathematical society, even an avowedly diverse and apolitical one, is really just an implicit claim that the existing societies, like the Mathematical Association of America (MAA) and the American Mathematical Society (AMS), have been co-opted by woke true-believers. But that’s paranoid and insane! I mean, it’s not as if an AMS blog has called for the mass resignation of white male mathematicians to make room for the marginalized, or the boycott of Israeli universities, or the abolition of the criminal justice system (what to do about Kyle Rittenhouse though?). Still, even though claims of that sort of co-option are obviously far-out, rabid fantasies, yeah, I did decide to give a new organization the benefit of the doubt. AMR might well fail or languish in obscurity, just like UATX might. On the other hand, the barriers to making a positive difference for the intellectual world, the world I love, the world under constant threat from the self-certain ideologues of every side, do strike me as orders of magnitude smaller for a new professional society than they do for a new university.

Q2B 2021

Monday, November 1st, 2021

This is a quick post to let people know that the 2021 Q2B (Quantum 2 Business) conference will be this December 7-9 at the Santa Clara Convention Center. (Full disclosure: Q2B is hosted by QC Ware, Inc., to which I’m the scientific adviser.) Barring a dramatic rise in cases or the like, I’m planning to attend to do my Ask-Me-Anything session, in what’s become an annual tradition. Notably, this will be my first in-person conference, and in fact my first professional travel of any kind, since before covid shut down the US in late March 2020. I hope to see many of you there! And if you won’t be at Q2B, but you’ll be in the Bay Area and would like to meet otherwise, let me know and we’ll try to work something out.

Gaussian BosonSampling, higher-order correlations, and spoofing: An update

Sunday, October 10th, 2021

In my last post, I wrote (among other things) about an ongoing scientific debate between the group of Chaoyang Lu at USTC in China, which over the past year has been doing experiments that seek to demonstrate quantum supremacy via Gaussian BosonSampling; and the group of Sergio Boixo at Google, which had a recent paper on a polynomial-time classical algorithm to sample approximately from the same distributions.  I reported the facts as I understood them at the time.  Since then, though, a long call with the Google team gave me a new and different understanding, and I feel duty-bound to share that here.

A week ago, I considered it obvious that if, using a classical spoofer, you could beat the USTC experiment on a metric like total variation distance from the ideal distribution, then you would’ve completely destroyed USTC’s claim of quantum supremacy.  The reason I believed that, in turn, is a proposition that I hadn’t given a name but needs one, so let me call it Hypothesis H:

The only way a classical algorithm to spoof BosonSampling can possibly do well in total variation distance, is by correctly reproducing the high-order correlations (correlations among the occupation numbers of large numbers of modes) — because that’s where the complexity of BosonSampling lies (if it lies anywhere).

Hypothesis H had important downstream consequences.  Google’s algorithm, by the Google team’s own admission, does not reproduce the high-order correlations.  Furthermore, because of limitations on both samples and classical computation time, Google’s paper calculates the total variation distance from the ideal distribution only on the marginal distribution on roughly 14 out of 144 modes.  On that marginal distribution, Google’s algorithm does do better than the experiment in total variation distance.  Google presents a claimed extrapolation to the full 144 modes, but eyeballing the graphs, it was far from clear to me what would happen: like, maybe the spoofing algorithm would continue to win, but maybe the experiment would turn around and win; who knows?

Chaoyang, meanwhile, made a clear prediction that the experiment would turn around and win, because of

  1. the experiment’s success in reproducing the high-order correlations,
  2. the admitted failure of Google’s algorithm in reproducing the high-order correlations, and
  3. the seeming impossibility of doing well on BosonSampling without reproducing the high-order correlations (Hypothesis H).

Given everything my experience told me about the central importance of high-order correlations for BosonSampling, I was inclined to agree with Chaoyang.

Now for the kicker: it seems that Hypothesis H is false.  A classical spoofer could beat a BosonSampling experiment on total variation distance from the ideal distribution, without even bothering to reproduce the high-order correlations correctly.

This is true because of a combination of two facts about the existing noisy BosonSampling experiments.  The first fact is that the contribution from the order-k correlations falls off like 1/exp(k).  The second fact is that, due to calibration errors and the like, the experiments already show significant deviations from the ideal distribution on the order-1 and order-2 correlations.

Put these facts together and what do you find?  Well, suppose your classical spoofing algorithm takes care to get the low-order contributions to the distribution exactly right.  Just for that reason alone, it could already win over a noisy BosonSampling experiment, as judged by benchmarks like total variation distance from the ideal distribution, or for that matter linear cross-entropy.  Yes, the experiment will beat the classical simulation on the higher-order correlations.  But because those higher-order correlations are exponentially attenuated anyway, they won’t be enough to make up the difference.  The experiment’s lack of perfection on the low-order correlations will swamp everything else.

Granted, I still don’t know for sure that this is what happens — that depends on whether I believe Sergio or Chaoyang about the extrapolation of the variation distance to the full 144 modes (my own eyeballs having failed to render a verdict!).  But I now see that it’s logically possible, maybe even plausible.

So, let’s imagine for the sake of argument that Google’s simulation wins on variation distance, even though the experiment wins on the high-order correlations.  In that case, what would be our verdict: would USTC have achieved quantum supremacy via BosonSampling, or not?

It’s clear what each side could say.

Google could say: by a metric that Scott Aaronson, the coinventor of BosonSampling, thought was perfectly adequate as late as last week — namely, total variation distance from the ideal distribution — we won.  We achieved lower variation distance than USTC’s experiment, and we did it using a fast classical algorithm.  End of discussion.  No moving the goalposts after the fact.

Google could even add: BosonSampling is a sampling task; it’s right there in the name!  The only purpose of any benchmark — whether Linear XEB or high-order correlation — is to give evidence about whether you are or aren’t sampling from a distribution close to the ideal one.  But that means that, if you accept that we are doing the latter better than the experiment, then there’s nothing more to argue about.

USTC could respond: even if Scott Aaronson is the coinventor of BosonSampling, he’s extremely far from an infallible oracle.  In the case at hand, his lack of appreciation for the sources of error in realistic experiments caused him to fixate inappropriately on variation distance as the success criterion.  If you want to see the quantum advantage in our system, you have to deliberately subtract off the low-order correlations and look at the high-order correlations.

USTC could add: from the very beginning, the whole point of quantum supremacy experiments was to demonstrate a clear speedup on some benchmark — we never particularly cared which one!  That horse is out of the barn as soon as we’re talking about quantum supremacy at all — something the Google group, which itself reported the first quantum supremacy experiment in Fall 2019, again for a completely artificial benchmark — knows as well as anyone else.  (The Google team even has experience with adjusting benchmarks: when, for example, Pan and Zhang pointed out that Linear XEB as originally specified is pretty easy to spoof for random 2D circuits, the most cogent rejoinder was: OK, fine then, add an extra check that the returned samples are sufficiently different from one another, which kills Pan and Zhang’s spoofing strategy.) In that case, then, why isn’t a benchmark tailored to the high-order correlations as good as variation distance or linear cross-entropy or any other benchmark?

Both positions are reasonable and have merit — though I confess to somewhat greater sympathy for the one that appeals to my doofosity rather than my supposed infallibility!

OK, but suppose, again for the sake of argument, that we accepted the second position, and we said that USTC gets to declare quantum supremacy as long as its experiment does better than any known classical simulation at reproducing the high-order correlations.  We’d still face the question: does the USTC experiment, in fact, do better on that metric?  It would be awkward if, having won the right to change the rules in its favor, USTC still lost even under the new rules.

Sergio tells me that USTC directly reported experimental data only for up to order-7 correlations, and at least individually, the order-7 correlations are easy to reproduce on a laptop (although sampling in a way that reproduces the order-7 correlations might still be hard—a point that Chaoyang confirms, and where further research would be great). OK, but USTC also reported that their experiment seems to reproduce up to order-19 correlations. And order-19 correlations, the Google team agrees, are hard to sample consistently with on a classical computer by any currently known algorithm.

So then, why don’t we have direct data for the order-19 correlations?  The trouble is simply that it would’ve taken USTC an astronomical amount of computation time.  So instead, they relied on a statistical extrapolation from the observed strength of the lower-order correlations — there we go again with the extrapolations!  Of course, if we’re going to let Google rest its case on an extrapolation, then maybe it’s only sporting to let USTC do the same.

You might wonder: why didn’t we have to worry about any of this stuff with the other path to quantum supremacy, the one via random circuit sampling with superconducting qubits?  The reason is that, with random circuit sampling, all the correlations except the highest-order ones are completely trivial — or, to say it another way, the reduced state of any small number of output qubits is exponentially close to the maximally mixed state.  This is a real difference between BosonSampling and random circuit sampling—and even 5-6 years ago, we knew that this represented an advantage for random circuit sampling, although I now have a deeper appreciation for just how great of an advantage it is.  For it means that, with random circuit sampling, it’s easier to place a “sword in the stone”: to say, for example, here is the Linear XEB score achieved by the trivial classical algorithm that outputs random bits, and lo, our experiment achieves a higher score, and lo, we challenge anyone to invent a fast classical spoofing method that achieves a similarly high score.

With BosonSampling, by contrast, we have various metrics with which to judge performance, but so far, for none of those metrics do we have a plausible hypothesis that says “here’s the best that any polynomial-time classical algorithm can possibly hope to do, and it’s completely plausible that even a noisy current or planned BosonSampling experiment can do better than that.”

In the end, then, I come back to the exact same three goals I would’ve recommended a week ago for the future of quantum supremacy experiments, but with all of them now even more acutely important than before:

  1. Experimentally, to increase the fidelity of the devices (with BosonSampling, for example, to observe a larger contribution from the high-order correlations) — a much more urgent goal, from the standpoint of evading classical spoofing algorithms, than further increasing the dimensionality of the Hilbert space.
  2. Theoretically, to design better ways to verify the results of sampling-based quantum supremacy experiments classically — ideally, even ways that could be applied via polynomial-time tests.
  3. For Gaussian BosonSampling in particular, to get a better understanding of the plausible limits of classical spoofing algorithms, and exactly how good a noisy device needs to be before it exceeds those limits.

Thanks so much to Sergio Boixo and Ben Villalonga for the conversation, and to Chaoyang Lu and Jelmer Renema for comments on this post. Needless to say, any remaining errors are my own.

The Physics Nobel, Gaussian BosonSampling, and Dorian Abbot

Tuesday, October 5th, 2021

1. Huge congratulations to the winners of this year’s Nobel Prize in Physics: Syukuro Manabe and Klaus Hasselmann for climate modelling, and separately, Giorgio Parisi for statistical physics. While I don’t know the others, I had the great honor to get to know Parisi three years ago, when he was chair of the committee that awarded me the Tomassoni-Chisesi Prize in Physics, and when I visited Parisi’s department at Sapienza University of Rome to give the prize lecture and collect the award. I remember Parisi’s kindness, a lot of good food, and a lot of discussion of the interplay between theoretical computer science and physics. Note that, while much of Parisi’s work is beyond my competence to comment on, in computer science he’s very well-known for applying statistical physics methods to the analysis of survey propagation—an algorithm that revolutionized the study of random 3SAT when it was introduced two decades ago.


2. Two weeks ago, a group at Google put out a paper with a new efficient classical algorithm to simulate the recent Gaussian BosonSampling experiments from USTC in China. They argued that this algorithm called into question USTC’s claim of BosonSampling-based quantum supremacy. Since then, I’ve been in contact with Sergio Boixo from Google, Chaoyang Lu from USTC, and Jelmer Renema, a Dutch BosonSampling expert and friend of the blog, to try to get to the bottom of this. Very briefly, the situation seems to be that Google’s new algorithm outperforms the USTC experiment on one particular metric: namely, total variation distance from the ideal marginal distribution, if (crucially) you look at only a subset of the optical modes, say 14 modes out of 144 total. Meanwhile, though, if you look at the kth-order correlations for large values of k, then the USTC experiment continues to win. With the experiment, the correlations fall off exponentially with k but still have a meaningful, detectable signal even for (say) k=19, whereas with Google’s spoofing algorithm, you choose the k that you want to spoof (say, 2 or 3), and then the correlations become nonsense for larger k.

Now, given that you were only ever supposed to see a quantum advantage from BosonSampling if you looked at the kth-order correlations for large values of k, and given that we already knew, from the work of Leonid Gurvits, that very small marginals in BosonSampling experiments would be easy to reproduce on a classical computer, my inclination is to say that USTC’s claim of BosonSampling-based quantum supremacy still stands. On the other hand, it’s true that, with BosonSampling especially, more so than with qubit-based random circuit sampling, we currently lack an adequate theoretical understanding of what the target should be. That is, which numerical metric should an experiment aim to maximize, and how well does it have to score on that metric before it’s plausibly outperforming any fast classical algorithm? One thing I feel confident about is that, whichever metric is chosen—Linear Cross-Entropy or whatever else—it needs to capture the kth-order correlations for large values of k. No metric that’s insensitive to those correlations is good enough.


3. Like many others, I was outraged and depressed that MIT uninvited Dorian Abbot (see also here), a geophysicist at the University of Chicago, who was slated to give the Carlson Lecture in the Department of Earth, Atmospheric, and Planetary Sciences about the atmospheres of extrasolar planets. The reason for the cancellation was that, totally unrelatedly to his scheduled lecture, Abbot had argued in Newsweek and elsewhere that Diversity, Equity, and Inclusion initiatives should aim for equality for opportunity rather than equality of outcomes, a Twitter-mob decided to go after him in retaliation, and they succeeded. It should go without saying that it’s perfectly reasonable to disagree with Abbot’s stance, to counterargue—if those very concepts haven’t gone the way of floppy disks. It should also go without saying that the MIT EAPS department chair is free to bow to social-media pressure, as he did, rather than standing on principle … just like I’m free to criticize him for it. To my mind, though, cancelling a scientific talk because of the speaker’s centrist (!) political views completely, 100% validates the right’s narrative about academia, that it’s become a fanatically intolerant echo chamber. To my fellow progressive academics, I beseech thee in the bowels of Bertrand Russell: why would you commit such an unforced error?

Yes, one can imagine views (e.g., open Nazism) so hateful that they might justify the cancellation of unrelated scientific lectures by people who hold those views, as many physicists after WWII refused to speak to Werner Heisenberg. But it seems obvious to me—as it would’ve been obvious to everyone else not long ago—that no matter where a reasonable person draws the line, Abbot’s views as he expressed them in Newsweek don’t come within a hundred miles of it. To be more explicit still: if Abbot’s views justify deplatforming him as a planetary scientist, then all my quantum computing and theoretical computer science lectures deserve to be cancelled too, for the many attempts I’ve made on this blog over the past 16 years to share my honest thoughts and life experiences, to write like a vulnerable human being rather than like a university press office. While I’m sure some sneerers gleefully embrace that implication, I ask everyone else to consider how deeply they believe in the idea of academic freedom at all—keeping in mind that such a commitment only ever gets tested when there’s a chance someone might denounce you for it.

Update: Princeton’s James Madison Program has volunteered to host Abbot’s Zoom talk in place of MIT. The talk is entitled “Climate and the Potential for Life on Other Planets.” Like probably hundreds of others who heard about this only because of the attempted cancellation, I plan to attend!

Unrelated Bonus Update: Here’s a neat YouTube video put together by the ACM about me as well as David Silver of AlphaGo and AlphaZero, on the occasion of our ACM Prizes in Computing.

“Is China Ahead in the Quantum Computing Race?”

Sunday, September 26th, 2021

Please enjoy an hourlong panel discussion of that question on YouTube, featuring yours truly, my former MIT colleague Will Oliver, and political scientist and China scholar Elsa Kania. If you’re worried that the title sounds too sensationalistic, I hope my fellow panelists and I will pleasantly surprise you with our relative sobriety! Thanks so much to QC Ware for arranging the panel (full disclosure: I’m QC Ware’s scientific adviser).

My ACM TechTalk on quantum supremadvantage

Wednesday, September 15th, 2021

This Erev Yom Kippur, I wish to repent for not putting enough quantum computing content on this blog. Of course, repentance is meaningless unless accompanied by genuine reform. That being the case, please enjoy the YouTube video of my ACM TechTalk from last week about quantum supremacy—although, as you’ll see if you watch the thing, I oscillate between quantum supremacy and other terms like “quantum advantage” and even “quantum supremadvantage.” This represents the first time ever that I got pushback about a talk before I’d delivered it for political reasons—the social-justice people, it turns out, are actually serious about wanting to ban the term “quantum supremacy”—but my desire to point out all the difficulties with their proposal competed with my desire not to let that issue overshadow my talk.

And there’s plenty to talk about! While regular Shtetl-Optimized readers will have already heard (or read) most of what I say, I make some new comments, including about the new paper from last week, the night before my talk (!), by the USTC group in China, where they report a quantum supremacy experiment based on random circuit sampling with a superconducting chip, this time with a record-setting 60 qubits and 24 layers of gates. On the other hand, I also stress how increasing the circuit fidelity has become a much more urgent issue than further increasing the number of qubits (or in the case of BosonSampling, the number of photons), if our goal is for these experiments to remain a couple steps ahead of classical spoofing algorithms.

Anyway, I hope you enjoy my lovingly handcrafted visuals. Over the course of this pandemic, I’ve become a full convert to writing out my talks with a stylus pen rather than PowerPointing them—not only is it faster for me, not only does it allow for continuous scrolling rather than arbitrary divisions into slides, but it enforces simplicity and concision in ways they should be enforced.

While there was only time for me to field a few questions at the end of the talk, I later supplied written answers to 52 questions (!!) that I hadn’t gotten to. If you have a question, please check to see if it’s already there, and otherwise ask away in the comments!

Thanks so much to Yan Timanovsky for inviting and organizing this talk, and to whurley for hosting it.

Open Problems Related to Quantum Query Complexity

Tuesday, September 14th, 2021

Way back in 2005, I posed Ten Semi-Grand Challenges for Quantum Computing Theory, on at least half of which I’d say there’s been dramatic progress in the 16 years since (most of the challenges were open-ended, so that it’s unclear when to count them as “solved”). I posed more open quantum complexity problems in 2010, and some classical complexity problems in 2011. In the latter cases, I’d say there’s been dramatic progress on about a third of the problems. I won’t go through the problems one by one, but feel free to ask in the comments about any that interest you.

Shall I push my luck as a problem-poser? Shall or shall not, I have.

My impetus, this time around, was a kind invitation by Travis Humble, the editor-in-chief of the new ACM Transactions on Quantum Computing, to contribute a perspective piece to that journal on the occasion of my ACM Prize. I agreed—but only on the condition that, rather than ponderously pontificate about the direction of the field, I could simply discuss a bunch of open problems that I wanted to see solved. The result is below. It’s coming soon to an arXiv near you, but Shtetl-Optimized readers get it first.

Open Problems Related to Quantum Query Complexity (11 pages, PDF)

by Scott Aaronson

Abstract: I offer a case that quantum query complexity still has loads of enticing and fundamental open problems—from relativized QMA versus QCMA and BQP versus IP, to time/space tradeoffs for collision and element distinctness, to polynomial degree versus quantum query complexity for partial functions, to the Unitary Synthesis Problem and more.

Some of the problems on my new hit-list are ones that I and others have flogged for years or even decades, but others, as far as I know, appear here for the first time. If your favorite quantum query complexity open problem, or a problem I’ve discussed in the past, is missing, that doesn’t mean that it’s been solved or is no longer interesting—it might mean I simply ran out of time or energy before I got to it.

Enjoy! And tell me what I missed or got wrong or has a trivial solution that I overlooked.

Stephen Wiesner (1942-2021)

Friday, August 13th, 2021

Photo credit: Lev Vaidman

These have not been an auspicious few weeks for Jewish-American-born theoretical physicists named Steve who made epochal contributions to human knowledge in the late 1960s, and who I had the privilege to get to know a bit when they were old.

This morning, my friend and colleague Or Sattath brought me the terrible news that Stephen Wiesner has passed away in Israel. [Because people have asked: I’ve now also heard directly from Wiesner’s daughter Sarah.]

Decades ago, Wiesner left academia, embraced Orthodox Judaism, moved from the US to Israel, and took up work there as a construction laborer—believing (or so he told me) that manual labor was good for the soul. In the late 1960s, however, Wiesner was still a graduate student in physics at Columbia University, when he wrote Conjugate Coding: arguably the foundational document of the entire field of quantum information science. Famously, this paper was so far ahead of its time that it was rejected over and over from journals, taking nearly 15 years to get published. (Fascinatingly, Gilles Brassard tells me that this isn’t true: it was rejected once, from IEEE Transactions on Information Theory, and then Wiesner simply shelved it.) When it finally appeared, in 1983, it was in SIGACT News—a venue that I know and love, where I’ve published too, but that’s more like the house newsletter for theoretical computer scientists than an academic journal.

But it didn’t matter. By the early 1980s, Wiesner’s ideas had been successfully communicated to Charlie Bennett and Gilles Brassard, who refashioned them into the first scheme for quantum key distribution—what we now call BB84. Even as Bennett and Brassard received scientific acclaim for the invention of quantum cryptography—including, a few years ago, the Wolf Prize (often considered second only to the Nobel Prize), at a ceremony in the Knesset in Jerusalem that I attended—the two B’s were always careful to acknowledge their massive intellectual debt to Steve Wiesner.


Let me explain what Wiesner does in the Conjugate Coding paper. As far as I know, this is the first paper ever to propose that quantum information—what Wiesner called “polarized light” or “spin-1/2 particles” but we now simply call qubits—works differently than classical bits, in ways that could actually be useful for achieving cryptographic tasks that are impossible in a classical world. What could enable these cryptographic applications, wrote Wiesner, is the fact that there’s no physical means for an attacker or eavesdropper to copy an unknown qubit, to produce a second qubit in the same quantum state. This observation—now called the No-Cloning Theorem—would only be named and published in 1982, but Wiesner treats it in his late-1960s manuscript as just obvious background.

Wiesner went further than these general ideas, though, to propose an explicit scheme for quantum money that would be physically impossible to counterfeit—a scheme that’s still of enormous interest half a century later (I teach it every year in my undergraduate course). In what we now call the Wiesner money scheme, a central bank prints “quantum bills,” each of which contains a classical serial number as well as a long string of qubits. Each qubit is prepared in one of four possible quantum states:

  • |0⟩,
  • |1⟩,
  • |+⟩ = (|0⟩+|1⟩)/√2, or
  • |-⟩ = (|0⟩-|1⟩)/√2.

The bank, in a central database, stores the serial number of every bill in circulation, as well as the preparation instructions for each of the bill’s qubits. If you want to verify a bill as genuine—this, as Wiesner knew, is the big drawback—you have to bring it back to the bank. The bank, using its secret knowledge of how each qubit was prepared, measures each qubit in the appropriate basis—the {|0⟩,|1⟩} basis for |0⟩ or |1⟩ qubits, the {|+⟩,|-⟩} basis for |+⟩ or |-⟩ qubits—and checks that it gets the expected outcomes. If even one qubit yields the wrong outcome, the bill is rejected as counterfeit.

Now consider the situation of a counterfeiter, who holds a quantum bill but lacks access to the bank’s secret database. When the counterfeiter tries to copy the bill, they won’t know the right basis in which to measure each qubit—and if they make the wrong choice, then it’s not only that they fail to make a copy; it’s that the measurement destroys even the original copy! For example, measuring a |+⟩ or |-⟩ qubit in the {|0⟩,|1⟩} basis will randomly collapse the qubit to either |0⟩ or |1⟩—so that, when the bank later measures the same qubit in the correct {|+⟩,|-⟩} basis, it will see the wrong outcome, and realize that the bill has been compromised, with 1/2 probability (with the probability increasing to nearly 1 as we repeat across hundreds or thousands of qubits).

Admittedly, the handwavy argument above, which Wiesner offered, is far from a security proof by cryptographers’ standards. In 2011, I pointed that out on StackExchange. My post, I’m happy to say, spurred Molina, Vidick, and Watrous to write a beautiful 2012 paper, where they rigorously proved for the first time that in Wiesner’s money scheme, no counterfeiter consistent with the laws of quantum mechanics can turn a single n-qubit bill into two bills that both pass the bank’s verification with success probability greater than (3/4)n (and this is tight). But the intuition was already clear enough to Wiesner in the 1960s.

In 2003—when I was already a PhD student in quantum information, but incredibly, had never heard of Stephen Wiesner or his role in founding my field—I rediscovered the idea of quantum states |ψ⟩ that you could store, measure, and feed into a quantum computer, but that would be usefully uncopyable. (My main interest was in whether you could create “unpiratable quantum software programs.”) Only in 2006, at the University of Waterloo, did Michele Mosca and his students make the connection for me to quantum money, Stephen Wiesner, and his Conjugate Coding paper, which I then read with amazement—along with a comparably amazing followup work by Bennett, Brassard, Breidbart, and Wiesner.

But it was clear that there was still a great deal to do. Besides unpiratable software, Wiesner and his collaborators had lacked the tools in the early 1980s seriously to tackle the problem of secure quantum money that anybody could verify, not only the bank that had created the money. I realized that, if such a thing was possible at all, then just like unpiratable software, it would require cryptographic hardness assumptions, a restriction to polynomial-time counterfeiters, and (hence) ideas from quantum computational complexity. The No-Cloning Theorem couldn’t do the job on its own.

That realization led to my 2009 paper Quantum Copy-Protection and Quantum Money, and from there, to the “modern renaissance” of Wiesner’s old idea of quantum money, with well over a hundred papers (e.g., my 2012 paper with Paul Christiano, Farhi et al.’s quantum state restoration paper, their quantum money from knots paper, Mark Zhandry’s 2017 quantum lightning paper, Dmitry Gavinsky’s improvement of Wiesner’s scheme wherein the money is verified by classical communication with the bank, Broduch et al.’s adaptive attack on Wiesner’s original scheme, my shadow tomography paper proving the necessity for the bank to keep a giant database in information-theoretic quantum money schemes like Wiesner’s, Daniel Kane’s strange scheme based on modular forms…). The purpose of many of these papers was either to break the quantum money schemes proposed in previous papers, or to patch the schemes that were previously broken.

After all this back-and-forth, spanning more than a decade, I’d say that Wiesner’s old idea of quantum money is now in good enough theoretical shape that the main obstacle to its practical realization is merely the “engineering difficulty”—namely, how to get the qubits in a bill, sitting in your pocket or whatever, to maintain their quantum coherence for more than a few nanoseconds! (Or possibly a few hours, if you’re willing to schlep a cryogenic freezer everywhere you go.) It’s precisely because quantum key distribution doesn’t have this storage problem—because there the qubits are simply sent across a channel and then immediately measured on arrival—that QKD is actually practical today, although the market for it has proven to be extremely limited so far.

In the meantime, while the world waits for the quantum error-correction that could keep qubits alive indefinitely, there’s Bitcoin. The latter perversely illustrates just how immense the demand for quantum money might someday be: the staggering lengths to which people will go, diverting the electricity to power whole nations into mining rigs, to get around our current inability to realize Wiesner’s elegant quantum-mechanical solution to the same problem. When I first learned about Bitcoin, shortly after its invention, it was in the context of: “here’s something I’d better bring up in my lectures on quantum money, in order to explain how much better WiesnerCoin could eventually be, when it’s the year 2200 or whatever and we all have quantum computers wired up by a quantum Internet!” It never occurred to me that I should forget about the year 2200, liquidate my life savings, and immediately buy up all the Bitcoin I could. [Added: I’ve since learned that Wiesner’s daughter Sarah is a professional in the Bitcoin space.]


Photo credit: Or Sattath

In his decades as a construction laborer, Wiesner had (as far as I know) no Internet presence; many of my colleagues didn’t even realize he was still alive. Even then, though, Wiesner never turned his back so far on his previous life, his academic life, that the quantum information faculty at Hebrew University in Jerusalem couldn’t entice him to participate in some seminars there. Those seminars are where I had the privilege to meet and talk to him several times over the last decade. He was thoughtful and kind, listening with interest as I told him how I and others were trying to take quantum money into the modern era by making it publicly verifiable.

I also vividly remember a conversation in 2013 where Steve shared his fears about the American physics establishment and military-industrial complex, and passionately urged me to

  1. quit academia and get a “real job,” and
  2. flee the US immediately and move my family to Israel, because of a wave of fascism and antisemitism that was about to sweep the US, just like with Germany in the 1930s.

I politely nodded along, pointing out that my Israeli wife and I had considered living in Israel but the job opportunities were better in US, silently wondering when Steve had gone completely off his rocker. Today, Steve’s urgent warning about an impending fascist takeover of the US seems … uh, slightly less crazy than in 2013? Maybe, just like with quantum money, Wiesner was simply too far ahead of his time to sound sane.

Wiesner also talked to me about his father, Jerome Wiesner, who was a legendary president of MIT—still spoken about in reverent tones when I taught there—as well as the chief science advisor to John F. Kennedy. One of JFK’s most famous decisions was to override the elder Wiesner’s fervent opposition to sending humans to the moon (Wiesner thought it a waste of money, as robots could do the same science for vastly cheaper).

While I don’t know all the details (I hope someone someday researches it and writes a book), Steve Wiesner made it clear to me that he did not get along with his famous father at all—in fact they became estranged. Steve told me that his embrace of Orthodox Judaism was, at least in part, a reaction against everything his father had stood for, including militant scientific atheism. I suppose that in the 1960s, millions of young Americans defied their parents via sex, drugs, and acoustic guitar; only a tiny number did so by donning tzitzit and moving to Israel to pray and toil with their hands. The two groups of rebels did, however, share a tendency to grow long beards.

Wiesner’s unique, remarkable, uncloneable life trajectory raises the question: who are the young Stephen Wiesners of our time? Will we be faster to recognize their foresight than Wiesner’s contemporaries were to recognize his?


Feel free to share any other memories of Stephen Wiesner or his influence in the comments.


Update (Aug. 14): See also Or Sattath’s memorial post, which (among other things) points out something that my narrative missed: namely, besides quantum money, Wiesner also invented superdense coding in 1970, although he and Bennett only published the idea 22 years later (!).

And I have more photos! Here’s Wiesner with an invention of his and another photo (thanks to his daughter Sarah). Here’s another photo from 1970 and Charlie Bennett’s handwritten notes (!) after first meeting Wiesner in 1970 (thanks to Charlie Bennett).

Another Update: Stephen’s daughter Sarah gave me the following fascinating information to share.

In the 70’s he lived in California where he worked in various Silicon Valley startups while also working weekends as part of a produce (fruits and vegetables) distribution co-op. During this time he became devoted to the ideas of solar energy, clean energy and space migration and exploration. He also became interested in Judaism. He truly wanted to help and make our world more peaceful and safe with his focus being on clean energy and branching out into space. He also believed that instead of fighting over the temple mount in Jerusalem, the Third Temple should be built in outer-space or in a structure above the original spot, an idea he tried to promote to prevent wars over land.