Shtetl-Optimized

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

With that in mind:

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

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

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

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

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

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

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

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Yesterday I attended a lecture by George Mason University economist Bryan Caplan, who’s currently visiting UT Austin, about his new book entitled Don’t Be a Feminist. (See also here for previous back-and-forth between me and Bryan about his book.) A few remarks:

(1) Maybe surprisingly, there were no protesters storming the lectern, no security detail, not even a single rotten vegetable thrown. About 30 people showed up, majority men but women too. They listened politely and asked polite questions afterward. One feminist civilly challenged Bryan during the Q&A about his gender pay gap statistics.

(2) How is it that I got denounced by half the planet for saying once, in a blog comment, that I agreed with 97% of feminism but had concerns with one particular way it was operationalized, whereas Bryan seems to be … not denounced in the slightest for publishing a book and going on a lecture tour about how he rejects feminism in its entirety as angry and self-pitying in addition to factually false? Who can explain this to me?

(3) For purposes of his argument, Bryan defines feminism as “the view that women are generally treated less fairly than men,” rather than (say) “the view that men and women ought to be treated equally,” or “the radical belief that women are people,” or other formulations that Bryan considers too obvious to debate. He then rebuts feminism as he’s defined it, by taking the audience on a horror tour of all the ways society treats men less fairly than women (expectations of doing dirty and dangerous work, divorce law, military drafts as in Ukraine right now, …), as well as potentially benign explanations for apparent unfairness toward women, to argue that it’s at least debatable which sex gets the rawer deal on average.

During the Q&A, I raised what I thought was the central objection to Bryan’s relatively narrow definition of feminism. Namely that, by the standards of 150 years ago, Bryan is obviously a feminist, and so am I, and so is everyone in the room. (Whereupon a right-wing business school professor interjected: “please don’t make assumptions about me!”)

I explained that this is why I call myself a feminist, despite agreeing with many of Bryan’s substantive points: because I want no one to imagine for a nanosecond that, if I had the power, I’d take gender relations back to how they were generations ago.

Bryan replied that >60% of Americans call themselves non-feminists in surveys. So, he asked me rhetorically, do all those Americans secretly yearn to take us back to the 19th century? Such a position, he said, seemed so absurdly uncharitable as not to be worth responding to.

Reflecting about it on my walk home, I realized: actually, give or take the exact percentages, this is precisely the progressive thesis. I.e., that just like at least a solid minority of Germans turned out to be totally fine with Nazism, however much they might’ve denied it beforehand, so too at least a solid minority of Americans would be fine with—if not ecstatic about—The Handmaid’s Tale made real. Indeed, they’d add, it’s only vociferous progressive activism that stands between us and that dystopia.

And if anyone were tempted to doubt this, progressives might point to the election of Donald Trump, the failed insurrection to maintain his power, and the repeal of Roe as proof enough to last for a quadrillion years.

Bryan would probably reply: why even waste time engaging with such a hysterical position? To me, though, the hysterical position sadly has more than a grain of truth to it. I wish we lived in a world where there was no point in calling oneself a pro-democracy anti-racist feminist and a hundred other banal and obvious things. I just don’t think that we do.

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Here’s an observation that’s mathematically trivial but might not be widely appreciated. In kindergarten, we all learned Gödel’s First Incompleteness Theorem, which given a formal system F, constructs an arithmetical encoding of

G(F) = “This sentence is not provable in F.”

If G(F) is true, then it’s an example of a true arithmetical sentence that’s unprovable in F. If, on the other hand, G(F) is false, then it’s provable, which means that F isn’t arithmetically sound. Therefore F is either incomplete or unsound.

Many have objected: “but despite Gödel’s Theorem, it’s still easy to explain why G(F) is true. In fact, the argument above basically already did it!”

[Note: Please stop leaving comments explaining to me that G(F) follows from F’s consistency. I understand that: the “heuristic” part of the argument is F’s consistency! I made a pedagogical choice to elide that, which nerd-sniping has now rendered untenable.]

You might make a more general point: there are many, many mathematical statements for which we currently lack a proof, but we do seem to have a fully convincing heuristic explanation: one that “proves the statement to physics standards of rigor.” For example:

• The Twin Primes Conjecture (there are infinitely many primes p for which p+2 is also prime).
• The Collatz Conjecture (the iterative process that maps each positive integer n to n/2 if n is even, or to 3n+1 if n is odd, eventually reaches 1 regardless of which n you start at).
• π is a normal number (or even just: the digits 0-9 all occur with equal limiting frequencies in the decimal expansion of π).
• π+e is irrational.

And so on. No one has any idea how to prove any of the above statements—and yet, just on statistical grounds, it seems clear that it would require a ludicrous conspiracy to make any of them false.

Conversely, one could argue that there are statements for which we do have a proof, even though we lack a “convincing explanation” for the statements’ truth. Maybe the Four-Color Theorem or Hales’s Theorem, for which every known proof requires a massive computer enumeration of cases, belong to this class. Other people might argue that, given a proof, an explanation could always be extracted with enough time and effort, though resolving this dispute won’t matter for what follows.

You might hope that, even if some true mathematical statements can’t be proved, every true statement might nevertheless have a convincing heuristic explanation. Alas, a trivial adaptation of Gödel’s Theorem shows that, if (1) heuristic explanations are to be checkable by computer, and (2) only true statements are to have convincing heuristic explanations, then this isn’t possible either. I mean, let E be a program that accepts or rejects proposed heuristic explanations, for statements like the Twin Prime Conjecture or the Collatz Conjecture. Then construct the sentence

S(E) = “This sentence has no convincing heuristic explanation accepted by E.”

If S(E) is true, then it’s an example of a true arithmetical statement without even a convincing heuristic explanation for its truth (!). If, on the other hand, S(E) is false, then there’s a convincing heuristic explanation of its truth, which means that something has gone wrong.

What’s happening, of course, is that given the two conditions we imposed, our “heuristic explanation system” was a proof system, even though we didn’t call it one. This is my point, though: when we use the word “proof,” it normally invokes a specific image, of a sequence of statements that marches from axioms to a theorem, with each statement following from the preceding ones by rigid inference rules like those of first-order logic. None of that, however, plays any direct role in the proof of the Incompleteness Theorem, which cares only about soundness (inability to prove falsehoods) and checkability by a computer (what, with hindsight, Gödel’s “arithmetization of syntax” was all about). The logic works for “heuristic explanations” too.

Now we come to something that I picked up from my former student (and now AI alignment leader) Paul Christiano, on a recent trip to the Bay Area, and which I share with Paul’s kind permission. Having learned that there’s no way to mechanize even heuristic explanations for all the true statements of arithmetic, we could set our sights lower still, and ask about mere plausibility arguments—arguments that might be overturned on further reflection. Is there some sense in which every true mathematical statement at least has a good plausibility argument?

Maybe you see where this is going. Letting P be a program that accepts or rejects proposed plausibility arguments, we can construct

S(P) = “This sentence has no argument for its plausibility accepted by P.”

If S(P) is true, then it’s an example of a true arithmetical statement without even a plausibility argument for its truth (!). If, on the other hand, S(P) is false, then there is a plausibility argument for it. By itself, this is not at all a fatal problem: all sorts of false statements (IP≠PSPACE, switching doors doesn’t matter in Monty Hall, Trump couldn’t possibly become president…) have had decent plausibility arguments. Having said that, it’s pretty strange that you can have a plausibility argument that’s immediately contradicted by its own existence! This rules out some properties that you might want your “plausibility system” to have, although maybe a plausibility system exists that’s still nontrivial and that has weaker properties.

Anyway, I don’t know where I’m going with this, or even why I posted it, but I hope you enjoyed it! And maybe there’s something to be discovered in this direction.

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I’m proud to say that Nick Hunter-Jones and Matteo Ippoliti—both of whom work at the interface between quantum information science and condensed-matter physics (Nick closer to the former and Matteo to the latter)—have joined the physics faculty at UT Austin this year. And Nick, Matteo, and I are jointly seeking postdocs to start in Fall 2023! Please check out our call for applications here. The deadline is December 1; you apply through AcademicJobsOnline rather than by emailing me as in past years.

The big news in AI and complexity theory this week was DeepMind’s AlphaTensor, and its automated discovery of new algorithms for matrix multiplication. (See here for the Nature paper.) More concretely, they’ve used AI to discover (among other things) an algorithm for multiplying 4×4 matrices, over finite fields of characteristic 2, using only 47 scalar multiplications. This beats the 49=7×7 that you’d get from Strassen’s algorithm. There are other improvements for other matrix dimensions, many of which work over fields of other characteristics.

Since I’ve seen confusion about the point on social media: this does not improve over the best known asymptotic exponent for matrix multiplication, which over any field, still stands at the human-discovered 2.373 (meaning, we know how to multiply two N×N matrices in O(N^2.373) time, but not faster). But it does asymptotically improve over Strassen’s O(N^2.81) algorithm from 1968, conceivably even in a way that could have practical relevance for multiplying hundreds-by-hundreds or thousands-by-thousands matrices over F₂.

Way back in 2007, I gave a talk at MIT CSAIL’s “Wild and Crazy Ideas Session,” where I explicitly proposed to use computer search to look for faster algorithms for 4×4 and 5×5 matrix multiplication. The response I got at the time was that it was hopeless, since the search space was already too huge. Of course, that was before the deep learning revolution.

This morning, the Wall Street Journal published an article by Karen Hao about competition between China and the US in quantum computing. Unfortunately paywalled, but includes the following passage:

Meanwhile, American academics say it’s gotten harder for Chinese students to obtain visas to conduct quantum research in the U.S. “It’s become common knowledge that when Chinese students or postdocs come to the U.S., they can’t say they’re doing quantum computing,” says Scott Aaronson, director of the Quantum Information Center at the University of Texas, Austin.

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

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

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

(2) Live long enough.

Huge congratulations to Aspect, Clauser, and Zeilinger!

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

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(1) Since I didn’t blog about this before: huge congratulations to David Deutsch, Charles Bennett, Gilles Brassard, and my former MIT colleague Peter Shor, and separately to Dan Spielman, for their well-deserved Breakthrough Prizes! Their contributions are all so epochal, so universally known to all of us in quantum information and theoretical computer science, that there’s little I can write to gild the lily, except to say how much I’ve learned by interacting with all five of them personally. I did enjoy this comment on the Breakthrough Prizes by someone on Twitter: “As long as that loudmouth Scott Aaronson keeps getting ignored, I’ll be happy.”

(2) My former UT colleague Ila Fiete brought to my attention an important scientists’ petition to the White House. The context is that the Biden administration has announced new rules requiring federally-funded research papers to be freely available to the public without delay. This is extremely welcome—in fact, I’ve advocated such a step since I first became aware of the scourge of predatory journals around 2004. But the looming danger is that publishers will just respond by leaning more heavily on the “author pays” model—i.e., hitting up authors or their institutions for thousands of dollars in page fees—and we’ll go from only the credentialed few being able to read papers that aren’t on preprint archives or the like, to only the credentialed few being able to publish them. The petition urges the White House to build, or fund the research community to build, an infrastructure that will make scientific publishing truly open to everyone. I’ve signed it, and I hope you’ll consider signing too.

(3) Bill Gasarch asked me to announce that he, my former MIT colleague Erik Demaine, and Mohammad Hajiaghayi have written a brand-new book entitled Computational Intractability: A Guide to Algorithmic Lower Bounds, and a free draft is available online. It looks excellent, like a Garey & Johnson for the 21st century. Bill and his coauthors are looking for feedback. I was happy to help them by advertising this—after all, it’s not as if Bill’s got his own complexity blog for such things!

(4) Back when Google was still a novelty—maybe 2000 or so—I had my best friend, the now-famous computer security researcher Alex Halderman, over for Rosh Hashanah dinner with my family. Alex and I were talking about how Google evaded the limitations of all the previous decades’ worth of information retrieval systems. One of my relatives, however, misheard “Google” as “kugel” (basically a dense block of noodles held together with egg), and so ended up passing the latter to Alex. “What is this?” Alex asked. Whereupon my uncle deadpanned, “it’s a noodle retrieval system.” Since then, every single Rosh Hashanah dinner, I think about querying the kugel to retrieve the noodles within, and how the desired search result is just the trivial “all of them.”

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As I slept fitfully, still recovering from COVID, I had one of the more interesting dreams of my life:

I was desperately trying to finish some PowerPoint slides in time to give a talk. Uncharacteristically for me, one of the slides displayed actual code. This was a dream, so nothing was as clear as I’d like, but the code did something vaguely reminiscent of Rosser’s Theorem—e.g., enumerating all proofs in ZFC until it finds the lexicographically first proof or disproof of a certain statement, then branching into cases depending on whether it’s a proof or a disproof. In any case, it was simple enough to fit on one slide.

Suddenly, though, my whole presentation was deleted. Everything was ruined!

One of the developers of PowerPoint happened to be right there in the lecture hall (of course!), so I confronted him with my laptop and angrily demanded an explanation. He said that I must have triggered the section of Microsoft Office that tries to detect and prevent any discussion of logical paradoxes that are too dangerous for humankind—the ones that would cause people to realize that our entire universe is just an illusion, a sandbox being run inside an AI, a glitch-prone Matrix. He said it patronizingly, as if it should’ve been obvious: “you and I both know that the Paradoxes are not to be talked about, so why would you be so stupid as to put one in your presentation?”

My reaction was to jab my finger in the guy’s face, shove him, scream, and curse him out. At that moment, I wasn’t concerned in the slightest about the universe being an illusion, or about glitches in the Matrix. I was concerned about my embarrassment when I’d be called in 10 minutes to give my talk and would have nothing to show.

My last thought, before I woke with a start, was to wonder whether Greg Kuperberg was right and I should give my presentations in Beamer, or some other open-source software, and then I wouldn’t have had this problem.

A coda: I woke a bit after 7AM Central and started to write this down. But then—this is now real life (!)—I saw an email saying that a dozen people were waiting for me in a conference room in Europe for an important Zoom meeting. We’d gotten the time zones wrong; I’d thought that it wasn’t until 8AM my time. If not for this dream causing me to wake up, I would’ve missed the meeting entirely.

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1. The same thing Salman Rushdie learned: either you spend your entire life in hiding, or eventually it’ll come for you. Years might pass. You might emerge from hiding once, ten times, a hundred times, be fine, and conclude (emotionally if not intellectually) that the danger must now be over, that if it were going to come at all then it already would have, that maybe you’re even magically safe. But this is just the nature of a Poisson process: 0, 0, 0, followed by 1.
2. First comes the foreboding (in my case, on the flight back home from the wonderful CQIQC meeting in Toronto)—“could this be COVID?”—the urge to reassure yourself that it isn’t, the premature relief when the test is negative. Only then, up to a day later, comes the second vertical line on the plastic cartridge.
3. I’m grateful for the vaccines, which have up to a 1% probability of having saved my life. My body was as ready for this virus as my brain would’ve been for someone pointing a gun at my head and demanding to know a proof of the Karp-Lipton Theorem. All the same, I wish I also could’ve taken a nasal vaccine, to neutralize the intruder at the gate. Through inaction, through delays, through safetyism that’s ironically caused millions of additional deaths, the regulatory bureaucracies of the US and other nations have a staggering amount to answer for.
4. Likewise, Paxlovid should’ve been distributed like candy, so that everyone would have a supply and could start the instant they tested positive. By the time you’re able to book an online appointment and send a loved one to a pharmacy, a night has likely passed and the Paxlovid is less effective.
5. By the usual standards of a cold, this is mild. But the headaches, the weakness, the tiredness … holy crap the tiredness. I now know what it’s like to be a male lion or a hundred-year-old man, to sleep for 20 hours per day and have that feel perfectly appropriate and normal. I can only hope I won’t be one of the long-haulers; if I were, this could be the end of my scientific career. Fortunately the probability seems small.
6. You can quarantine in your bedroom, speak to your family only through the door, have meals passed to you, but your illness will still cast a penumbra on everyone around you. Your spouse will be stuck watching the kids alone. Other parents won’t let their kids play with your kids … and you can’t blame them; you’d do the same in their situation.
7. It’s hard to generalize from a sample size of 1 (or 2 if you count my son Daniel, who recovered from a thankfully mild case half a year ago). Readers: what are your COVID stories?

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Back in January, you might recall, Skype cofounder Jaan Tallinn’s Survival and Flourishing Fund (SFF) was kind enough to earmark $200,000 for me to donate to any charitable organizations of my choice. So I posted a call for proposals on this blog. You “applied” to my “foundation” by simply sending me an email, or leaving a comment on this blog, with a link to your organization’s website and a 1-paragraph explanation of what you wanted the grant for, and then answering any followup questions that I had.

After receiving about 20 awesome proposals in diverse areas, in the end I decided to split the allotment among organizations around the world doing fantastic, badly-needed work in math and science enrichment at the precollege level. These included Canada/USA Mathcamp, AddisCoder, a magnet school in Maine, a math circle in Oregon, a math enrichment program in Ghana, and four others. I chose to focus on advanced precollege STEM education both because I have some actual knowledge and experience there, and because I wanted to make a strong statement about an underfunded cause close to my heart that’s recently suffered unjust attacks.

To quote the immortal Carl Sagan, from shortly before his death:

[C]hildren with special abilities and skills need to be nourished and encouraged. They are a national treasure. Challenging programs for the “gifted” are sometimes decried as “elitism.” Why aren’t intensive practice sessions for varsity football, baseball, and basketball players and interschool competition deemed elitism? After all, only the most gifted athletes participate. There is a self-defeating double standard at work here, nationwide.

Anyway, the thank-you notes from the programs I selected were some of the most gratifying emails I’ve ever received.

But wait, it gets better! After reading about the Scott Aaronson Speculation Grants on this blog, representatives from a large, reputable family foundation contacted me to say that they wanted to be involved too. This foundation, which wishes to remain anonymous at this stage although not to the potential grant recipient, intends to make a single US$250,000 grant in the area of advanced precollege STEM education. They wanted my advice on where their grant should go.

Of course, I could’ve simply picked one of the same wonderful organizations that SFF and I helped in the first round. On reflection, though, I decided that it would be more on the up-and-up to issue a fresh call for proposals.

So: do you run a registered 501(c)(3) nonprofit dedicated to advanced precollege STEM education? If so, email me or leave a comment here by Friday, September 9, telling me a bit about what your organization does and what more it could do with an extra $250K. Include a rough budget, if that will help convince me that you can actually make productive use of that amount, that it won’t just sit in your bank account. Organizations that received a Scott Aaronson Speculation Grant the last time are welcome to reapply; newcomers are also welcome.

I’ll pass up to three finalists along to the funder, which will then make a final decision as to the recipient. The funder will be directly in touch with the potential grantee(s) and will proceed with its intake, review and due diligence process.

We expect to be able to announce a recipient on or around October 24. Can’t wait to see what people come up with!

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Here they are [PDF].

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

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

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

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

Extra bonus topics include:

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

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

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

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

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

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

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