On the IBM Qiskit blog, there’s an interview with me about the role of complexity theory in the early history of quantum computing. Not much new for regular readers, but I’m very happy with how it came out—thanks so much to Robert Davis and Olivia Lanes for making it happen! My only quibble is with the sketch of my face, which might create the inaccurate impression that I no longer have teeth.
Boaz Barak pointed me to a Twitter thread of DALL-E paintings of people using quantum computers, in the styles of many of history’s famous artists. While the motifs are unsurprising (QCs look like regular computers but glowing, or maybe like giant glowing atoms), highly recommended as another demonstration of the sort of thing DALL-E does best.
Dan Spielman asked me to announce that the National Academy of Sciences is seeking nominations for the Held Prize in combinatorial and discrete optimization. The deadline is October 3.
I’m at the NSF Workshop on Quantum Advantage and Next Steps at the University of Chicago. My talk yesterday was entitled “Verifiable Quantum Advantage: What I Hope Will Be Done” (yeah yeah, I decided to call it “advantage” rather than “supremacy” in deference to the name of the workshop). My PowerPoint slides are here. Meanwhile, this morning was the BosonSampling session. The talk by Chaoyang Lu, leader of USTC’s experimental BosonSampling effort, was punctuated by numerous silly memes and videos, as well as the following striking sentence: “only by putting the seven dragon balls together can you unlock the true quantum computational power.”
Gavin Leech lists and excerpts his favorite writings of mine over the past 25 years, while complaining that I spend “a lot of time rebutting fleeting manias” and “obsess[ing] over political flotsam.”
I promise you: this post is going to tell a scientifically coherent story that involves all five topics listed in the title. Not one can be omitted.
My story starts with a Zoom talk that the one and only Lenny Susskind delivered for the Simons Institute for Theory of Computing back in May. There followed a panel discussion involving Lenny, Edward Witten, Geoffrey Penington, Umesh Vazirani, and your humble shtetlmaster.
Lenny’s talk led up to a gedankenexperiment involving an observer, Alice, who bravely jumps into a specially-prepared black hole, in order to see the answer to a certain computational problem in her final seconds before being ripped to shreds near the singularity. Drawing on earlier work by Bouland, Fefferman, and Vazirani, Lenny speculated that the computational problem could be exponentially hard even for a (standard) quantum computer. Despite this, Lenny repeatedly insisted—indeed, he asked me again to stress here—that he was not claiming to violate the Quantum Extended Church-Turing Thesis (QECTT), the statement that all of nature can be efficiently simulated by a standard quantum computer. Instead, he was simply investigating how the QECTT needs to be formulated in order to be a true statement.
I didn’t understand this, to put it mildly. If what Lenny was saying was right—i.e., if the infalling observer could see the answer to a computational problem not in BQP, or Bounded-Error Quantum Polynomial-Time—then why shouldn’t we call that a violation of the QECTT? Just like we call Shor’s quantum factoring algorithm a likely violation of the classical Extended Church-Turing Thesis, the thesis saying that nature can be efficiently simulated by a classical computer? Granted, you don’t have to die in order to run Shor’s algorithm, as you do to run Lenny’s experiment. But why should such implementation details matter from the lofty heights of computational complexity?
Alas, not only did Lenny never answer that in a way that made sense to me, he kept trying to shift the focus from real, physical black holes to “silicon spheres” made of qubits, which would be programmed to simulate the process of Alice jumping into the black hole (in a dual boundary description). Say what? Granting that Lenny’s silicon spheres, being quantum computers under another name, could clearly be simulated in BQP, wouldn’t this still leave the question about the computational powers of observers who jump into actual black holes—i.e., the question that we presumably cared about in the first place?
Confusing me even further, Witten seemed almost dismissive of the idea that Lenny’s gedankenexperiment raised any new issue for the QECTT—that is, any issue that wouldn’t already have been present in a universe without gravity. But as to Witten’s reasons, the most I understood from his remarks was that he was worried about various “engineering” issues with implementing Lenny’s gedankenexperiment, involving gravitational backreaction and the like. Ed Witten, now suddenly the practical guy! I couldn’t even isolate the crux of disagreement between Susskind and Witten, since after all, they agreed (bizarrely, from my perspective) that the QECTT wasn’t violated. Why wasn’t it?
Anyway, shortly afterward I attended the 28th Solvay Conference in Brussels, where one of the central benefits I got—besides seeing friends after a long COVID absence and eating some amazing chocolate mousse—was a dramatically clearer understanding of the issues in Lenny’s gedankenexperiment. I owe this improved understanding to conversations with many people at Solvay, but above all Daniel Gottesman and Daniel Harlow. Lenny himself wasn’t there, other than in spirit, but I ran the Daniels’ picture by him afterwards and he assented to all of its essentials.
The Daniels’ picture is what I want to explain in this post. Needless to say, I take sole responsibility for any errors in my presentation, as I also take sole responsibility for not understanding (or rather: not doing the work to translate into terms that I understood) what Susskind and Witten had said to me before.
The first thing you need to understand about Lenny’s gedankenexperiment is that it takes place entirely in the context of AdS/CFT: the famous holographic duality between two types of physical theories that look wildly different. Here AdS stands for anti-de-Sitter: a quantum theory of gravity describing a D-dimensional universe with a negative cosmological constant (i.e. hyperbolic geometry), one where black holes can form and evaporate and so forth. Meanwhile, CFT stands for conformal field theory: a quantum field theory, with no apparent gravity (and hence no black holes), that lives on the (D-1)-dimensional boundary of the D-dimensional AdS space. The staggering claim of AdS/CFT is that every physical question about the AdS bulk can be translated into an equivalent question about the CFT boundary, and vice versa, with a one-to-one mapping from states to states and observables to observables. So in that sense, they’re actually the same theory, just viewed in two radically different ways. AdS/CFT originally came out of string theory, but then notoriously “swallowed its parent,” to the point where nowadays, if you go to what are still called “string theory” meetings, you’re liable to hear vastly more discussion of AdS/CFT than of actual strings.
Thankfully, the story I want to tell won’t depend on fine details of how AdS/CFT works. Nevertheless, you can’t just ignore the AdS/CFT part as some technicality, in order to get on with the vivid tale of Alice jumping into a black hole, hoping to learn the answer to a beyond-BQP computational problem in her final seconds of existence. The reason you can’t ignore it is that the whole beyond-BQP computational problem we’ll be talking about, involves the translation (or “dictionary”) between the AdS bulk and the CFT boundary. If you like, then, it’s actually the chasm between bulk and boundary that plays the starring role in this story. The more familiar chasm within the bulk, between the interior of a black hole and its exterior (the two separated by an event horizon), plays only a subsidiary role: that of causing the AdS/CFT dictionary to become exponentially complex, as far as anyone can tell.
Pause for a minute. Previously I led you to believe that we’d be talking about an actual observer Alice, jumping into an actual physical black hole, and whether Alice could see the answer to a problem that’s intractable even for quantum computers in her last moments before hitting the singularity, and if so whether we should take that to refute the Quantum Extended Church-Turing Thesis. What I’m saying now is so wildly at variance with that picture, that it had to be repeated to me about 10 times before I understood it. Once I did understand, I then had to repeat it to others about 10 times before they understood. And I don’t care if people ridicule me for that admission—how slow Scott and his friends must be, compared to string theorists!—because my only goal right now is to get you to understand it.
To say it again: Lenny has not proposed a way for Alice to surpass the complexity-theoretic power of quantum computers, even for a brief moment, by crossing the event horizon of a black hole. If that was Alice’s goal when she jumped into the black hole, then alas, she probably sacrificed her life for nothing! As far as anyone knows, Alice’s experiences, even after crossing the event horizon, ought to continue to be described extremely well by general relativity and quantum field theory (at least until she nears the singularity and dies), and therefore ought to be simulatable in BQP. Granted, we don’t actually know this—you can call it an open problem if you like—but it seems like a reasonable guess.
In that case, though, what beyond-BQP problem was Lenny talking about, and what does it have to do with black holes? Building on the Bouland-Fefferman-Vazirani paper, Lenny was interested in a class of problems of the following form: Alice is given as input a pure quantum state |ψ⟩, which encodes a boundary CFT state, which is dual to an AdS bulk universe that contains a black hole. Alice’s goal is, by examining |ψ⟩, to learn something about what’s inside the black hole. For example: does the black hole interior contain “shockwaves,” and if so how many and what kind? Does it contain a wormhole, connecting it to a different black hole in another universe? If so, what’s the volume of that wormhole? (Not the first question I would ask either, but bear with me.)
Now, when I say Alice is “given” the state |ψ⟩, this could mean several things: she could just be physically given a collection of n qubits. Or, she could be given a gigantic table of 2n amplitudes. Or, as a third possibility, she could be given a description of a quantum circuit that prepares |ψ⟩, say from the all-0 initial state |0n⟩. Each of these possibilities leads to a different complexity-theoretic picture, and the differences are extremely interesting to me, so that’s what I mostly focused on in my remarks in the panel discussion after Lenny’s talk. But it won’t matter much for the story I want to tell in this post.
However |ψ⟩ is given to Alice, the prediction of AdS/CFT is that |ψ⟩ encodes everything there is to know about the AdS bulk, including whatever is inside the black hole—but, and this is crucial, the information about what’s inside the black hole will be pseudorandomly scrambled. In other words, it works like this: whatever simple thing you’d like to know about parts of the bulk that aren’t hidden behind event horizons—is there a star over here? some gravitational lensing over there? etc.—it seems that you could not only learn it by measuring |ψ⟩, but learn it in polynomial time, the dictionary between bulk and boundary being computationally efficient in that case. (As with almost everything else in this subject, even that hasn’t been rigorously proven, though my postdoc Jason Pollack and I made some progress this past spring by proving a piece of it.) On the other hand, as soon as you want to know what’s inside an event horizon, the fact that there are no probes that an “observer at infinity” could apply to find out, seems to translate into the requisite measurements on |ψ⟩ being exponentially complex to apply. (Technically, you’d have to measure an ensemble of poly(n) identical copies of |ψ⟩, but I’ll ignore that in what follows.)
In more detail, the relevant part of |ψ⟩ turns into a pseudorandom, scrambled mess: a mess that it’s plausible that no polynomial-size quantum circuit could even distinguish from the maximally mixed state. So, while in principle the information is all there in |ψ⟩, getting it out seems as hard as various well-known problems in symmetric-key cryptography, if not literally NP-hard. This is way beyond what we expect even a quantum computer to be able to do efficiently: indeed, after 30 years of quantum algorithms research, the best quantum speedup we know for this sort of task is typically just the quadratic speedup from Grover’s algorithm.
So now you understand why there was some hope that Alice, by jumping into a black hole, could solve a problem that’s exponentially hard for quantum computers! Namely because, once she’s inside the black hole, she can just see the shockwaves, or the volume of the wormhole, or whatever, and no longer faces the exponentially hard task of decoding that information from |ψ⟩. It’s as if the black hole has solved the problem for her, by physically instantiating the otherwise exponentially complex transformation between the bulk and boundary descriptions of |ψ⟩.
Having now gotten your hopes up, the next step in the story is to destroy them.
Here’s the fundamental problem: |ψ⟩ does not represent the CFT dual of a bulk universe that contains the black hole with the shockwaves or whatever, and that also contains Alice herself, floating outside the black hole, and being given |ψ⟩ as an input. Indeed, it’s unclear what the latter state would even mean: how do we get around the circularity in its definition? How do we avoid an infinite regress, where |ψ⟩ would have to encode a copy of |ψ⟩ which would have to encode a copy of … and so on forever? Furthermore, who created this |ψ⟩ to give to Alice? We don’t normally imagine that an “input state” contains a complete description of the body and brain of the person whose job it is to learn the output.
By contrast, a scenario that we can define without circularity is this: Alice is given (via physical qubits, a giant table of amplitudes, an obfuscated quantum circuit, or whatever) a pure quantum state |ψ⟩, which represents the CFT dual of a hypothetical universe containing a black hole. Alice wants to learn what shockwaves or wormholes are inside the black hole, a problem plausibly conjectured not to have any ordinary polynomial-size quantum circuit that takes copies of |ψ⟩ as input. To “solve” the problem, Alice sets into motion the following sequence of events:
Alice scans and uploads her own brain into a quantum computer, presumably destroying the original meat brain in the process! The QC represents Alice, who now exists only virtually, via a state |φ⟩.
The QC performs entangling operations on |φ⟩ and |ψ⟩, which correspond to inserting Alice into the bulk of the universe described by |ψ⟩, and then having her fall into the black hole.
Now in simulated form, “Alice” (or so we assume, depending on our philosophical position) has the subjective experience of falling into the black hole and observing what’s inside. Success! Given |ψ⟩ as input, we’ve now caused “Alice” (for some definition of “Alice”) to have observed the answer to the beyond-BQP computational problem.
In the panel discussion, I now model Susskind as having proposed scenario 1-3, Witten as going along with 1-2 but rejecting 3 or not wanting to discuss it, and me as having made valid points about the computational complexity of simulating Alice’s experience in 1-3, yet while being radically mistaken about what the scenario was (I still thought an actual black hole was involved).
An obvious question is whether, having learned the answer, “Alice” can now get the answer back out to the “real, original” world. Alas, the expectation is that this would require exponential time. Why? Because otherwise, this whole process would’ve constituted a subexponential-time algorithm for distinguishing random from pseudorandom states using an “ordinary” quantum computer! Which is conjectured not to exist.
And what about Alice herself? In polynomial time, could she return from “the Matrix,” back to a real-world biological body? Sure she could, in principle—if, for example, the entire quantum computation were run in reverse. But notice that reversing the computation would also make Alice forget the answer to the problem! Which is not at all a coincidence: if the problem is outside BQP, then in general, Alice can know the answer only while she’s “inside the Matrix.”
Now that hopefully everything is crystal-clear and we’re all on the same page, what can we say about this scenario? In particular: should it cause us to reject or modify the QECTT itself?
Daniel Gottesman, I thought, offered a brilliant reductio ad absurdum of the view that the simulated black hole scenario should count as a refutation of the QECTT. Well, he didn’t call it a “reductio,” but I will.
For the reductio, let’s forget not only about quantum gravity but even about quantum mechanics itself, and go all the way back to classical computer science. A fully homomorphic encryption scheme, the first example of which was discovered by Craig Gentry 15 years ago, lets you do arbitrary computations on encrypted data without ever needing to decrypt it. It has both an encryption key, for encrypting the original plaintext data, and a separate decryption key, for decrypting the final answer.
Now suppose Alice has some homomorphically encrypted top-secret emails, which she’d like to read. She has the encryption key (which is public), but not the decryption key.
If the homomorphic encryption scheme is secure against quantum computers—as the schemes discovered by Gentry and later researchers currently appear to be—and if the QECTT is true, then Alice’s goal is obviously infeasible: decrypting the data will take her exponential time.
Now, however, a classical version of Lenny comes along, and explains to Alice that she simply needs to do the following:
Upload her own brain state into a classical computer, destroying the “meat” version in the process (who needed it?).
Using the known encryption key, homomorphically encrypt a computer program that simulates (and thereby, we presume, enacts) Alice’s consciousness.
Using the homomorphically encrypted Alice-brain, together with the homomorphically encrypted input data, do the homomorphic computations that simulate the process of Alice’s brain reading the top-secret emails.
The claim would now be that, inside the homomorphic encryption, the simulated Alice has the subjective experience of reading the emails in the clear. Aha, therefore she “broke” the homomorphic encryption scheme! Therefore, assuming that the scheme was secure even against quantum computers, the QECTT must be false!
According to Gottesman, this is almost perfectly analogous to Lenny’s black hole scenario. In particular, they share the property that “encryption is easy but decryption is hard.” Once she’s uploaded her brain, Alice can efficiently enter the homomorphically encrypted world to see the solution to a hard problem, just like she can efficiently enter the black hole world to do the same. In both cases, however, getting back to her normal world with the answer would then take Alice exponential time. Note that in the latter case, the difficulty is not so much about “escaping from a black hole,” as it is about inverting the AdS/CFT dictionary.
Going further, we can regard the AdS/CFT dictionary for regions behind event horizons as, itself, an example of a fully homomorphic encryption scheme—in this case, of course, one where the ciphertexts are quantum states. This strikes me as potentially an important insight about AdS/CFT itself, even if that wasn’t Gottesman’s intention. It complements many other recent connections between AdS/CFT and theoretical computer science, including the view of AdS/CFT as a quantum error-correcting code, and the connection between AdS/CFT and the Max-Flow/Min-Cut Theorem (see also my talk about my work with Jason Pollack).
So where’s the reductio? Well, when it’s put so starkly, I suspect that not many would regard Gottesman’s classical homomorphic encryption scenario as a “real” challenge to the QECTT. Or rather, people might say: yes, this raises fascinating questions for the philosophy of mind, but at any rate, we’re no longer talking about physics. Unlike with (say) quantum computing, no new physical phenomenon is being brought to light that lets an otherwise intractable computational problem be solved. Instead, it’s all about the user herself, about Alice, and which physical systems get to count as instantiating her.
It’s like, imagine Alice at the computer store, weighing which laptop to buy. Besides weight, battery life, and price, she definitely does care about processing power. She might even consider a quantum computer, if one is available. Maybe even a computer with a black hole, wormhole, or closed timelike curve inside: as long as it gives the answers she wants, what does she care about the innards? But a computer whose normal functioning would (pessimistically) kill her or (optimistically) radically change her own nature, trapping her in a simulated universe that she can escape only by forgetting the computer’s output? Yeah, I don’t envy the computer salesman.
Anyway, if we’re going to say this about the homomorphic encryption scenario, then shouldn’t we say the same about the simulated black hole scenario? Again, from an “external” perspective, all that’s happening is a giant BQP computation. Anything beyond BQP that we consider to be happening, depends on adopting the standpoint of an observer who “jumps into the homomorphic encryption on the CFT boundary”—at which point, it would seem, we’re no longer talking about physics but about philosophy of mind.
So, that was the story! I promised you that it would integrally involve black holes, holography, the Quantum Extended Church-Turing Thesis, fully homomorphic encryption, and brain uploading, and I hope to have delivered on my promise.
Of course, while this blog post has forever cleared up all philosophical confusions about AdS/CFT and the Quantum Extended Church-Turing Thesis, many questions of a more technical nature remain. For example: what about the original scenario? can we argue that the experiences of bulk observers can be simulated in BQP, even when those observers jump into black holes? Also, what can we say about the complexity class of problems to which the simulated Alice can learn the answers? Could she even solve NP-complete problems in polynomial time this way, or at least invert one-way functions? More broadly, what’s the power of “BQP with an oracle for applying the AdS/CFT dictionary”—once or multiple times, in one direction or both directions?
Lenny himself described his gedankenexperiment as exploring the power of a new complexity class that he called “JI/poly,” where the JI stands for “Jumping In” (to a black hole, that is). The nomenclature is transparently ridiculous—“/poly” means “with polynomial-size advice,” which we’re not talking about here—and I’ve argued in this post that the “JI” is rather misleading as well. If Alice is “jumping” anywhere, it’s not into a black hole per se, but into a quantum computer that simulates a CFT that’s dual to a bulk universe containing a black hole.
In a broader sense, though, to contemplate these questions at all is clearly to “jump in” to … something. It’s old hat by now that one can start in physics and end up in philosophy: what else is the quantum measurement problem, or the Boltzmann brain problem, or anthropic cosmological puzzles like whether (all else equal) we’re a hundred times as likely to find ourselves in a universe with a hundred times as many observers? More recently, it’s also become commonplace that one can start in physics and end in computational complexity theory: quantum computing itself is the example par excellence, but over the past decade, the Harlow-Hayden argument about decoding Hawking radiation and the complexity = action proposal have made clear that it can happen even in quantum gravity.
Lenny’s new gedankenexperiment, however, is the first case I’ve seen where you start out in physics, and end up embroiled in some of the hardest questions of philosophy of mind and computational complexity theory simultaneously.
(1) Fellow CS theory blogger (and, 20 years ago, member of my PhD thesis committee) Luca Trevisan interviews me about Shtetl-Optimized, for the Bulletin of the European Association for Theoretical Computer Science. Questions include: what motivates me to blog, who my main inspirations are, my favorite posts, whether blogging has influenced my actual research, and my thoughts on the role of public intellectuals in the age of social-media outrage.
(3) The Microsoft team has finally released its promised paper about the detection of Majorana zero modes (“this time for real”), a major step along the way to creating topological qubits. See also this live YouTube peer review—is that a thing now?—by Vincent Mourik and Sergey Frolov, the latter having been instrumental in the retraction of Microsoft’s previous claim along these lines. I’ll leave further discussion to people who actually understand the experiments.
(4) I’m looking forward to the 2022 Conference on Computational Complexity less than two weeks from now, in my … safe? clean? beautiful? awe-inspiring? … birth-city of Philadelphia. There I’ll listen to a great lineup of talks, including one by my PhD student William Kretschmer on his joint work with me and DeVon Ingram on The Acrobatics of BQP, and to co-receive the CCC Best Paper Award (wow! thanks!) for that work. I look forward to meeting some old and new Shtetl-Optimized readers there.
In Steven Pinker’s guest post from last week, there’s one bit to which I never replied. Steve wrote:
After all, in many areas Einstein was no Einstein. You [Scott] above all could speak of his not-so-superintelligence in quantum physics…
While I can’t speak “above all,” OK, I can speak. Now that we’re closing in on a century of quantum physics, can we finally adjudicate what Einstein and Bohr were right or wrong about in the 1920s and 1930s? (Also, how is it still even a thing people argue about?)
The core is this: when confronted with the phenomena of entanglement—including the ability to measure one qubit of an EPR pair and thereby collapse the other in a basis of one’s choice (as we’d put it today), as well as the possibility of a whole pile of gunpowder in a coherent superposition of exploding and not exploding (Einstein’s example in a letter to Schrödinger, which the latter then infamously transformed into a cat)—well, there are entire conferences and edited volumes about what Bohr and Einstein said, didn’t say, meant to say or tried to say about these matters, but in cartoon form:
Einstein said that quantum mechanics can’t be the final answer, it has ludicrous implications for reality if you actually take it seriously, the resolution must be that it’s just a statistical approximation to something deeper, and at any rate there’s clearly more to be said.
Bohr (translated from Ponderousness to English) said that quantum mechanics sure looks like a final answer and not an approximation to anything deeper, there’s not much more to be said, we don’t even know what the implications are for “reality” (if any) so we shouldn’t hyperventilate about it, and mostly we need to change the way we use words and think about our own role as observers.
A century later, do we know anything about these questions that Einstein and Bohr didn’t? Well, we now know the famous Bell inequality, the experiments that have demonstrated Bell inequality violation with increasing finality (most recently, in 2015, closing both the detector and the locality loopholes), other constraints on hidden-variable theories (e.g. Kochen-Specker and PBR), decoherence theory, and the experiments that have manufactured increasingly enormous superpositions (still, for better or worse, not exploding piles of gunpowder or cats!), while also verifying detailed predictions about how such superpositions decohere due to entanglement with the environment rather than some mysterious new law of physics.
So, if we were able to send a single short message back in time to the 1927 Solvay Conference, adjudicating between Einstein and Bohr without getting into any specifics, what should the message say? Here’s my attempt:
In 2022, quantum mechanics does still seem to be a final answer—not an approximation to anything deeper as Einstein hoped. And yet, contra Bohr, there was considerably more to say about the matter! The implications for reality could indeed be described as “ludicrous” from a classical perspective, arguably even more than Einstein realized. And yet the resolution turns out simply to be that we live in a universe where those implications are true.
OK, here’s the point I want to make. Even supposing you agree with me (not everyone will) that the above would be a reasonable modern summary to send back in time, it’s still totally unclear how to use it to mark the Einstein vs. Bohr scorecard!
Indeed, it’s not surprising that partisans have defended every possible scoring, from 100% for Bohr (quantum mechanics vindicated! Bohr called it from the start!), to 100% for Einstein (he put his finger directly on the implications that needed to be understood, against the evil Bohr who tried to shut everyone up about them! Einstein FTW!).
Personally, I’d give neither of them perfect marks, in part because they not only both missed Bell’s Theorem, but failed even to ask the requisite question (namely: what empirically verifiable tasks can Alice and Bob use entanglement to do, that they couldn’t have done without entanglement?). But I’d give both of them very high marks for, y’know, still being Albert Einstein and Niels Bohr.
And with that, I’m proud to have said the final word about precisely what Einstein and Bohr got right and wrong about quantum physics. I’m relieved that no one will ever need to debate that tiresome historical question again … certainly not in the comments section of this post.
Thanks so much to everyone who sent messages of support following my last post! I vowed there that I’m going to stop letting online trolls and sneerers occupy so much space in my mental world. Truthfully, though, while there are many trolls and sneerers who terrify me, there are also some who merely amuse me. A good example of the latter came a few weeks ago, when an anonymous commenter calling themselves “String Theorist” submitted the following:
It’s honestly funny to me when you [Scott] call yourself a “nerd” or a “prodigy” or whatever [I don’t recall ever calling myself a “prodigy,” which would indeed be cringe, though “nerd” certainly —SA], as if studying quantum computing, which is essentially nothing more than glorified linear algebra, is such an advanced intellectual achievement. For what it’s worth I’m a theoretical physicist, I’m in a completely different field, and I was still able to learn Shor’s algorithm in about half an hour, that’s how easy this stuff is. I took a look at some of your papers on arXiv and the math really doesn’t get any more advanced than linear algebra. To understand quantum circuits about the most advanced concept is a tensor product which is routinely covered in undergraduate linear algebra. Wheras in my field of string theory grasping, for instance, holographic dualities relating confirmal field theories and gravity requires vastly more expertise (years of advanced study). I actually find it pretty entertaining that you’ve said yourself you’re still struggling to understand QFT, which most people I’m working with in my research group were first exposed to in undergrad 😉 The truth is we’re in entirely different leagues of intelligence (“nerdiness”) and any of your qcomputing papers could easily be picked up by a first or second year math major. It’s just a joke that this is even a field (quantum complexity theory) with journals and faculty when the results in your papers that I’ve seen are pretty much trivial and don’t require anything more than undergraduate level maths.
Why does this sort of trash-talk, reminiscent of Luboš Motl, no longer ruffle me? Mostly because the boundaries between quantum computing theory, condensed matter physics, and quantum gravity, which were never clear in the first place, have steadily gotten fuzzier. Even in the 1990s, the field of quantum computing attracted amazing physicists—folks who definitely do know quantum field theory—such as Ed Farhi, John Preskill, and Ray Laflamme. Decades later, it would be fair to say that the physicists have banged their heads against many of the same questions that we computer scientists have banged our heads against, oftentimes in collaboration with us. And yes, there were cases where actual knowledge of particle physics gave physicists an advantage—with some famous examples being the algorithms of Farhi and collaborators (the adiabatic algorithm, the quantum walk on conjoined trees, the NAND-tree algorithm). There were other cases where computer scientists’ knowledge gave them an advantage: I wouldn’t know many details about that, but conceivably shadow tomography, BosonSampling, PostBQP=PP? Overall, it’s been what you wish every indisciplinary collaboration could be.
What’s new, in the last decade, is that the scientific conversation centered around quantum information and computation has dramatically “metastasized,” to encompass not only a good fraction of all the experimentalists doing quantum optics and sensing and metrology and so forth, and not only a good fraction of all the condensed-matter theorists, but even many leading string theorists and quantum gravity theorists, including Susskind, Maldacena, Bousso, Hubeny, Harlow, and yes, Witten. And I don’t think it’s just that they’re too professional to trash-talk quantum information people the way commenter “String Theorist” does. Rather it’s that, because of the intellectual success of “It from Qubit,” we’re increasingly participating in the same conversations and working on the same technical questions. One particularly exciting such question, which I’ll have more to say about in a future post, is the truth or falsehood of the Quantum Extended Church-Turing Thesis for observers who jump into black holes.
Not to psychoanalyze, but I’ve noticed a pattern wherein, the more secure a scientist is about their position within their own field, the readier they are to admit ignorance about the neighboring fields, to learn about those fields, and to reach out to the experts in them, to ask simple or (as it usually turns out) not-so-simple questions.
I can’t imagine any better illustration of these tendencies better than the 28th Solvay Conference on the Physics of Quantum Information, which I attended two weeks ago in Brussels on my 41st birthday.
As others pointed out, the proportion of women is not as high as we all wish, but it’s higher than in 1911, when there was exactly one: Madame Curie herself.
It was my first trip out of the US since before COVID—indeed, I’m so out of practice that I nearly missed my flights in both directions, in part because of my lack of familiarity with the COVID protocols for transatlantic travel, as well as the immense lines caused by those protocols. My former adviser Umesh Vazirani, who was also at the Solvay Conference, was proud.
The Solvay Conference is the venue where, legendarily, the fundamentals of quantum mechanics got hashed out between 1911 and 1927, by the likes of Einstein, Bohr, Planck, and Curie. (Einstein complained, in a letter, about being called away from his work on general relativity to attend a “witches’ sabbath.”) Remarkably, it’s still being held in Brussels every few years, and still funded by the same Solvay family that started it. The once-every-few-years schedule has, we were constantly reminded, been interrupted only three times in its 110-year history: once for WWI, once for WWII, and now once for COVID (this year’s conference was supposed to be in 2020).
This was the first ever Solvay conference organized around the theme of quantum information, and apparently, the first ever that counted computer scientists among its participants (me, Umesh Vazirani, Dorit Aharonov, Urmila Mahadev, and Thomas Vidick). There were four topics: (1) many-body physics, (2) quantum gravity, (3) quantum computing hardware, and (4) quantum algorithms. The structure, apparently unchanged since the conference’s founding, is this: everyone attends every session, without exception. They sit around facing each other the whole time; no one ever stands to lecture. For each topic, two “rapporteurs” introduce the topic with half-hour prepared talks; then there are short prepared response talks as well as an hour or more of unstructured discussion. Everything everyone says is recorded in order to be published later.
Daniel Gottesman and I were the two rapporteurs for quantum algorithms: Daniel spoke about quantum error-correction and fault-tolerance, and I spoke about “How Much Structure Is Needed for Huge Quantum Speedups?” The link goes to my PowerPoint slides, if you’d like to check them out. I tried to survey 30 years of history of that question, from Simon’s and Shor’s algorithms, to huge speedups in quantum query complexity (e.g., glued trees and Forrelation), to the recent quantum supremacy experiments based on BosonSampling and Random Circuit Sampling, all the way to the breakthrough by Yamakawa and Zhandry a couple months ago. The last slide hypothesizes a “Law of Conservation of Weirdness,” which after all these decades still remains to be undermined: “For every problem that admits an exponential quantum speedup, there must be some weirdness in its detailed statement, which the quantum algorithm exploits to focus amplitude on the rare right answers.” My title slide also shows DALL-E2‘s impressionistic take on the title question, “how much structure is needed for huge quantum speedups?”:
The discussion following my talk was largely a debate between me and Ed Farhi, reprising many debates he and I have had over the past 20 years: Farhi urged optimism about the prospect for large, practical quantum speedups via algorithms like QAOA, pointing out his group’s past successes and explaining how they wouldn’t have been possible without an optimistic attitude. For my part, I praised the past successes and said that optimism is well and good, but at the same time, companies, venture capitalists, and government agencies are right now pouring billions into quantum computing, in many cases—as I know from talking to them—because of a mistaken impression that QCs are already known to be able to revolutionize machine learning, finance, supply-chain optimization, or whatever other application domains they care about, and to do so soon. They’re genuinely surprised to learn that the consensus of QC experts is in a totally different place. And to be clear: among quantum computing theorists, I’m not at all unusually pessimistic or skeptical, just unusually willing to say in public what others say in private.
Afterwards, one of the string theorists said that Farhi’s arguments with me had been a highlight … and I agreed. What’s the point of a friggin’ Solvay Conference if everyone’s just going to agree with each other?
Besides quantum algorithms, there was naturally lots of animated discussion about the practical prospects for building scalable quantum computers. While I’d hoped that this discussion might change the impressions I’d come with, it mostly confirmed them. Yes, the problem is staggeringly hard. Recent ideas for fault-tolerance, including the use of LDPC codes and bosonic codes, might help. Gottesman’s talk gave me the insight that, at its core, quantum fault-tolerance is all about testing, isolation, and contact-tracing, just for bit-flip and phase-flip errors rather than viruses. Alas, we don’t yet have the quantum fault-tolerance analogue of a vaccine!
At one point, I asked the trapped-ion experts in open session if they’d comment on the startup company IonQ, whose stock price recently fell precipitously in the wake of a scathing analyst report. Alas, none of them took the bait.
On a different note, I was tremendously excited by the quantum gravity session. Netta Engelhardt spoke about her and others’ celebrated recent work explaining the Page curve of an evaporating black hole using Euclidean path integrals—and by questioning her and others during coffee breaks, I finally got a handwavy intuition for how it works. There was also lots of debate, again at coffee breaks, about Susskind’s recent speculations on observers jumping into black holes and the quantum Extended Church-Turing Thesis. One of my main takeaways from the conference was a dramatically better understanding of the issues involved there—but that’s a big enough topic that it will need its own post.
Toward the end of the quantum gravity session, the experimentalist John Martinis innocently asked what actual experiments, or at least thought experiments, had been at issue for the past several hours. I got a laugh by explaining to him that, while the gravity experts considered this too obvious to point out, the thought experiments in question all involve forming a black hole in a known quantum pure state, with total control over all the Planck-scale degrees of freedom; then waiting outside the black hole for ~1070 years; collecting every last photon of Hawking radiation that comes out and routing them all into a quantum computer; doing a quantum computation that might actually require exponential time; and then jumping into the black hole, whereupon you might either die immediately at the event horizon, or else learn something in your last seconds before hitting the singularity, which you could then never communicate to anyone outside the black hole. Martinis thanked me for clarifying.
Anyway, I had a total blast. Here I am amusing some of the world’s great physicists by letting them mess around with GPT-3.
Back: Ahmed Almheiri, Juan Maldacena, John Martinis, Aron Wall. Front: Geoff Penington, me, Daniel Harlow. Thanks to Michelle Simmons for the photo.
I also had the following exchange at my birthday dinner:
Physicist: So I don’t get this, Scott. Are you a physicist who studied computer science, or a computer scientist who studied physics?
Me: I’m a computer scientist who studied computer science.
Physicist: But then you…
Me: Yeah, at some point I learned what a boson was, in order to invent BosonSampling.
Physicist: And your courses in physics…
Me: They ended at thermodynamics. I couldn’t handle PDEs.
Physicist: What are the units of h-bar?
Me: Uhh, well, it’s a conversion factor between energy and time. (*)
Physicist: Good. What’s the radius of the hydrogen atom?
Me: Uhh … not sure … maybe something like 10-15 meters?
Physicist: OK fine, he’s not one of us.
(The answer, it turns out, is more like 10-10 meters. I’d stupidly substituted the radius of the nucleus—or, y’know, a positively-charged hydrogen ion, i.e. proton. In my partial defense, I was massively jetlagged and at most 10% conscious.)
(*) Actually h-bar is a conversion factor between energy and 1/time, i.e. frequency, but the physicist accepted this answer.
Anyway, I look forward to attending more workshops this summer, seeing more colleagues who I hadn’t seen since before COVID, and talking more science … including branching out in some new directions that I’ll blog about soon. It does beat worrying about online trolls.
Thanks to everyone who asked whether I’m OK! Yeah, I’ve been living, loving, learning, teaching, worrying, procrastinating, just not blogging.
Last week, Takashi Yamakawa and Mark Zhandry posted a preprint to the arXiv, “Verifiable Quantum Advantage without Structure,” that represents some of the most exciting progress in quantum complexity theory in years. I wish I’d thought of it. tl;dr they show that relative to a random oracle (!), there’s an NP search problem that quantum computers can solve exponentially faster than classical ones. And yet this is 100% consistent with the Aaronson-Ambainis Conjecture!
A student brought my attention to Quantle, a variant of Wordle where you need to guess a true equation involving 1-qubit quantum states and unitary transformations. It’s really well-done! Possibly the best quantum game I’ve seen.
Last month, Microsoft announced on the web that it had achieved an experimental breakthrough in topological quantum computing: not quite the creation of a topological qubit, but some of the underlying physics required for that. This followed their needing to retract their previous claim of such a breakthrough, due to the criticisms of Sergey Frolov and others. One imagines that they would’ve taken far greater care this time around. Unfortunately, a research paper doesn’t seem to be available yet. Anyone with further details is welcome to chime in.
Woohoo! Maximum flow, maximum bipartite matching, matrix scaling, and isotonic regression on posets (among many others)—all algorithmic problems that I was familiar with way back in the 1990s—are now solvable in nearly-linear time, thanks to a breakthrough by Chen et al.! Many undergraduate algorithms courses will need to be updated.
For those interested, Steve Hsu recorded a podcast with me where I talk about quantum complexity theory.
In the past few months, I’ve twice injured the same ankle while playing with my kids. This, perhaps combined with covid, led me to several indisputable realizations:
I am mortal.
Despite my self-conception as a nerdy little kid awaiting the serious people’s approval, I am now firmly middle-aged. By my age, Einstein had completed general relativity, Turing had founded CS, won WWII, and proposed the Turing Test, and Galois, Ramanujan, and Ramsey had been dead for years.
Thus, whatever I wanted to accomplish in my intellectual life, I should probably get started on it now.
Hence today’s post. I’m feeling a strong compulsion to write an essay, or possibly even a book, surveying and critically evaluating a century of ideas about the following question:
Q: Why should the universe have been quantum-mechanical?
If you want, you can divide Q into two subquestions:
Q1: Why didn’t God just make the universe classical and be done with it? What would’ve been wrong with that choice?
Q2: Assuming classical physics wasn’t good enough for whatever reason, why this specific alternative? Why the complex-valued amplitudes? Why unitary transformations? Why the Born rule? Why the tensor product?
Despite its greater specificity, Q2 is ironically the question that I feel we have a better handle on. I could spend half a semester teaching theorems that admittedly don’t answer Q2, as satisfyingly as Einstein answered the question “why the Lorentz transformations?,” but that at least render this particular set of mathematical choices (the 2-norm, the Born Rule, complex numbers, etc.) orders-of-magnitude less surprising than one might’ve thought they were a priori. Q1 therefore stands, to me at least, as the more mysterious of the two questions.
So, I want to write something about the space of credible answers to Q, and especially Q1, that humans can currently conceive. I want to do this for my own sake as much as for others’. I want to do it because I regard Q as one of the biggest questions ever asked, for which it seems plausible to me that there’s simply an answer that most experts would accept as valid once they saw it, but for which no such answer is known. And also because, besides having spent 25 years working in quantum information, I have the following qualifications for the job:
I don’t dismiss either Q1 or Q2 as silly; and
crucially, I don’t think I already know the answers, and merely need better arguments to justify them. I’m genuinely uncertain and confused.
The purpose of this post is to invite you to share your own answers to Q in the comments section. Before I embark on my survey project, I’d better know if there are promising ideas that I’ve missed, and this blog seems like as good a place as any to crowdsource the job.
Any answer is welcome, no matter how wild or speculative, so long as it honestly grapples with the actual nature of QM. To illustrate, nothing along the lines of “the universe is quantum because it needs to be holistic, interconnected, full of surprises, etc. etc.” will cut it, since such answers leave utterly unexplained why the world wasn’t simply endowed with those properties directly, rather than specifically via generalizing the rules of probability to allow interference and noncommuting observables.
Relatedly, whatever “design goal” you propose for the laws of physics, if the goal is satisfied by QM, but satisfied even better by theories that provide even more power than QM does—for instance, superluminal signalling, or violations of Tsirelson’s bound, or the efficient solution of NP-complete problems—then your explanation is out. This is a remarkably strong constraint.
Oh, needless to say, don’t try my patience with anything about the uncertainty principle being due to floating-point errors or rendering bugs, or anything else that relies on a travesty of QM lifted from a popular article or meme! 🙂
OK, maybe four more comments to enable a more productive discussion, before I shut up and turn things over to you:
I’m aware, of course, of the radical uncertainty about what form an answer to Q should even take. Am I asking you to psychoanalyze the will of God in creating the universe? Or, what perhaps amounts to the same thing, am I asking for the design objectives of the giant computer simulation that we’re living in? (As in, “I’m 100% fine with living inside a Matrix … I just want to understand why it’s a unitary matrix!”) Am I instead asking for an anthropic explanation, showing why of course QM would be needed if you wanted life or consciousness like ours? Am I “merely” asking for simpler or more intuitive physical principles from which QM is to be derived as a consequence? Am I asking why QM is the “most elegant choice” in some space of mathematical options … even to the point where, with hindsight, a 19th-century mathematician or physicist could’ve been convinced that of course this must be part of Nature’s plan? Am I asking for something else entirely? You get to decide! Should you take up my challenge, this is both your privilege and your terrifying burden.
I’m aware, of course, of the dizzying array of central physical phenomena that rely on QM for their ultimate explanation. These phenomena range from the stability of matter itself, which depends on the Pauli exclusion principle; to the nuclear fusion that powers the sun, which depends on a quantum tunneling effect; to the discrete energy levels of electrons (and hence, the combinatorial nature of chemistry), which relies on electrons being waves of probability amplitude that can only circle nuclei an integer number of times if their crests are to meet their troughs. Important as they are, though, I don’t regard any of these phenomena as satisfying answers to Q in themselves. The reason is simply that, in each case, it would seem like child’s-play to contrive some classical mechanism to produce the same effect, were that the goal. QM just seems far too grand to have been the answer to these questions! An exponentially larger state space for all of reality, plus the end of Newtonian determinism, just to overcome the technical problem that accelerating charges radiate energy in classical electrodynamics, thereby rendering atoms unstable? It reminds me of the Simpsons episode where Homer uses a teleportation machine to get a beer from the fridge without needing to get up off the couch.
I’m aware of Gleason’s theorem, and of the specialness of the 1-norm and 2-norm in linear algebra, and of the arguments for complex amplitudes as opposed to reals or quaternions, and of the beautiful work of Lucien Hardy and of Chiribella et al. and others on axiomatic derivations of quantum theory. As some of you might remember, I even discussed much of this material in Quantum Computing Since Democritus! There’s a huge amount to say about these fascinating justifications for the rules of QM, and I hope to say some of it in my planned survey! For now, I’ll simply remark that every axiomatic reconstruction of QM that I’ve seen, impressive though it was, has relied on one or more axioms that struck me as weird, in the sense that I’d have little trouble dismissing the axioms as totally implausible and unmotivated if I hadn’t already known (from QM, of course) that they were true. The axiomatic reconstructions do help me somewhat with Q2, but little if at all with Q1.
To keep the discussion focused, in this post I’d like to exclude answers along the lines of “but what if QM is merely an approximation to something else?,” to say nothing of “a century of evidence for QM was all just a massive illusion! LOCAL HIDDEN VARIABLES FOR THE WIN!!!” We can have those debates another day—God knows that, here on Shtetl-Optimized, we have and we will. Here I’m asking instead: imagine that, as fantastical as it sounds, QM were not only exactly true, but (along with relativity, thermodynamics, evolution, and the tastiness of chocolate) one of the profoundest truths our sorry species had ever discovered. Why should I have expected that truth all along? What possible reasons to expect it have I missed?
(Hopefully no one has taken taken that title yet!)
I waste a large fraction of my existence just reading about what’s happening in the world, or discussion and analysis thereof, in an unending scroll of paralysis and depression. On the first anniversary of the January 6 attack, I read the recent revelations about just how close the seditionists actually came to overturning the election outcome (e.g., by pressuring just one Republican state legislature to “decertify” its electors, after which the others would likely follow in a domino effect), and how hard it now is to see a path by which democracy in the United States will survive beyond 2024. Or I read about Joe Manchin, who’s already entered the annals of history as the man who could’ve halted the slide to the abyss and decided not to. Of course, I also read about the wokeists, who correctly see the swing of civilization getting pushed terrifyingly far out of equilibrium to the right, so their solution is to push the swing terrifyingly far out of equilibrium to the left, and then they act shocked when their own action, having added all this potential energy to the swing, causes it to swing back even further to the right, as swings tend to do. (And also there’s a global pandemic killing millions, and the correct response to it—to authorize and distribute new vaccines as quickly as the virus mutates—is completely outside the Overton Window between Obey the Experts and Disobey the Experts, advocated by no one but a few nerds. When I first wrote this post, I forgot all about the global pandemic.) And I see all this and I am powerless to stop it.
In such a dark time, it’s easy to forget that I’m a theoretical computer scientist, mainly focused on quantum computing. It’s easy to forget that people come to this blog because they want to read about quantum computing. It’s like, who gives a crap about that anymore? What doth it profit a man, if he gaineth a few thousand fault-tolerant qubits with which to calculateth chemical reaction rates or discrete logarithms, and he loseth civilization?
Nevertheless, in the rest of this post I’m going to share some quantum-related debunking updates—not because that’s what’s at the top of my mind, but in an attempt to find my way back to sanity. Picture that: quantum mechanics (and specifically, the refutation of outlandish claims related to quantum mechanics) as the part of one’s life that’s comforting, normal, and sane.
There’s been lots of online debate about the claim to have entangled a tardigrade (i.e., water bear) with a superconducting qubit; see also this paper by Vlatko Vedral, this from CNET, this from Ben Brubaker on Twitter. So, do we now have Schrödinger’s Tardigrade: a living, “macroscopic” organism maintained coherently in a quantum superposition of two states? How could such a thing be possible with the technology of the early 21st century? Hasn’t it been a huge challenge to demonstrate even Schrödinger’s Virus or Schrödinger’s Bacterium? So then how did this experiment leapfrog (or leaptardigrade) over those vastly easier goals?
Short answer: it didn’t. The experimenters couldn’t directly measure the degree of freedom in the tardigrade that’s claimed to be entangled with the qubit. But it’s consistent with everything they report that whatever entanglement is there, it’s between the superconducting qubit and a microscopic part of the tardigrade. It’s also consistent with everything they report that there’s no entanglement at all between the qubit and any part of the tardigrade, just boring classical correlation. (Or rather that, if there’s “entanglement,” then it’s the Everett kind, involving not merely the qubit and the tardigrade but the whole environment—the same as we’d get by just measuring the qubit!) Further work would be needed to distinguish these possibilities. In any case, it’s of course cool that they were able to cool a tardigrade to near absolute zero and then revive it afterwards.
I thank the authors of the tardigrade paper, who clarified a few of these points in correspondence with me. Obviously the comments section is open for whatever I’ve misunderstood.
People also asked me to respond to Sabine Hossenfelder’s recent video about superdeterminism, a theory that holds that quantum entanglement doesn’t actually exist, but the universe’s initial conditions were fine-tuned to stop us from choosing to measure qubits in ways that would make its nonexistence apparent: even when we think we’re applying the right measurements, we’re not, because the initial conditions messed with our brains or our computers’ random number generators. (See, I tried to be as non-prejudicial as possible in that summary, and it still came out sounding like a parody. Sorry!)
Sabine sets up the usual dichotomy that people argue against superdeterminism only because they’re attached to a belief in free will. She rejects Bell’s statistical independence assumption, which she sees as a mere dogma rather than a prerequisite for doing science. Toward the end of the video, Sabine mentions the objection that, without statistical independence, a demon could destroy any randomized controlled trial, by tampering with the random number generator that decides who’s in the control group and who isn’t. But she then reassures the viewer that it’s no problem: superdeterministic conspiracies will only appear when quantum mechanics would’ve predicted a Bell inequality violation or the like. Crucially, she never explains the mechanism by which superdeterminism, once allowed into the universe (including into macroscopic devices like computers and random number generators), will stay confined to reproducing the specific predictions that quantum mechanics already told us were true, rather than enabling ESP or telepathy or other mischief. This is stipulated, never explained or derived.
To say I’m not a fan of superdeterminism would be a super-understatement. And yet, nothing I’ve written previously on this blog—about superdeterminism’s gobsmacking lack of explanatory power, or about how trivial it would be to cook up a superdeterministic “mechanism” for, e.g., faster-than-light signaling—none of it seems to have made a dent. It’s all come across as obvious to the majority of physicists and computer scientists who think as I do, and it’s all fallen on deaf ears to superdeterminism’s fans.
So in desperation, let me now try another tack: going meta. It strikes me that no one who saw quantum mechanics as a profound clue about the nature of reality could ever, in a trillion years, think that superdeterminism looked like a promising route forward given our current knowledge. The only way you could think that, it seems to me, is if you saw quantum mechanics as an anti-clue: a red herring, actively misleading us about how the world really is. To be a superdeterminist is to say:
OK, fine, there’s the Bell experiment, which looks like Nature screaming the reality of ‘genuine indeterminism, as predicted by QM,’ louder than you might’ve thought it even logically possible for that to be screamed. But don’t listen to Nature, listen to us! If you just drop what you thought were foundational assumptions of science, we can explain this away! Not explain it, of course, but explain it away. What more could you ask from us?
Here’s my challenge to the superdeterminists: when, in 400 years from Galileo to the present, has such a gambit ever worked? Maxwell’s equations were a clue to special relativity. The Hamiltonian and Lagrangian formulations of classical mechanics were clues to quantum mechanics. When has a great theory in physics ever been grudgingly accommodated by its successor theory in a horrifyingly ad-hoc way, rather than gloriously explained and derived?
Update: Oh right, and the QIP’2022 list of accepted talks is out! And I was on the program committee! And they’re still planning to hold QIP in person, in March at Caltech, will you fancy that! actually I have no idea—but if they’re going to move to virtual, I’m awaiting an announcement just like everyone else.
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.
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.
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