Shtetl-Optimized

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.

Way back in the covid-filled summer of 2020, I wrote a survey article about the ridiculously-rapidly-growing Busy Beaver function. My survey then expanded to nearly twice its original length, with the ideas, observations, and open problems of commenters on this blog. Ever since, I’ve felt a sort of duty to blog later developments in BusyBeaverology as well. It’s like, I’ve built my dam, I’ve built my lodge, I’m here in the pond to stay!

So without further ado:

• This May, Pavel Kropitz found a machine demonstrating that $$ BB(6) \ge 10^{10^{10^{10^{10^{10^{10^{10^{10^{10^{10^{10^{10^{10^{10}}}}}}}}}}}}}} $$ (15 times)—thereby blasting through his own 2010 record, that BB(6)≥10^36,534. Or, for those tuning in from home: Kropitz constructed a 6-state, 2-symbol, 1-tape Turing machine that runs for at least the above number of steps, when started on an initially blank tape, and then halt. The machine was analyzed and verified by Pascal Michel, the modern keeper of Busy Beaver lore. In my 2020 survey, I’d relayed an open problem posed by my then 7-year-old daughter Lily: namely, what’s the first n such that BB(n) exceeds A(n), the n^th value of the Ackermann function? All that’s been proven is that this n is at least 5 and at most 18. Kropitz and Michel’s discovery doesn’t settle the question—titanic though it is, the new lower bound on BB(6) is still less than A(6) (!!)—but in light of this result, I now strongly conjecture that the crossover happens at either n=6 or n=7. Huge congratulations to Pavel and Pascal!

• Tristan Stérin and Damien Woods wrote to tell me about a new collaborative initiative they’ve launched called BB Challenge. With the participation of other leading figures in the neither-large-nor-sprawling Busy Beaver world, Tristan and Damien are aiming, not only to pin down the value of BB(5)—proving or disproving the longstanding conjecture that BB(5)=47,176,870—but to do so in a formally verified way, with none of the old ambiguities about which Turing machines have or haven’t been satisfactorily analyzed. In my survey article, I’d passed along a claim that, of all the 5-state machines, only 25 remained to be analyzed, to understand whether or not they run forever—the “default guess” being that they all do, but that proving it for some of them might require fearsomely difficult number theory. With their more formal and cautious approach, Tristan and Damien still count 1.5 million (!) holdout machines, but they hope to cut down that number extremely rapidly. If you’re feeling yourself successfully nerd-sniped, please join the quest and help them out!

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

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

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Hi Nikita,

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

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

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

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

Anyway, thanks again for sharing!

Best,
Scott

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

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

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

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

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

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

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

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

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

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

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

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

1. 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.
2. 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.
3. 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.
4. 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.”
5. 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.”