The first paper, with Guy Rothblum, is Gentle Measurement of Quantum States and Differential Privacy (85 pages, to appear in STOC’2019). This is Guy’s first paper that has anything to do with quantum, and also my first paper that has anything to do with privacy. (What do I care about privacy? I just share everything on this blog…) The paper has its origin when I gave a talk at the Weizmann Institute about “shadow tomography” (a task where you have to measure quantum states very carefully to avoid destroying them), and Guy was in the audience, and he got all excited that the techniques sounded just like what they use to ensure privacy in data-mining, and I figured it was just some wacky coincidence and brushed him off, but he persisted, and it turned out that he was 100% right, and our two fields were often studying the same problems from different angles and we could prove it. Anyway, here’s the abstract:

Indifferential privacy (DP), we want to query a database about n users, in a way that “leaks at most ε about any individual user,” even conditioned on any outcome of the query. Meanwhile, ingentle measurement, we want to measure n quantum states, in a way that “damages the states by at most α,” even conditioned on any outcome of the measurement. In both cases, we can achieve the goal by techniques like deliberately adding noise to the outcome before returning it. This paper proves a new and general connection between the two subjects. Specifically, we show that on products of n quantum states, any measurement that is α-gentle for small α is also O(α)-DP, and any product measurement that is ε-DP is also O(ε√n)-gentle.Illustrating the power of this connection, we apply it to the recently studied problem of

shadow tomography. Given an unknown d-dimensional quantum state ρ, as well as known two-outcome measurements E_{1},…,E_{m}, shadow tomography asks us to estimate Pr[E_{i}accepts ρ], foreveryi∈[m], by measuring few copies of ρ. Using our connection theorem, together with a quantum analog of the so-calledprivate multiplicative weightsalgorithm of Hardt and Rothblum, we give a protocol to solve this problem using O((log m)^{2}(log d)^{2}) copies of ρ, compared to Aaronson’s previous bound of ~O((log m)^{4}(log d)). Our protocol has the advantages of beingonline(that is, the E_{i}‘s are processed one at a time), gentle, and conceptually simple.Other applications of our connection include new

lowerbounds for shadow tomography from lower bounds on DP, and a result on the safe use of estimation algorithms as subroutines inside larger quantum algorithms.

The second paper, with Robin Kothari, UT Austin PhD student William Kretschmer, and Justin Thaler, is Quantum Lower Bounds for Approximate Counting via Laurent Polynomials. Here’s the abstract:

Given only a membership oracle for S, it is well-known that approximate counting takes Θ(√(N/|S|)) quantum queries. But what if a quantum algorithm is also given “QSamples”—i.e., copies of the state |S⟩=Σ_{i∈S}|i⟩—or even the ability to apply reflections about |S⟩? Our first main result is that, even then, the algorithm needs either Θ(√(N/|S|)) queries or else Θ(min{|S|^{1/3},√(N/|S|)}) reflections or samples. We also give matching upper bounds.We prove the lower bound using a novel generalization of the polynomial method of Beals et al. to

Laurent polynomials, which can have negative exponents. We lower-bound Laurent polynomial degree using two methods: a new “explosion argument” that pits the positive- and negative-degree parts of the polynomial against each other, and a new formulation of the dual polynomials method.Our second main result rules out the possibility of a black-box Quantum Merlin-Arthur (or QMA) protocol for proving that a set is large. More precisely, we show that, even if Arthur can make T quantum queries to the set S⊆[N], and also receives an m-qubit quantum witness from Merlin in support of S being large, we have Tm=Ω(min{|S|,√(N/|S|)}). This resolves the open problem of giving an oracle separation between SBP, the complexity class that captures approximate counting, and QMA.

Note that QMA is “stronger” than the queries+QSamples model in that Merlin’s witness can be anything, rather than just the specific state |S⟩, but also “weaker” in that Merlin’s witness cannot be trusted. Intriguingly, Laurent polynomials

alsoplay a crucial role in our QMA lower bound, but in a completely different manner than in the queries+QSamples lower bound. This suggests that the “Laurent polynomial method” might be broadly useful in complexity theory.

I need to get ready for our family’s Seder now, but after that, I’m happy to answer any questions about either of these papers in the comments.

Meantime, the biggest breakthrough in quantum complexity theory of the past month isn’t either of the above: it’s the paper by Anand Natarajan and John Wright showing that MIP*, or multi-prover interactive proof systems with entangled provers, contains NEEXP, or nondeterministic **doubly**-exponential time (!!). I’ll try to blog about this later, but if you can’t wait, check out this excellent post by Thomas Vidick.

It occurred to me that we could do something analogous for quantum computing. While my own deep-seated aversion to Twitter prevents me from doing it myself, which of my readers is up for starting an account that just reposts one overhyped QC article after another, while appending the words “A CLASSICAL COMPUTER COULD ALSO DO THIS” to each one?

]]>(If, for example, you were designing the US Constitution, how would you guard against a presidential candidate who *openly* supported and was supported by a hostile foreign power, and won anyway? Would it even occur to you to include such possibilities in your definitions of concepts like “treason” or “collusion”?)

The original Zionist project—the secular, democratic vision of Herzl and Ben-Gurion and Weizmann and Einstein, which the Nazis turned from a dream to a necessity—matters more to me than most things in this world, and that was true even before I’d spent time in Israel and before I had a wife and kids who are Israeli citizens. It would be depressing if, after a century of wildly improbable victories against external threats, Herzl’s project were finally to eat itself from the inside. Of course I have analogous worries (scaled up by two orders of magnitude) for the US—not to mention the UK, India, Brazil, Poland, Hungary, the Philippines … the world is now engaged in a massive test of whether Enlightenment liberalism can long endure, or whether it’s just a metastable state between one Dark Age and the next. (And to think that people have accused me of uncritical agreement with Steven Pinker, the world’s foremost optimist!)

In times like this, one takes one’s happiness where one can find it.

So, yeah: I’m happy that there’s now an “image of a black hole” (or rather, of radio waves being bent around a black hole’s silhouette). If you really want to understand what the now-ubiquitous image is showing, I strongly recommend this guide by Matt Strassler.

I’m happy that integer multiplication can apparently now be done in O(n log n), capping a decades-long quest (see also here).

I’m happy that there’s now apparently a spectacular fossil record of the first minutes after the asteroid impact that ended the Cretaceous period. Even better will be if this finally proves that, yes, some non-avian dinosaurs were still alive on impact day, and had not gone extinct due to unrelated climate changes slightly earlier. (Last week, my 6-year-old daughter sang a song in a school play about how “no one knows what killed the dinosaurs”—the verses ran through the asteroid and several other possibilities. I wonder if they’ll retire that song next year.) If you haven’t yet read it, the *New Yorker* piece on this is a must.

I’m happy that my good friend Zach Weinersmith (legendary author of SMBC Comics), as well as the GMU economist Bryan Caplan (rabblerousing author of *The Case Against Education*, which I reviewed here), have a new book out: a graphic novel that makes a moral and practical case for open borders (!). Their book is now available for pre-order, and at least at some point cracked Amazon’s top 10. Just last week I met Bryan for the first time, when he visited UT Austin to give a talk based on the book. He said that meeting me (having known me only from the blog) was like meeting a fictional character; I said the feeling was mutual. And as for Bryan’s talk? It felt great to spend an hour visiting a fantasyland where the world’s economies are run by humane rationalist technocrats, and where walls are going down rather than up.

I’m happy that, according to this *Vanity Fair* article, Facebook will still ban you for writing that “men are scum” *or* that “women are scum”—having ultimately rejected the demands of social-justice activists that it ban only the latter sentence, not the former. According to the article, everyone on Facebook’s Community Standards committee agreed with the activists that this was the right result: dehumanizing comments about women have no place on the platform, while (superficially) dehumanizing comments about men are an important part of feminist consciousness-raising that require protection. The problem was simply that the committee couldn’t come up with any *general principle* that would yield that desired result, without also yielding bad results in other cases.

I’m happy that the 737 Max jets are grounded and will hopefully be fixed, with no thanks to the FAA. Strangely, while this was still the top news story, I gave a short talk about quantum computing to a group of Boeing executives who were visiting UT Austin on a previously scheduled trip. The title of the session they put me in? “Disruptive Computation.”

I’m happy that Greta Thunberg, the 15-year-old Swedish climate activist, has attracted a worldwide following and might win the Nobel Peace Prize. I hope she does—and more importantly, I hope her personal story will help galvanize the world into accepting things that it already knows are true, as with the example of Anne Frank (or for that matter, Gandhi or MLK). Confession: back when I was an adolescent, I often daydreamed about doing exactly what Thunberg is doing right now, leading a worldwide children’s climate movement. I didn’t, of course. In my defense, I wouldn’t have had the charisma for it anyway—and also, I got sidetracked by even more pressing problems, like working out the quantum query complexity of recursive Fourier sampling. But fate seems to have made an excellent choice in Thunberg. She’s not the prop of any adult—just a nerdy girl with Asperger’s who formed the idea to sit in front of Parliament every day to protest the destruction of the world, because she couldn’t understand why everyone else wasn’t.

I’m happy that the college admissions scandal has finally forced Americans to confront the farcical injustice of our current system—a system where elite colleges pretend to peer into applicants’ souls (or the souls of the essay consultants hired by the applicants’ parents?), and where they preen about the moral virtue of their “holistic, multidimensional” admissions criteria, which amount in practice to shutting out brilliant working-class Asian kids in favor of legacies and rich badminton players. Not to horn-toot, but Steven Pinker and I tried to raise the alarm about this travesty five years ago (see for example this post), and were both severely criticized for it. I do worry, though, that people will draw *precisely* the wrong lessons from the scandal. If, for example, a few rich parents resorted to outright cheating on the SAT—all their other forms of gaming and fraud apparently being insufficient—then the SAT itself must be to blame so we should get rid of it. In reality, the SAT (whatever its flaws) is almost the only bulwark we have against the *complete* collapse of college admissions offices into nightclub bouncers. This *Quillette* article says it much better than I can.

I’m happy that there will a symposium from May 6-9 at the University of Toronto, to honor Stephen Cook and the (approximate) 50^{th} anniversary of the discovery of NP-completeness. I’m happy that I’ll be attending and speaking there. If you’re interested, there’s still time to register!

Finally, I’m happy about the following “Sierpinskitaschen” baked by CS grad student and friend-of-the-blog Jess Sorrell, and included with her permission (Jess says she got the idea from Debs Gardner).

]]>Congrats to Avi Wigderson for winning the Knuth Prize. When I was asked to write a supporting nomination letter, my first suggestion was to submit a blank sheet of paper—since for anyone in theoretical computer science, there’s nothing that needs to be said about why Avi should win any awards we have. I hope Avi remains a guiding light of our community for many years to come.

And congrats to Mark Braverman for winning the Alan T. Waterman Award, one that I have some personal fondness for, along with materials scientist Jennifer Dionne. As Sasha Razborov once put it, after he (Sasha), I, and others recoiled from the task of proving the Linial-Nisan Conjecture, that polylog-wise independent distributions are indistinguishable from uniform by AC^{0} circuits, a “braver man” stepped in to do the job.

Anyway, I’ve been thrashing for several weeks—just barely escaping spaghettification at the Email Event Horizon—but I hope to be back shortly with your regularly scheduled programming.

]]>If you’re simulating a time-reversible process on your computer, then you can ‘reverse the direction of time’ by simply reversing the direction of your simulation. From a quick look at the paper, I confess that I didn’t understand how this becomes more profound if the simulation is being done on IBM’s quantum computer.

Incredibly, the time-reversal claim has now gotten uncritical attention in *Newsweek*, *Discover*, *Cosmopolitan*, my Facebook feed, and elsewhere—hence this blog post, which has basically no content except “the claim to have ‘reversed time,’ by running a simulation backwards, is exactly as true and as earth-shattering as a layperson might think it is.”

If there’s anything interesting here, I suppose it’s just that “scientists use a quantum computer to reverse time” is one of the purest examples I’ve ever seen of a scientific claim that basically amounts to a mind-virus or meme optimized for sharing on social media—discarding all nontrivial “science payload” as irrelevant to its propagation.

]]>Today Manolis will be celebrating his 42nd birthday, with a symposium on the meaning of life (!). He asked his friends and colleagues to contribute talks and videos reflecting on that weighty topic.

Here’s a 15-minute video interview that Manolis and I recorded last night, where he asks me to pontificate about the implications of quantum mechanics for consciousness and free will and whether the universe is a computer simulation—and also about, uh, how to balance blogging with work and family.

Also, here’s a 2-minute birthday video that I made for Manolis before I really understood what he wanted. Unlike the first video, this one has no academic content, but it does involve me wearing a cowboy hat and swinging a makeshift “lasso.”

Happy birthday Manolis!

]]>Now Horgan—who you could variously describe as a wonderful sport, or a ham, or a sucker for punishment—has written a 26-year retrospective on his “death of proof” article. The prompt for this was Horgan’s recent discovery that, back in the 90s, David Hoffman and Hermann Karcher, two mathematicians annoyed by the “death of proof” article, had named a nonexistent mathematical object after its author. The so-called *Horgan surface* is a minimal surface that numerical computations strongly suggested should exist, but that can be rigorously proven not to exist after all. “The term was intended as an insult, but I’m honored anyway,” Horgan writes.

As a followup to his blog post, Horgan then decided to solicit commentary from various people he knew, including yours truly, about “how proofs are faring in an era of increasing computerization.” He wrote, “I’d love to get a paragraph or two from you.” Alas, I didn’t have the time to do as requested, but only to write *eight* paragraphs. So Horgan suggested that I make the result into a post on my own blog, which he’d then link to. Without further ado, then:

John, I like you so I hate to say it, but the last quarter century has not been kind to your thesis about “the death of proof”! Those mathematicians sending you the irate letters had a point: there’s been no fundamental change to mathematics that deserves such a dramatic title. Proof-based math remains quite healthy, with (e.g.) a solution to the Poincaré conjecture since your article came out, as well as to the Erdős discrepancy problem, the Kadison-Singer conjecture, Catalan’s conjecture, bounded gaps in primes, testing primality in deterministic polynomial time, etc. — just to pick a few examples from the tiny subset of areas that I know anything about.

There are evolutionary changes to mathematical practice, as there always have been. Since 2009, the website MathOverflow has let mathematicians query the global hive-mind about an obscure reference or a recalcitrant step in a proof, and get near-instant answers. Meanwhile “polymath” projects have, with moderate success, tried to harness blogs and other social media to make advances on long-standing open math problems using massive collaborations.

While humans remain in the driver’s seat, there are persistent efforts to increase the role of computers, with some notable successes. These include Thomas Hales’s 1998 computer-assisted proof of the Kepler Conjecture (about the densest possible way to pack oranges) — now fully machine-verified from start to finish, after ~~the ~~~~Annals of Mathematics~~~~ refused to publish a mixture of traditional mathematics and computer code~~ (seems this is not exactly what happened; see the comment section for more). It also includes William McCune’s 1996 solution to the Robbins Conjecture in algebra (the computer-generated proof was only half a page, but involved substitutions so strange that for 60 years no human had found them); and at the “opposite extreme,” the 2016 solution to the Pythagorean triples problem by Marijn Heule and collaborators, which weighed in at 200 terabytes (at that time, “the longest proof in the history of mathematics”).

It’s conceivable that someday, computers will replace humans at all aspects of mathematical research — but it’s also conceivable that, by the time they can do that, they’ll be able to replace humans at music and science journalism and everything else!

New notions of proof — including probabilistic, interactive, zero-knowledge, and even quantum proofs — have seen further development by theoretical computer scientists since 1993. So far, though, these new types of proof remain either entirely theoretical (as with quantum proofs), or else they’re used for cryptographic protocols but not for mathematical research. (For example, zero-knowledge proofs now play a major role in certain cryptocurrencies, such as Zcash.)

In many areas of math (including my own, theoretical computer science), proofs have continued to get longer and harder for any one person to absorb. This has led some to advocate a split approach, wherein human mathematicians would talk to each other only about the handwavy intuitions and high-level concepts, while the tedious verification of details would be left to computers. So far, though, the huge investment of time needed to write proofs in machine-checkable format — for almost no return in new insight — has prevented this approach’s wide adoption.

Yes, there are non-rigorous approaches to math, which continue to be widely used in physics and engineering and other fields, as they always have been. But none of these approaches have displaced proof as the gold standard whenever it’s available. If I had to speculate about why, I’d say: if you use non-rigorous approaches, then even if it’s clear to you under what conditions your results can be trusted, it’s probably much less clear to others. Also, even if only one segment of a research community cares about rigor, whatever earlier work that segment builds on will need to be rigorous as well — thereby exerting constant pressure in that direction. Thus, the more collaborative a given research area becomes, the more important is rigor.

For my money, the elucidation of the foundations of mathematics a century ago, by Cantor, Frege, Peano, Hilbert, Russell, Zermelo, Gödel, Turing, and others, still stands as one of the greatest triumphs of human thought, up there with evolution or quantum mechanics or anything else. It’s true that the ideal set by these luminaries remains mostly aspirational. When mathematicians say that a theorem has been “proved,” they still mean, as they always have, something more like: “we’ve reached a social consensus that all the ideas are now in place for a strictly formal proof that could be verified by a machine … with the only task remaining being massive rote coding work that none of us has any intention of ever doing!” It’s also true that mathematicians, being human, are subject to the full panoply of foibles you might expect: claiming to have proved things they haven’t, squabbling over who proved what, accusing others of lack of rigor while hypocritically taking liberties themselves. But just like love and honesty remain fine ideals no matter how often they’re flouted, so too does mathematical rigor.

**Update:** Here’s Horgan’s new post (entitled “Okay, Maybe Proofs Aren’t Dying After All”), which also includes a contribution from Peter Woit.

Ironically, *the SneerClubbers themselves* begged me to stop reading them (!), so presumably for once they’ll be okay with something I did (but if not, I don’t care). If any of them still have something to say to me, they can come to *this* blog, or email me, or if they pass through Austin, set up a time to hash it out over chips and queso (my treat). What I’ll no longer do is spend hours every week binge-reading a forum of people who’ve adopted nastiness and bad faith as their explicit principles. I’ll no longer toss and turn at night wondering how it came about that two thousand Redditors hate Scott Aaronson so much, and what I could say or do (short of total self-abnegation) that would make them hate me less. I plan to spend the freed-up time *being* Scott Aaronson.

Resolving to ignore one particular online hate pit—and then *sticking* to the resolution, as so far I have—has been a pure, unmitigated improvement to my quality of life. If you don’t believe me, ask my wife and kids. I recommend this course to anyone.

You could sensibly ask: why did I *ever* spend time worrying about an anti-nerds-like-me forum that’s so poisonous for its targets and participants alike? After long introspection, I think the answer is: there’s a part of me, perhaps a gift from the childhood bullies, that’s so obsessed with “society’s hatred of STEM nerds,” that it *constantly seeks out evidence to confirm that its fears are justified—*evidence that it can then wave in front of the rest of my brain to say “you see?? what did I always tell you?” And alas, whenever that part of my brain seeks such evidence, the world dutifully supplies mountains of it. It’s never once disappointed.

Now the SneerClubbers—who are perceptive and talented in their cruelty, if in nothing else—notice this about me, and gleefully ridicule me for it. But they’re oblivious to the central irony: that unlike the vast majority of humankind, or even the vast majority of social justice activists, they (the SneerClubbers) *really do* hate everyone like me. They’re *precisely* what the paranoid part of my brain wrongly fears that everyone else I meet is secretly like. They’re like someone who lectures you about your hilariously overblown fear of muggers, *while simultaneously mugging you*.

But at least they’re not the contented and self-confident bullies of my childhood nightmares, kicking dirt down at nerds from atop their pinnacle of wokeness and social adeptness. If you spend enough time studying them, they themselves come across as angry, depressed, pathetic. So for example: here’s one of my most persistent attackers, popping up on a math thread commemorating Michael Atiyah (one of the great mathematicians of the 20th century), just to insult Atiyah—randomly, gratuitously, and a few days after Atiyah had died. Almost everything posted all over Reddit by this individual—who uses the accurate tagline “unpleasantly radical”—has the same flavor. Somehow seeing this made it click for me: wait a second, *these* are the folks are lecturing *me* about my self-centeredness and arrogance and terrible social skills? Like, at least I *try* to be nice.

Scott Alexander, who writes the world’s best blog and is a more central target of SneerClub than I’ve been, recently announced that he asked the moderators of r/ssc to close its notorious “Culture War” thread, and they’ve done so—moving the thread to a new home on Reddit called “TheMotte.”

For those who don’t know: r/ssc is the place on Reddit to discuss Scott’s SlateStarCodex blog, though Scott himself was never too involved as more than a figurehead. The Culture War thread was the place within r/ssc to discuss race, gender, immigration, and other hot-button topics. The thread, which filled up with a bewildering thousands of comments per week (!), attracted the, err … *full range* of political views, including leftists, libertarians, and moderates but also alt-righters, neoreactionaries, and white nationalists. Predictably, SneerClub treated the thread as a gift from heaven: a constant source of inflammatory material that they could use to smear Scott personally (even if most of the time, Scott hadn’t even *seen* the offending content, let alone endorsing it).

Four months ago, I was one of the apparently many friends who told Scott that I felt he should dissociate the Culture War thread from his brand. So I congratulate him on his decision, which (despite his eloquently-expressed misgivings) I feel confident was the right one. Think about it this way: nobody’s freedom of speech has been curtailed—the thread continues full steam at TheMotte, for anyone who enjoys it—but meanwhile, the sneerers have been deprived of a golden weapon with which to slime Scott. Meanwhile, while the sneerers themselves might never change their minds about anything, Scott has demonstrated to third parties that he’s open and reasonable and ready to compromise, like the debater who happily switches to his opponent’s terminology. What’s not to like?

A couple weeks ago, while in Albuquerque for the SQuInT conference, I visited the excellent National Museum of Nuclear Science and History. It was depressing, as it should have been, to tour the detailed exhibits about the murderous events surrounding the birth of the nuclear era: the Holocaust, the Rape of Nanking, the bombings of Hiroshima and Nagasaki. It was depressing in a different way to tour the exhibits about the early Atomic Age, and see the boundless optimism that ‘unleashing the power of the atom’ would finally usher in a near-utopia of space travel and clean energy—and then to compare that vision to where we are now, with climate change ravaging the planet and (in a world-historic irony) the people who care most about the environment having denounced and marginalized the most reliable source of carbon-free energy, the one that probably had the best chance to avert our planet’s terrifying future.

But on the bright side: how wonderful to have born into a time and place when, for the most part, those who hate you have only the power to destroy your life that you yourself grant them. How wonderful when one can blunt their knives by simply refusing to open a browser tab.

]]>There’s an interview with me at the website “GigaOm,” conducted by Byron Reese and entitled Quantum Computing: Capabilities and Limits. I didn’t proofread the transcript and it has some errors in it, but hopefully the meaning comes through. In other interview news, if you were interested in my podcast with Adam Ford in Melbourne but don’t like YouTube, Adam has helpfully prepared transcripts of the two longest segments: The Ghost in the Quantum Turing Machine and The Winding Road to Quantum Supremacy.

The *New York Times* ran an article entitled The Hard Part of Computer Science? Getting Into Class, about the surge in computer science majors all over the US, and the shortage of professors to teach them. The article’s go-to example of a university where this is happening is UT Austin, and there’s extensive commentary from my department chair, Don Fussell.

The STOC’2019 accepted papers list is finally out. Lots of cool stuff!

]]>