Beyond fiction

August 8th, 2018

I now know firsthand what it’s like to be arrested by armed police officers, handcuffed, and sharply interrogated, while one’s wife and children look on helplessly.  This is not a prank post.

It happened in Philadelphia International Airport.  As someone who was born in Philadelphia, and who’s since visited ~40 countries on 6 continents and flies every week or two, I’ve long considered PHL possibly the most depressing airport on the planet (and the competition is fierce).

I’d just eaten dinner with my wife Dana and our two kids in a food court—after a day of travel that had already, before this happened, involved a missed flight and a lost suitcase, owing to a chain of mishaps that I’d (probably melodramatically) been describing to Dana as insane beyond the collective imagination of Homer and Shakespeare and Tolstoy and the world’s other literary giants to invent.  Again, that was before my arrest.

Two large uniformed men with holstered pistols saw me as we were exiting the airport, surrounded and handcuffed me, and demanded that I confess.

“I’m … sorry, officers,” I managed.  “I don’t understand what this is about.”

“Stop the games.  You know exactly what you took.  We have it all on video.  Where is it?”

Me, a thief?  I felt terrified to be at the beginning of a Kafka story.  But if I’m going to be brutally honest about it, I also felt … secretly vindicated in my irrational yet unshakeable beliefs that

  1. the laws of probability are broken, capricious horribleness reigning supreme over the universe,
  2. I’m despised by a large fraction of the world just for being who I am, and
  3. it’s only a matter of time until big, scary armed guys come for me, as they came for so many other nerdy misfits.

I almost wanted to say to the police: where have you been?  I’ve been expecting you my whole life.  And I wanted to say to Dana: you see??  see what I’ve been telling you all these years, about the nature of the universe we were born into?

Dana, for her part, was remonstrating with the officers that there must be some misunderstanding, that her husband was often absentminded but it’s completely impossible that he stole anything.  The officers brushed her away, told her to remove the kids from the situation.

“Are you gonna come clean?” one of the cops barked at me.  “We know you took it.”

“I didn’t take anything.”  Then I thought it over more.  “Or if somehow I did … then I’m certain that it would’ve been an accident, and I’d be more than happy to fix the…”

“Wait, if you did?  It sounds like you just confessed!”

“No, I definitely didn’t steal anything.  I’m just saying it’s possible that I might have mistakenly…”

“Your answers are rambling and all over the place.  Stop making up stories.  We know you did it.”

I’m not proud of myself for the next part, but the officers were so serious, and somehow I had to make them realize the sheer comical absurdity of what was happening.  “Look, I’m a computer science professor,” I said.  “I’ve never stolen a penny in my life, and it’s not something I’d ever…”

“Yeah, well I’m a police officer.  I’ve seen a lot in my thirty years in this job.  This is not about who you are, it’s about what you did.”

But what did I do?  After many more attempts to intimidate me, I was finally informed of the charge: “that smoothie place over there says you stole cash from their tip jar.”  Huh? How much?  One of the officers returned from the smoothie bar, and said, a bit sheepishly: “they say it was $4.”

Now a vague recollection came into sharper focus.  Yes, I had bought a berry smoothie for my daughter and a sparkling grapefruit juice for me.  I’d paid with a debit card, for reasons that I don’t remember, even though I normally pay cash.  My mind was elsewhere: on the missed flight, the lost suitcase, the brazen behavior of American Airlines (about which more later).  Then, completely forgetting I hadn’t paid cash this time, I looked down for my change: $4 in an unmarked plastic change cup.  I collected the change, put it in my wallet, then completely forgot about it.

After a minute, an employee angrily pointed down at a tray that the plastic cup was on (though not clearly at the cup itself), and said “hey, the tips go here!”  So I took a dollar from my wallet and put it on the tray.  I thought: this guy has some chutzpah, to demand a tip, and for an over-the-counter smoothie!  But whatever, he probably needs the dollar more than I do.  So if it will make him stop being angry…

But he was still angry.  He repeated: “this here is for tips!”

I said something to the effect of: “yeah, I know–that’s what you just told me, isn’t it?  So that’s why I just left you a tip!”  Sheesh.

At no point did he ever say, “you accidentally took from the tip jar,” or any other statement that would’ve clarified his meaning.

As I turned and walked away, I thought: yes, this is the strange world I was born into.  A world where people yell at me for not tipping at a smoothie bar–is that expected? I didn’t think it was–and then continue yelling even after I do.  But what did I expect?  Did I expect, as a nerdy outsider, to be able to buy normal people’s toleration with mere money?

As soon as I figured out what had happened, of course I offered to pay back the smoothie bar, not merely the $3 I still owed them, but $40 or whatever other amount would express my goodwill and compensate them for their trouble.  But the smoothie bar returned the $40 that I’d asked Dana to give them—I was unable to bring it myself on account of being handcuffed—and refused to press charges.  (In fact, apparently the employees hadn’t wanted to involve the police at all.  It was the manager, who hadn’t seen what happened, who’d insisted on it.)

So with no case, the police finally had no choice but to let me go–though not before giving me a stern lecture about never again putting my hands on stuff that’s not mine.

A week later, I’m still processing the experience.  In the rest of the post, I’d like to reflect on some lessons I think I learned from it.

First, it’s said that “a conservative is a liberal who’s been mugged; a liberal is a conservative who’s been arrested.”  It’s true: there are aspects of being arrested that are hard to understand until you’ve been through it.  While I’m white (well, insofar as Ashkenazim are), and while both officers who interrogated me happened to be African-Americans, what I went through further increased my sympathy for the many minority victims of aggressive policing.  Sitting in your armchair, it’s easy to think: in a liberal democracy, as long you know you did nothing wrong, even if you got arrested, frisked, detained, there’d probably be no real need to panic.  All you’d need to do is calmly clear up the misunderstanding and be back on your merry way.

But at least in my experience, an actual arrest isn’t like that.  The presumption of innocence, Miranda rights, all the things you might learn about in civics class—none of it seems to play any role.  From the very beginning, there’s an overwhelming presumption of guilt.  Everything you say gets interpreted as if you’re a red-handed criminal trying to fabricate a story, no matter how strained and how ludicrous such an interpretation might become.

And something strange happened: the officers seemed so certain I was guilty, that after only a few minutes I started to feel guilty.  I still had only a hazy sense of my “crime,” but I knew I was going to be punished for it, and I only hoped that the punishment wouldn’t tear me away from my family and previous life forever.

I came away from this incident with a visceral feel for just how easy it would be to procure a false confession from someone, which I didn’t have before but which will now stay with me as long as I live.

Second, it occurred to me that the sight of me, stuttering and potbellied complexity blogger, shackled and interrogated by armed policemen demanding that he confess to the theft of $3 from an airport stand, is a decent visual metaphor for much of my life.  If you doubt this, simply imagine Arthur Chu or Amanda Marcotte in place of the police officers.

It’s like: my accusers arrive on the scene committed to a specific, hostile theory of me: that I’m a petty thief of smoothie bars, let’s say, or a sexual-harassment-loving misogynist.  With all due modesty, people who know me might say that it’s not merely that I don’t fit the theory, that I happen to be innocent of the charge.  Rather, it’s that I’m one of the most astronomically, ridiculously unlikely people to fit the theory you could ever meet.  Not because I’m especially saintly, but simply because I already walk around all day feeling like my right to exist is conditional and might be revoked at any minute.  Breaking the normal people’s rules is the last thing on my agenda!  And yes, I still often feel that way, even as a professor with an endowed chair and awards and whatever.  The only times when I really relax, among strangers, is when everyone’s there to discuss ideas.

But my accusers don’t know any of that, or they refuse to believe it.  Everything I say gets interpreted in the light of the hostile theory, and therefore serves only as further confirmation of it.  Ironically—and this is key—the very unusual personality traits that make me so unlikely to be an offender, are also what throw off my accusers’ detection algorithms, and make them double down on their wrong theory.  When I’m trapped, I tend to fall back on the only tools I know: argument, openness, frank confession of my mistakes and failings, sometimes a little self-deprecating humor.  Unfortunately, I find this often backfires, as my accusers see in my vulnerability a golden opportunity to mount another wretched evildoer above their fireplace.

Or, to go even further out on a psychoanalytic limb: I sometimes get the sense that it gradually does dawn on my accusers that I’m not who they thought I was.  And then, far from prompting an apology, that realization seems to make my accusers even angrier, as if my throwing off their model of reality so badly, was an even worse offense than actually being guilty of whatever they thought!  A thief, a misogynist, they know how to handle.  But a living, breathing adversarial example for their worldview?

Dana, who watched the entire arrest, tells me that the central mistake I made was to try to reason with the police officers: “you say I took $3 that wasn’t mine?  If so, then I’m sure it was an accident, so let’s try to figure out what happened so we can fix it…”  In Dana’s view, what I saw as an earnest desire to get to the bottom of things, came across to grizzled cops only as evasiveness and guilt.  She says it would’ve been far better if I’d categorically denied: “no, I did not steal.  That’s completely absurd.  Please release me immediately.”

I’ve asked myself: how do you live in a world where, again and again, you can choose the hard right path over the easy wrong one, and then see your choice gleefully wielded against you?  Where you can spill your guts out to your accusers, in a desperate attempt to talk with them not as hardened warriors, but one confused and vulnerable human to another–and your reward is (to take one example) your picture in Salon above the headline “The Plight of the Bitter Nerd”?

The only way to live in such a world, as far as I can see, is to remind yourself that sometimes openness and vulnerability work.  In the course of my arrest, the two officers gradually differentiated themselves into a “good cop” and a “bad cop.”  While the “bad cop” treated me till the end like an unrepentant kleptomaniac being freed on a technicality, the “good cop,” who talked to me and Dana much more, became almost apologetic: “look man, when we get a call that someone stole money, we have to treat it like that’s the situation, you understand what I’m saying?  And then if it’s not, well then it’s not.”  Likewise, Arthur Chu recently tweeted that he’s “unhappy about [my] continued existence”–i.e., on a straightforward reading, that he wants me to die.  But I try to remind myself every day that the human race doesn’t consist solely of Arthur Chus (or Amanda Marcottes, or Lubos Motls, or SneerClub posters, or Paul Manaforts or Donald Trumps).  The world contains millions of women and men of every background and ideology who want actual dialogue, many of whom I’m lucky to count as friends, many of whom I met through this blog.  Vulnerability is possible because the world is not uniformly evil.

Third, I emerged from my arrest with a self-help technique that’s probably well-known to somebody, but that was new to me, and that I hope others will find as useful as I’m finding it.  Here it is: when something freakishly bad happens to you, draw a directed graph of all the known causes of the event, and the causes of the causes, and so forth as far back as you can trace them.  Also draw all the known measures that could have blocked the causal path leading to the bad event, and what prevented those measures from working or from being tried.

For example: why did I end up in handcuffs?  Firstly because, earlier in the day, Lily threw a temper tantrum that prevented us from packing and leaving for Logan Airport on time.  Because there was also heavy traffic on the way there.  Because we left from Harvard Square, and failed to factor in the extra 10 minutes to reach the airport, compared to if we’d left from MIT.  Because online check-in didn’t work.  Because when we did arrive, (barely) on time, the contemptuous American Airlines counter staff deliberately refused to check us in, chatting as we stewed impotently, so that we’d no longer be on time and they could legally give our seats away to others, and strand us in an airport with two young kids.  Because the only replacement flight was in a different terminal.  Because, in the stress of switching terminals–everything is stressful with two kids in an airport–I lost our suitcase.  Because the only shuttle to get back to the terminal went around the long way, and was slow as molasses, and by the time I returned our suitcase had been taken by the bomb squad.  Because the stress of such events bears down on me like an iron weight, and makes me unable to concentrate on the reality in front of me.  Because the guy at the smoothie counter and I failed to communicate.  Because the police chose to respond (or were trained to respond), not by politely questioning me to try to understand what had happened, but by handcuffing me and presuming guilt.

I actually drew the graph, filled a notebook page with it–and when I searched it for answers, neither I nor the world got off easily.  Looking over the strange chain of events that led to my arrest, I could find much to support my “default narrative,” that the laws of probability are broken and the universe is grotesquely awful.  But also, my belief in the universe’s grotesque awfulness clearly played a role in the events.  Had I been able maintain a calm demeanor, I would not have made so many mistakes.

Again and again, I screwed up.  Again and again, airport personnel responded to my honest mistakes with a maximum of cold bureaucracy rather than commonsense discussion: the booting from the flight, the bomb squad, the handcuffs.

We tend to think of bureaucracy as a mere nuisance, the person behind the counter at the Department of Motor Vehicles who makes you wait all day and then sends you home to get a different form of ID.  In my view, though, the bureaucratic impulse is one of the worst evils of which the human mind is capable.  It is, after all, the impulse that once sent trainloads of Jewish children to their deaths because that was the policy and there were no documents stating that any exception should be made in this case.  Today it’s the impulse that rounds up and deports people who’ve lived in the US for decades, sometimes served in the army, etc., and that separates screaming children from their parents.  To me, the mindset that willingly carries out such orders is almost more terrifying than the mindset that gives the orders in the first place.  I don’t mean to suggest, of course, that my arrest was even a trillionth as bad as those other things; at most I got a tiny, accidental taste of many less fortunate people’s daily reality.  But it’s worth remembering: every time you exercise official power over another person without even trying to talk it over first, clear up any honest misunderstandings, find out if there’s a reasonable explanation, you’re surrendering to one of the most destructive impulses in the history of civilization.

May we each strive to kill the bureaucrat in us and nurture the human being.

Unrelated Announcements:

I’m in Mexico City this week, to participate in a computer science and philosophy conference at UNAM and then give a broad quantum computing talk at CViCom 2018.  Because of this, responses to this post might be delayed.

(Update: But I’m having a wonderful time in Mexico!  Lots of delicious mole and horchata, and no arrests so far.  Today I gave my survey talk on P vs. NP.  I opened with the following icebreaker: “As a computer scientist speaking in a philosophy institute, I apologize that my talk will contain very little philosophy  Also, as an American speaking in Mexico, I apologize for our president.”)

My friend Elette Boyle asked me to announce that the 2018 CRYPTO conference, to be held in Santa Barbara, will be preceded by exciting workshops, including one that I’ll be speaking at myself entitled Beyond Crypto: A Theory Perspective.  Register now if you’re interested.

Huge congratulations to Costis Daskalakis, my former MIT colleague, for winning the Nevanlinna Prize for his work in algorithmic game theory!  While I don’t pretend to understand their work, congratulations to the four new Fields Medalists as well.

I put a new preprint online: Quantum Lower Bound for Approximate Counting Via Laurent Polynomials.

I’ve added a new blog to my blogroll: The Unit of Caring. I’ve been impressed by the author’s moral adeptness: when she addresses contentious debates among nerds, rationalists, feminists, SJWs, etc. etc., she often seems perfectly balanced on an atom-thin tightrope, even as some of us are plummetting left and right.

I forgot to mention this earlier, but I’m now a donor to the campaign of Beto O’Rourke, as he strives to unseat the quisling Ted Cruz in my adopted home state of Texas.  Americans: please consider donating as well!

Further Thoughts (Aug. 9):

  1. I wholeheartedly endorse an observation that many commenters (on this blog and elsewhere) made independently: that what really happened, is that I was forced to live out an episode of Seinfeld or Curb Your Enthusiasm.  To my detractors, I say the following: try for one minute to imagine how pathological, narcissistic, far outside the human norm, etc. etc. you could make Seinfeld or George or Kramer or Elaine seem, if their misadventures from any given episode were described and analyzed with clinical detachment.  (Or you were never a Seinfeld fan, then I guess this argument fails and we have nothing to say to each other.)
  2. I feel like some commenters are imposing their own after-the-fact knowledge (“c’mon, it was obviously a tip jar, he must be lying!”).  Dana, who’s generally more grounded than I am, saw their whole setup and agreed it was profoundly non-obvious that the tiny, unmarked plastic cup was supposed to be for tips, particularly to someone who was extremely stressed and not concentrating.  And when the employee later talked about tips, he didn’t indicate the cup so I didn’t make a connection.
  3. Most importantly: I wish to clarify that I don’t regard the police officers who handcuffed and interrogated me as having been “evil” in any sense.  I even took a liking to the “good cop,” the one who implicitly acknowledged the situation’s surreal absurdity by the end (although maybe that’s the whole point of a “good cop”?).  Having said that, I’m still rattled by the way the “bad cop” treated me as an unrepentant thief even to the end, even after the situation had been cleared up to everyone else’s satisfaction.  And I stand by my view that there was no need to handcuff me in front of my wife and young children, when I’d shown not a single subatomic particle of resistance.
  4. Speaking of which, let me now relate the most interesting and unexpected part of the reaction to my story.  Again and again, I found that fellow Americans, even nominally left-wing ones, sided with the police, said that I was crazy and guilty as charged and should’ve expected much worse, etc.  And again and again, commenters from Australia and New Zealand sided with me 300%, said that handcuffing someone over such a trivial mishap was a ludicrous overreaction, which would be totally unheard of in their countries and which confirms all the bad things they’ve heard about the US.  So maybe the rational conclusion is that I should be learning to enjoy vegemite in preparation for a move down under?

Summer recapitulates life

July 24th, 2018

Last week, I was back at the IAS in Princeton, to speak at a wonderful PITP summer school entitled “From Qubits to Spacetime,” co-organized by Juan Maldacena and Edward Witten. This week, I’ll be back in Waterloo, to visit old and new friends at the Perimeter Institute and Institute for Quantum Computing and give a couple talks.  Then, over the weekend, I’ll be back in Boston to see old friends, colleagues, and students.  After some other miscellaneous travel, I’ll then return to Austin in late August when the semester begins.  The particular sequence IAS → Waterloo → Boston → Austin is of course one that I’ve followed before, over a longer timescale.

Two quick announcements:

First, at the suggestion of reader Sanketh Menda, I’m thinking of holding a Shtetl-Optimized meetup in Waterloo this week.  Please send me an email if you’re interested, and we’ll figure out a time and place that work for everyone.

Second, many of the videos from the IAS summer school are now available, including mine: Part I and Part II.  I cover some basics of complexity theory, the complexity of quantum states and unitary transformations, the Harlow-Hayden argument about the complexity of turning a black hole event horizon into a firewall (with my refinement), and my and Lenny Susskind’s work on circuit complexity, wormholes, and AdS/CFT.  As a special bonus, check out the super-embarrassing goof at the beginning of my first lecture—claiming a mistaken symmetry of conditional entropy and even attributing it to Edward Witten’s lecture!  (But Witten, who I met for the first time on this visit, was kind enough to call my talk “lots of fun” anyway, and give me other positive comments, which I should put on my CV or something.)

Addendum: Many of the PITP videos are well worth watching!  As one example, I found Witten’s talks to be shockingly accessible.  I’d been to a previous talk of his involving Khovanov homology, but beyond the first few minutes, it went so far over my head that I couldn’t tell you how it was for its intended audience.  I’d also been to a popular talk of Witten’s on string theory, but that’s something he could do with only 3 awake brain cells.  In these talks, by contrast, Witten proves some basic inequalities of classical and quantum information theory, then proves the Reeh-Schlieder Theorem of quantum field theory and the Hawking and Penrose singularity theorems of GR, and finally uses quantum information theory to prove positive energy conditions from quantum field theory that are often needed to make statements about GR.

Customers who liked this quantum recommendation engine might also like its dequantization

July 12th, 2018

I’m in Boulder, CO right now for the wonderful Boulder summer school on quantum information, where I’ll be lecturing today and tomorrow on introductory quantum algorithms.  But I now face the happy obligation of taking a break from all the lecture-preparing and schmoozing, to blog about a striking new result by a student of mine—a result that will probably make an appearance in my lectures as well.

Yesterday, Ewin Tang—an 18-year-old who just finished a bachelor’s at UT Austin, and who will be starting a PhD in CS at the University of Washington in the fall—posted a preprint entitled A quantum-inspired classical algorithm for recommendation systems. Ewin’s new algorithm solves the following problem, very loosely stated: given m users and n products, and incomplete data about which users like which products, organized into a convenient binary tree data structure; and given also the assumption that the full m×n preference matrix is low-rank (i.e., that there are not too many ways the users vary in their preferences), sample some products that a given user is likely to want to buy.  This is an abstraction of the problem that’s famously faced by Amazon and Netflix, every time they tell you which books or movies you “might enjoy.”  What’s striking about Ewin’s algorithm is that it uses only polylogarithmic time: that is, time polynomial in log(m), log(n), the matrix rank, and the inverses of the relevant error parameters.  Admittedly, the polynomial involves exponents of 33 and 24: so, not exactly “practical”!  But it seems likely to me that the algorithm will run much, much faster in practice than it can be guaranteed to run in theory.  Indeed, if any readers would like to implement the thing and test it out, please let us know in the comments section!

As the title suggests, Ewin’s algorithm was directly inspired by a quantum algorithm for the same problem, which Kerenidis and Prakash (henceforth KP) gave in 2016, and whose claim to fame was that it, too, ran in polylog(m,n) time.  Prior to Ewin’s result, the KP algorithm was arguably the strongest candidate there was for an exponential quantum speedup for a real-world machine learning problem.  The new result thus, I think, significantly changes the landscape of quantum machine learning, by killing off one of its flagship applications.  (Note that whether KP gives a real exponential speedup was one of the main open problems mentioned in John Preskill’s survey on the applications of near-term quantum computers.)  At the same time, Ewin’s result yields a new algorithm that can be run on today’s computers, that could conceivably be useful to those who need to recommend products to customers, and that was only discovered by exploiting intuition that came from quantum computing. So I’d consider this both a defeat and a victory for quantum algorithms research.

This result was the outcome of Ewin’s undergraduate thesis project (!), which I supervised. A year and a half ago, Ewin took my intro quantum information class, whereupon it quickly became clear that I should offer this person an independent project.  So I gave Ewin the problem of proving a poly(m,n) lower bound on the number of queries that any classical randomized algorithm would need to make to the user preference data, in order to generate product recommendations for a given user, in exactly the same setting that KP had studied.  This seemed obvious to me: in their algorithm, KP made essential use of quantum phase estimation, the same primitive used in Shor’s factoring algorithm.  Without phase estimation, you seemed to be stuck doing linear algebra on the full m×n matrix, which of course would take poly(m,n) time.  But KP had left the problem open, I didn’t know how to solve it either, and nailing it down seemed like an obvious challenge, if we wanted to establish the reality of quantum speedups for at least one practical machine learning problem.  (For the difficulties in finding such speedups, see my essay for Nature Physics, much of which is still relevant even though it was written prior to KP.)

Anyway, for a year, Ewin tried and failed to rule out a superfast classical algorithm for the KP problem—eventually, of course, discovering the unexpected reason for the failure!  Throughout this journey, I served as Ewin’s occasional sounding board, but can take no further credit for the result.  Indeed, I admit that I was initially skeptical when Ewin told me that phase estimation did not look essential after all for generating superfast recommendations—that a classical algorithm could get a similar effect by randomly sampling a tiny submatrix of the user preference matrix, and then carefully exploiting a variant of a 2004 result by Frieze, Kannan, and Vempala.  So when I was in Berkeley a few weeks ago for the Simons quantum computing program, I had the idea of flying Ewin over to explain the new result to the experts, including Kerenidis and Prakash themselves.  After four hours of lectures and Q&A, a consensus emerged that the thing looked solid.  Only after that gauntlet did I advise Ewin to put the preprint online.

So what’s next?  Well, one obvious challenge is to bring down the running time of Ewin’s algorithm, and (as I mentioned before) to investigate whether or not it could give a practical benefit today.  A different challenge is to find some other example of a quantum algorithm that solves a real-world machine learning problem with only a polylogarithmic number of queries … one for which the exponential quantum speedup will hopefully be Ewin-proof, ideally even provably so!  The field is now wide open.  It’s possible that my Forrelation problem, which Raz and Tal recently used for their breakthrough oracle separation between BQP and PH, could be an ingredient in such a separation.

Anyway, there’s much more to say about Ewin’s achievement, but I now need to run to lecture about quantum algorithms like Simon’s and Shor’s, which do achieve provable exponential speedups in query complexity!  Please join me in offering hearty congratulations, see Ewin’s nicely-written paper for details, and if you have any questions for me or (better yet) Ewin, feel free to ask in the comments.

Update: On the Hacker News thread, some commenters are lamenting that such a brilliant mind as Ewin’s would spend its time figuring out how to entice consumers to buy even more products that they don’t need. I confess that that’s an angle that hadn’t even occurred to me: I simply thought that it was a beautiful question whether you actually need a quantum computer to sample the rows of a partially-specified low-rank matrix in polylogarithmic time, and if the application to recommendation systems helped to motivate that question, then so much the better. Now, though, I feel compelled to point out that, in addition to the potentially lucrative application to Amazon and Netflix, research on low-rank matrix sampling algorithms might someday find many other, more economically worthless applications as well.

Another Update: For those who are interested, streaming video of my quantum algorithms lectures in Boulder are now available:

You can also see all the other lectures here.

My Y Combinator podcast

June 29th, 2018

Here it is, recorded last week at Y Combinator’s office in San Francisco.  For regular readers of this blog, there will be a few things that are new—research projects I’ve been working on this year—and many things that are old.  Hope you enjoy it!  Thanks so much to Craig Cannon of Y Combinator for inviting me.

Associated with the podcast, Hacker News will be doing an AMA with me later today.  I’ll post a link to that when it’s available.  Update: here it is.

I’m at STOC’2018 TheoryFest in Los Angeles right now, where theoretical computer scientists celebrated the 50th anniversary of the conference that in some sense was the birthplace of the P vs. NP problem.  (Two participants in the very first STOC in 1969, Richard Karp and Allan Borodin, were on a panel to share their memories, along with Ronitt Rubinfeld and Avrim Blum, who joined the action in the 1980s.)  There’s been a great program this year—if you’d like to ask me about it, maybe do so in the comments of this post rather than in the AMA.

Ask me anything: moral judgments edition

June 17th, 2018

Reader Lewikee asked when I’d do another “Ask Me Anything.”  So fine, let’s do one now (and for the next 24 hours or so, or until I get too fatigued).  The rules:

  • This time around, only questions that ask me to render a moral judgment on some issue, which could be personal, political, or both (I answer plenty of quantum and complexity questions in the comments sections of other posts…)
  • One question per person total; no multipart questions or questions that require me to watch a video or read a linked document
  • Anything nasty, sneering, or non-genuine will be left in the moderation queue at my discretion

Let me get things started with the following judgment:

It is morally wrong to lie to parents that you’re taking their children away from them for 20 minutes to give them a bath, but then instead separate the children from their parents indefinitely, imprison the parents, and confine the children in giant holding facilities where they can no longer be contacted, as United States border agents are apparently now doing.  And yes, I know that people sometimes make such proclamations not out of genuine moral concern, but simply to virtue-signal for their chosen tribe and attack a rival tribe.  However, as someone who’s angered and offended nearly every tribe on his blog, I hope I might be taken at face value if I simply say: this is wrong.

Update (June 18): OK, thanks to everyone who participated! I’ll circle back to the few questions I haven’t yet gotten to, but no new questions please.

Five announcements

June 12th, 2018
  1. For the next two weeks, I’m in Berkeley for the Simons program “Challenges in Quantum Computation” (awesome program, by the way).  If you’re in the Bay Area and wanted to meet, feel free to shoot me an email (easiest for me if you come to Berkeley, though I do have a couple planned trips to SF).  If enough people wanted, we could even do a first-ever dedicated Shtetl-Optimized meetup.
  2. More broadly: I’m finally finished my yearlong sabbatical in Israel.  At some point I’ll do a post with my reflections on the experience.  I’ll now be traveling around North America all summer, then returning to UT Austin in the fall.
  3. Longtime friend-of-the-blog Boaz Barak, from a university in Cambridge, MA known as Harvard, asks me to invite readers to check out his new free draft textbook Introduction to Theoretical Computer Science, and to post comments about “typos, bugs, confusing explanations and such” in the book’s GitHub repository.  It looks great!
  4. This is already almost a month old, but if you enjoy the quantum computing content on this blog and wish to see related content from our carefully selected partners, check out John Preskill’s Y Combinator interview.
  5. Here’s the text of Senator Kamala Harris’s bill, currently working its way through the Senate, to create a US Quantum Computing Research Consortium.  Apparently there’s now also a second, competing quantum computing bill (!)—has anyone seen the text of that one?

Update (June 16): Even though I said there wouldn’t be a meetup, enough people eventually emailed wanting to have coffee that we did do the first-ever dedicated Shtetl-Optimized meetup after all—appropriately, given the title of the blog, at Saul’s Delicatessen in Berkeley. It was awesome. I met people working on fascinating and important things, from cheap nuclear energy to data analytics for downballot Democrats, and who I felt very proud to count as readers. Thanks so much to everyone who came; we’ll have to do another one sometime!

Quantum computing for policymakers and philosopher-novelists

June 6th, 2018

Last week Rebecca Newberger Goldstein, the great philosopher and novelist who I’m privileged to call a friend, wrote to ask me whether I “see any particular political and security problems that are raised by quantum computing,” to help her prepare for a conference she’d be attending in which that question would be discussed.  So I sent her the response below, and then decided that it might be of broader interest.

Shtetl-Optimized regulars and QC aficionados will find absolutely nothing new here—move right along, you’ve been warned.  But I decided to post my (slightly edited) response to Rebecca anyway, for two reasons.  First, so I have something to send anyone who asks me the same question in the future—something that, moreover, as Feynman said about the Feynman Lectures on Physics, contains views “not far from my own.”  And second, because, while of course I’ve written many other popular-level quantum computing essays, with basically all of them, my goal was to get the reader to hear the music, so to speak.  On reflection, though, I think there might also be some value in a piece for business and policy people (not to mention humanist intellectuals) that sets aside the harmony of the interfering amplitudes, and just tries to convey some of the words to the song without egregious howlers—which is what Rebecca’s question about “political and security problems” forced me to do.  This being quantum computing, of course, much of what one finds in the press doesn’t even get the lyrics right!  So without further ado:

Dear Rebecca,

If you want something serious and thoughtful about your question, you probably won’t do much better than the recent essay “The Potential Impact of Quantum Computers on Society,” by my longtime friend and colleague Ronald de Wolf.

To elaborate my own thoughts, though: I feel like the political and security problems raised by quantum computing are mostly the usual ones raised by any new technology (national prestige competitions, haves vs have-nots, etc)—but with one added twist, coming from quantum computers’ famous ability to break our current methods for public-key cryptography.

As Ronald writes, you should think of a quantum computer as a specialized device, which is unlikely to improve all or even most of what we do with today’s computers, but which could give dramatic speedups for a few specific problems.  There are three most important types of applications that we know about today:

(1) Simulation of quantum physics and chemistry. This was Richard Feynman’s original application when he proposed quantum computing in 1981, and I think it’s still the most important one economically.  Having a fast, general-purpose quantum simulator could help a lot in designing new drugs, materials, solar cells, high-temperature superconductors, chemical reactions for making fertilizer, etc.  Obviously, these are not applications like web browsing or email that will directly affect the everyday computer user.  But they’re areas where you’d only need a few high-profile successes to generate billions of dollars of value.

(2) Breaking existing public-key cryptography.  This is the most direct political and security implication.  Every time you visit a website that begins with “https,” the authentication and encryption—including, e.g., protecting your credit card number—happen using a cryptosystem based on factoring integers or discrete logarithms or a few other related problems in number theory.  A full, universal quantum computer, if built, is known to be able to break all of this.

Having said that, we all know today that hackers, and intelligence agencies, can compromise people’s data in hundreds of more prosaic ways than by building a quantum computer!  Usually they don’t even bother trying to break the encryption, relying instead on poor implementations and human error.

And it’s also important to understand that a quantum computer wouldn’t mean the end of online security.  There are public-key cryptosystems currently under development—most notably, those based on lattices—that are believed to resist attack even by quantum computers; NIST is planning to establish standards for these systems over the next few years.  Switching to these “post-quantum” systems would be a significant burden, much like fixing the Y2K bug (and they’re also somewhat slower than our current systems), but hopefully it would only need to happen once.

As you might imagine, there’s some interest in switching to post-quantum cryptosystems even now—for example, if you wanted to encrypt messages today with some confidence they won’t be decrypted even 30 years from now.  Google did a trial of a post-quantum cryptosystem two years ago.  On the other hand, given that a large fraction of web servers still use 512-bit “export grade” cryptography that was already breakable in the 1990s (good news: commenter Viktor Dukhovni tells me that this has now been mostly fixed, since security experts, including my childhood friend Alex Halderman, raised a stink about it a few years ago), it’s a safe bet that getting everyone to upgrade would take quite a long time, even if the experts agreed (which they don’t yet) which of the various post-quantum cryptosystems should become the new standard.  And since, as I said, most attacks target mistakes in implementation rather than the underlying cryptography, we should expect any switch to post-quantum cryptography to make security worse rather than better in the short run.

As a radical alternative to post-quantum crypto, there’s also (ironically enough) quantum cryptography, which doesn’t work over the existing Internet—it requires setting up new communications infrastructure—but which has already been deployed in a tiny number of places, and which promises security based only on quantum physics (and, of course, on the proper construction of the hardware), as opposed to mathematical problems that a quantum computer or any other kind of computer could potentially solve.  According to a long-running joke (or not-quite-joke) in our field, one of the central applications of quantum computing will be to create demand for quantum cryptography!

Finally, there’s private-key cryptography—i.e., the traditional kind, where the sender and recipient meet in secret to agree on a key in advance—which is hardly threatened by quantum computing at all: you can achieve the same level of security as before, we think, by simply doubling the key lengths.  If there’s no constraint on key length, then the ultimate here is the one-time pad, which when used correctly, is theoretically unbreakable by anything short of actual physical access to the sender or recipient (e.g., hacking their computers, or beating down their doors with an ax).  But while private-key crypto might be fine for spy agencies, it’s impractical for widespread deployment on the Internet, unless we also have a secure way to distribute the keys.  This is precisely where public-key crypto typically gets used today, and where quantum crypto could in principle be used in the future: to exchange private keys that are then used to encrypt and decrypt the actual data.

I should also mention that, because it breaks elliptic-curve-based signature schemes, a quantum computer might let a thief steal billions of dollars’ worth of Bitcoin.  Again, this could in principle be fixed by migrating Bitcoin (and other cryptocurrencies) to quantum-resistant cryptographic problems, but that hasn’t been done yet.

(3) Optimization and machine learning.  These are obviously huge application areas for industry, defense, and pretty much anything else.  The main issue is just that we don’t know how to get as large a speedup from a quantum computer as we’d like for these applications.  A quantum computer, we think, will often be able to solve optimization and machine learning problems in something like the square root of the number of steps that would be needed classically, using variants of what’s called Grover’s algorithm.  So, that’s significant, but it’s not the exponential speedup and complete game-changer that we’d have for quantum simulation or for breaking public-key cryptography.  Most likely, a quantum computer will be able to achieve exponential speedups for these sorts of problems only in special cases, and no one knows yet how important those special cases will be in practice.  This is a still-developing research area—there might be further theoretical breakthroughs (in inventing new quantum algorithms, analyzing old algorithms, matching the performance of the quantum algorithms by classical algorithms, etc.), but it’s also possible that we won’t really understand the potential of quantum computers for these sorts of problems until we have the actual devices and can test them out.


As for how far away all this is: given the spectacular progress by Google and others over the last few years, my guess is that we’re at most a decade away from some small, special-purpose quantum computers (with ~50-200 qubits) that could be useful for quantum simulation.  These are what the physicist John Preskill called “Noisy Intermediate Scale Quantum” (NISQ) computers in his excellent recent essay.

However, my guess is also that it will take longer than that to get the full, error-corrected, universal quantum computers that would be needed for optimization and (most relevant to your question) for breaking public-key cryptography.  Currently, the engineering requirements for a “full, universal” quantum computer look downright scary—so we’re waiting either for further breakthroughs that would cut the costs by a few more orders of magnitude (which, by their very nature, can’t be predicted), or for some modern-day General Groves and Oppenheimer who’d be licensed to spend however many hundreds of billions of dollars it would take to make it happen sooner.

The race to build “NISQ” devices has been heating up, with a shift from pure academic research to venture capitalists and industrial efforts just within the last 4-5 years, noticeably changing the character of our field.

In this particular race, I think that the US is the clear world leader right now—specifically, Google, IBM, Intel, Microsoft, University of Maryland / NIST, and various startups—followed by Europe (with serious experimental efforts in the Netherlands, Austria, and the UK among other places).  Here I should mention that the EU has a new 1-billion-Euro initiative in quantum information.  Other countries that have made or are now making significant investments include Canada, Australia, China, and Israel.  Surprisingly, there’s been very little investment in Russia in this area, and less than I would’ve expected in Japan.

China is a very interesting case.  They’ve chosen to focus less on quantum computing than on the related areas of quantum communication and cryptography, where they’ve become the world leader.  Last summer, in a big upset, China launched the first satellite (“Micius”) specifically for quantum communications, and were able to use it to do quantum cryptography and to distribute entanglement over thousands of miles (from one end of China to the other), the previous record being maybe 100 miles.  If the US has anything comparable to this, it isn’t publicly known (my guess is that we don’t).

This past year, there were hearings in Congress about the need for the US to invest more in quantum information, for example to keep up with China, and it looks likely to happen.  As indifferent or hostile as the current administration has been toward science more generally, the government and defense people I’ve met are very much on board with quantum information—often more so than I am!  I’ve even heard China’s Micius satellite referred to as the “quantum Sputnik,” the thing that will spur the US and others to spend much more to keep up.

As you’d imagine, part of me is delighted that something so abstruse, and interesting for fundamental science, and close to my heart, is now getting attention and funding at this level.  But part of me is worried by how much of the current boom I know to be fueled by misconceptions, among policymakers and journalists and the general public, about what quantum computers will be able to do for us once we have them.  Basically, people think they’ll be magic oracles that will solve all problems faster, rather than just special classes of problems like the ones I enumerated above—and that they’ll simply allow the continuation of the Moore’s Law that we know and love, rather than being something fundamentally different.  I’ve been trying to correct these misconceptions, on my blog and elsewhere, to anyone who will listen, for all the good that’s done!  In any case, the history of AI reminds us that a crash could easily follow the current boom-time, if the results of quantum computing research don’t live up to people’s expectations.

I guess there’s one final thing I’ll say.  Quantum computers are sometimes analogized to nuclear weapons, as a disruptive technology with implications for global security that scientists theorized about decades before it became technically feasible.  But there are some fundamental differences.  Most obviously: the deterrent value of a nuclear weapon comes if everyone knows you have it but you never need to use it, whereas the intelligence value of a quantum computer comes if you use it but no one knows you have it.

(Which is related to how the Manhattan Project entered the world’s consciousness in August 1945, whereas Bletchley Park—which was much more important to the actual winning of WWII—remained secret until the 1970s.)

As I said before, once your adversaries realized that you had a universal quantum computer, or might have one soon, they could switch to quantum-resistant forms of encryption, at least for their most sensitive secrets—in which case, as far as encryption was concerned, everyone would be more-or-less back where they started.  Such a switch would be onerous, cost billions of dollars, and (in practice) probably open up its own security holes unrelated to quantum computing.  But we think we already basically understand how to do it.

This is one reason why, even in a hypothetical future where hostile powers got access to quantum computers (and despite the past two years, I still don’t think of the US as a “hostile power”—I mean, like, North Korea or ISIS or something…!)—even in that future, I’d still be much less concerned about the hostile powers having this brand-new technology, than I’d be about their having the generations-old technology of fission and fusion bombs.


Unrelated Update (June 8): Ian Tierney asked me to advertise a Kickstarter for a short film that he’s planning to make about Richard Feynman, and a letter that he wrote to his first wife Arlene after she died.

The relativized BQP vs. PH problem (1993-2018)

June 3rd, 2018

Update (June 4): OK, I think the blog formatting issues are fixed now—thanks so much to Jesse Kipp for his help!

True story.  A couple nights ago, I was sitting in the Knesset, Israel’s parliament building, watching Gilles Brassard and Charles Bennett receive the Wolf Prize in Physics for their foundational contributions to quantum computing and information.  (The other laureates included, among others, Beilinson and Drinfeld in mathematics; the American honeybee researcher Gene Robinson; and Sir Paul McCartney, who did not show up for the ceremony.)

Along with the BB84 quantum cryptography scheme, the discovery of quantum teleportation, and much else, Bennett and Brassard’s seminal work included some of the first quantum oracle results, such as the BBBV Theorem (Bennett, Bernstein, Brassard, Vazirani), which proved the optimality of Grover’s search algorithm, and thus the inability of quantum computers to solve NP-complete problems in polynomial time in the black-box setting.  It thereby set the stage for much of my own career.  Of course, the early giants were nice enough to bequeath to us a few problems they weren’t able to solve, such as: is there an oracle relative to which quantum computers can solve some problem outside the entire polynomial hierarchy (PH)?  That particular problem, in fact, had been open from 1993 all the way to the present, resisting sporadic attacks by me and others.

As I sat through the Wolf Prize ceremony — the speeches in Hebrew that I only 20% understood (though with these sorts of speeches, you can sort of fill in the inspirational sayings for yourself); the applause as one laureate after another announced that they were donating their winnings to charity; the ironic spectacle of far-right, ultranationalist Israeli politicians having to sit through a beautiful (and uncensored) choral rendition of John Lennon’s “Imagine” — I got an email from my friend and colleague Avishay Tal.  Avishay wrote that he and Ran Raz had just posted a paper online giving an oracle separation between BQP and PH, thereby putting to rest that quarter-century-old problem.  So I was faced with a dilemma: do I look up, at the distinguished people from the US, Canada, Japan, and elsewhere winning medals in Israel, or down at my phone, at the bombshell paper by two Israelis now living in the US?

For those tuning in from home, BQP, or Bounded-Error Quantum Polynomial Time, is the class of decision problems efficiently solvable by a quantum computer.  PH, or the Polynomial Hierarchy, is a generalization of NP to allow multiple quantifiers (e.g., does there exist a setting of these variables such that for every setting of those variables, this Boolean formula is satisfied?).  These are two of the most fundamental complexity classes, which is all the motivation one should need for wondering whether the former is contained in the latter.  If additional motivation is needed, though, we’re effectively asking: could quantum computers still solve problems that were classically hard, even in a hypothetical world where P=NP (and hence P=PH also)?  If so, the problems in question could not be any of the famous ones like factoring or discrete logarithms; they’d need to be stranger problems, for which a classical computer couldn’t even recognize a solution efficiently, let alone finding it.

And just so we’re on the same page: if BQP ⊆ PH, then one could hope for a straight-up proof of the containment, but if BQP ⊄ PH, then there’s no way to prove such a thing unconditionally, without also proving (at a minimum) that P ≠ PSPACE.  In the latter case, the best we can hope is to provide evidence for a non-containment—for example, by showing that BQP ⊄ PH relative to a suitable oracle.  What’s noteworthy here is that even the latter, limited goal remained elusive for decades.

In 1993, Bernstein and Vazirani defined an oracle problem called Recursive Fourier Sampling (RFS), and proved it was in BQP but not in BPP (Bounded-Error Probabilistic Polynomial-Time).  One can also show without too much trouble that RFS is not in NP or MA, though one gets stuck trying to put it outside AM.  Bernstein and Vazirani conjectured—at least verbally, I don’t think in writing—that RFS wasn’t even in the polynomial hierarchy.  In 2003, I did some work on Recursive Fourier Sampling, but was unable to find a version that I could prove was outside PH.

Maybe this is a good place to explain that, by a fundamental connection made in the 1980s, proving that oracle problems are outside the polynomial hierarchy is equivalent to proving lower bounds on the sizes of AC0 circuits—or more precisely, constant-depth Boolean circuits with unbounded fan-in and a quasipolynomial number of AND, OR, and NOT gates.  And proving lower bounds on the sizes of AC0 circuits is (just) within complexity theory’s existing abilities—that’s how, for example, Furst-Saxe-Sipser, Ajtai, and Yao managed to show that PH ≠ PSPACE relative to a suitable oracle (indeed, even a random oracle with probability 1).  Alas, from a lower bounds standpoint, Recursive Fourier Sampling is a horrendously complicated problem, and none of the existing techniques seemed to work for it.  And that wasn’t even the only problem: even if one somehow succeeded, the separation that one could hope for from RFS was only quasipolynomial (n versus nlog n), rather than exponential.

Ten years ago, as I floated in a swimming pool in Cambridge, MA, it occurred to me that RFS was probably the wrong way to go.  If you just wanted an oracle separation between BQP and PH, you should focus on a different kind of problem—something like what I’d later call Forrelation.  The Forrelation problem asks: given black-box access to two Boolean functions f,g:{0,1}n→{0,1}, are f and g random and independent, or are they random individually but with each one close to the Boolean Fourier transform of the other one?  It’s easy to give a quantum algorithm to solve Forrelation, even with only 1 query.  But the quantum algorithm really seems to require querying all the f- and g-inputs in superposition, to produce an amplitude that’s a global sum of f(x)g(y) terms with massive cancellations in it.  It’s not clear how we’d reproduce this behavior even with the full power of the polynomial hierarchy.  To be clear: to answer the question, it would suffice to show that no AC0 circuit with exp(poly(n)) gates could distinguish a “Forrelated” distribution over (f,g) pairs from the uniform distribution.

Using a related problem, I managed to show that, relative to a suitable oracle—in fact, even a random oracle—the relational version of BQP (that is, the version where we allow problems with many valid outputs) is not contained in the relational version of PH.  I also showed that a lower bound for Forrelation itself, and hence an oracle separation between the “original,” decision versions of BQP and PH, would follow from something that I called the “Generalized Linial-Nisan Conjecture.”  This conjecture talked about the inability of AC0 circuits to distinguish the uniform distribution from distributions that “looked close to uniform locally.”  My banging the drum about this, I’m happy to say, initiated a sequence of events that culminated in Mark Braverman’s breakthrough proof of the original Linial-Nisan Conjecture.  But alas, I later discovered that my generalized version is false.  This meant that different circuit lower bound techniques, ones more tailored to problems like Forrelation, would be needed to go the distance.

I never reached the promised land.  But my consolation prize is that Avishay and Ran have now done so, by taking Forrelation as their jumping-off point but then going in directions that I’d never considered.

As a first step, Avishay and Ran modify the Forrelation problem so that, in the “yes” case, the correlation between f and the Fourier transform of g is much weaker (though still detectable using a quantum algorithm that makes nO(1) queries to f and g).  This seems like an inconsequential change—sure, you can do that, but what does it buy you?—but it turns out to be crucial for their analysis.  Ultimately, this change lets them show that, when we write down a polynomial that expresses an AC0 circuit’s bias in detecting the forrelation between f and g, all the “higher-order contributions”—those involving a product of k terms of the form f(x) or g(y), for some k>2—get exponentially damped as a function of k, so that only the k=2 contributions still matter.

There are a few additional ideas that Raz and Tal need to finish the job.  First, they relax the Boolean functions f and g to real-valued, Gaussian-distributed functions—very similar to what Andris Ambainis and I did when we proved a nearly-tight randomized lower bound for Forrelation, except that they also need to truncate f and g so they take values in [-1,1]; they then prove that a multilinear polynomial has no way to distinguish their real-valued functions from the original Boolean ones.  Second, they exploit recent results of Tal about the Fourier spectra of AC0 functions.  Third, they exploit recent work of Chattopadhyay et al. on pseudorandom generators from random walks (Chattopadhyay, incidentally, recently finished his PhD at UT Austin).  A crucial idea turns out to be to think of the values of f(x) and g(y), in a real-valued Forrelation instance, as sums of huge numbers of independent random contributions.  Formally, this changes nothing: you end up with exactly the same Gaussian distributions that you had before.  Conceptually, though, you can look at how each tiny contribution changes the distinguishing bias, conditioned on the sum of all the previous contributions; and this leads to the suppression of higher-order terms that we talked about before, with the higher-order terms going to zero as the step size does.

Stepping back from the details, though, let me talk about a central conceptual barrier—one that I know from an email exchange with Avishay was on his and Ran’s minds, even though they never discuss it explicitly in their paper.  In my 2009 paper, I identified what I argued was the main reason why no existing technique was able to prove an oracle separation between BQP and PH.  The reason was this: the existing techniques, based on the Switching Lemma and so forth, involved arguing (often implicitly) that

  1. any AC0 circuit can be approximated by a low-degree real polynomial, but
  2. the function that we’re trying to compute can’t be approximated by a low-degree real polynomial.

Linial, Mansour, and Nisan made this fully explicit in the context of their learning algorithm for AC0.  And this is all well and good if, for example, we’re trying to prove the n-bit PARITY function is not in AC0, since PARITY is famously inapproximable by any polynomial of sublinear degree.  But what if we’re trying to separate BQP from PH?  In that case, we need to deal with the fundamental observation of Beals et al. 1998: that any function with a fast quantum algorithm, by virtue of having a fast quantum algorithm, is approximable by a low-degree real polynomial!  Approximability by low-degree polynomials giveth with the one hand and taketh away with the other.

To be sure, I pointed out that this barrier wasn’t necessarily insuperable.  For the precise meaning of “approximable by low-degree polynomials” that follows from a function’s being in BQP, might be different from the meaning that’s used to put the function outside of PH.  As one illustration, Razborov and Smolensky’s AC0 lower bound method relates having a small constant-depth circuit to being approximable by low-degree polynomials over finite fields, which is different from being approximable by low-degree polynomials over the reals.  But this didn’t mean I knew an actual way around the barrier: I had no idea how to prove that Forrelation wasn’t approximable by low-degree polynomials over finite fields either.

So then how do Raz and Tal get around the barrier?  Apparently, by exploiting the fact that Tal’s recent results imply much more than just that AC0 functions are approximable by low-degree real polynomials.  Rather, they imply approximability by low-degree real polynomials with bounded L1 norms (i.e., sums of absolute values) of their coefficients.  And crucially, these norm bounds even apply to the degree-2 part of a polynomial—showing that, even all the way down there, the polynomial can’t be “spread around,” with equal weight on all its coefficients.  But being “spread around” is exactly how the true polynomial for Forrelation—the one that you derive from the quantum algorithm—works.  The polynomial looks like this:

$$ p(f,g) = \frac{1}{2^{3n/2}} \sum_{x,y \in \left\{0,1\right\}^n} (-1)^{x \cdot y} f(x) g(y). $$

This still isn’t enough for Raz and Tal to conclude that Forrelation itself is not in AC0: after all, the higher-degree terms in the polynomial might somehow compensate for the failures of the lower-degree terms.  But this difference between the two different kinds of low-degree polynomial—the “thin” kind that you get from AC0 circuits, and the “thick” kind that you get from quantum algorithms—gives them an opening that they’re able to combine with the other ideas mentioned above, at least for their noisier version of the Forrelation problem.

This difference between “thin” and “thick” polynomials is closely related to, though not identical with, a second difference, which is that any AC0 circuit needs to compute some total Boolean function, whereas a quantum algorithm is allowed to be indecisive on many inputs, accepting them with a probability that’s close neither to 0 nor to 1.  Tal used the fact that an AC0 circuit computes a total Boolean function, in his argument showing that it gives rise to a “thin” low-degree polynomial.  His argument also implies that no low-degree polynomial that’s “thick,” like the above quantum-algorithm-derived polynomial for Forrelation, can possibly represent a total Boolean function: it must be indecisive on many inputs.

The boundedness of the L1 norm of the coefficients is related to a different condition on low-degree polynomials, which I called the “low-fat condition” in my Counterexample to the Generalized Linial-Nisan Conjecture paper.  However, the whole point of that paper was that the low-fat condition turns out not to work, in the sense that there exist depth-three AC0 circuits that are not approximable by any low-degree polynomials satisfying the condition.  Raz and Tal’s L1 boundedness condition, besides being simpler, also has the considerable advantage that it works.

As Lance Fortnow writes, in his blog post about this achievment, an obvious next step would be to give an oracle relative to which P=NP but P≠BQP.  I expect that this can be done.  Another task is to show that my original Forrelation problem is not in PH—or more generally, to broaden the class of problems that can be handled using Raz and Tal’s methods.  And then there’s one of my personal favorite problems, which seems closely related to BQP vs. PH even though it’s formally incomparable: give an oracle relative to which a quantum computer can’t always prove its answer to a completely classical skeptic via an interactive protocol.

Since (despite my journalist moratorium) a journalist already emailed to ask me about the practical implications of the BQP vs. PH breakthrough—for example, for the ~70-qubit quantum computers that Google and others hope to build in the near future—let me take the opportunity to say that, as far as I can see, there aren’t any.  This is partly because Forrelation is an oracle problem, one that we don’t really know how to instantiate explicitly (in the sense, for example, that factoring and discrete logarithm instantiate Shor’s period-finding algorithm).  And it’s partly because, even if you did want to run the quantum algorithm for Forrelation (or for Raz and Tal’s noisy Forrelation) on a near-term quantum computer, you could easily do that sans the knowledge that the problem sits outside the polynomial hierarchy.

Still, as Avi Wigderson never tires of reminding people, theoretical computer science is richly interconnected, and things can turn up in surprising places.  To take a relevant example: Forrelation, which I introduced for the purely theoretical purpose of separating BQP from PH (and which Andris Ambainis and I later used for another purely theoretical purpose, to prove a maximal separation between randomized and quantum query complexities), now furnishes one of the main separating examples in the field of quantum machine learning algorithms.  So it’s early to say what implications Avishay and Ran’s achievement might ultimately have.  In any case, huge congratulations to them.

PDQP/qpoly = ALL

May 19th, 2018

I’ve put up a new paper.  Unusually for me these days, it’s a very short and simple one (8 pages)—I should do more like this!  Here’s the abstract:

    We show that combining two different hypothetical enhancements to quantum computation—namely, quantum advice and non-collapsing measurements—would let a quantum computer solve any decision problem whatsoever in polynomial time, even though neither enhancement yields extravagant power by itself. This complements a related result due to Raz. The proof uses locally decodable codes.

I welcome discussion in the comments.  The real purpose of this post is simply to fulfill a request by James Gallagher, in the comments of my Robin Hanson post:

The probably last chance for humanity involves science progressing, can you apply your efforts to quantum computers, which is your expertise, and stop wasting many hours of you [sic] time with this [expletive deleted]

Indeed, I just returned to Tel Aviv, for the very tail end of my sabbatical, from a weeklong visit to Google’s quantum computing group in LA.  While we mourned tragedies—multiple members of the quantum computing community lost loved ones in recent weeks—it was great to be among so many friends, and great to talk and think for once about actual progress that’s happening in the world, as opposed to people saying mean things on Twitter.  Skipping over its plans to build a 49-qubit chip, Google is now going straight for 72 qubits.  And we now have some viable things that one can do, or try to do, with such a chip, beyond simply proving quantum supremacy—I’ll say more about that in subsequent posts.

Anyway, besides discussing this progress, the other highlight of my trip was going from LA to Santa Barbara on the back of Google physicist Sergio Boixo’s motorcycle—weaving in and out of rush-hour traffic, the tightness of my grip the only thing preventing me from flying out onto the freeway.  I’m glad to have tried it once, and probably won’t be repeating it.

Update: I posted a new version of the PDQP/qpoly=ALL paper, which includes an observation about communication complexity, and which—inspired by the comments section—clarifies that when I say “all languages,” I really do mean “all languages” (even the halting problem).

The stupidest story I ever wrote (it was a long flight)

May 18th, 2018

All the legal maneuvers, the decades of recriminations, came down in the end to two ambiguous syllables.  No one knew why old man Memeson had named his two kids “Laurel” and “Yanny,” or why his late wife had gone along with it.  Not Laura, not Lauren, but Laurel—like, the leaves that the complacent rest on?  Poor girl.  And yet she lucked out compared to her younger brother. “Yanny”? Rhymes with fanny, seriously?  If you got picked on in school half as much as Yanny did, you too might grow up angry enough to spend half your life locked in an inheritance fight.

But people mostly tolerated the old man’s eccentricities, because he clearly knew something. All through the 1930s, Memeson Audio was building the highest-end radios and record players that money could buy.  And long after he’d outdone the competition, Memeson continued to outdo himself. At the 1939 New York World’s Fair, he proudly unveiled a prototype of his finest record player yet, the one he’d been tinkering with in his personal workshop for a decade: the Unmistakable.  Interviewed about it later, people who attended the demo swore that you couldn’t mishear a single syllable that came out of the thing if you were 99% deaf. No one had ever heard a machine like it—or would, perhaps, until the advent of digital audio.  On Internet forums, audiophiles still debate how exactly Memeson managed to do it with the technology of the time.  Alas, just like the other Memeson debate—about which more shortly—this one might continue indefinitely, since only one Unmistakable was ever built, and that World’s Fair was the last time anyone heard it.

The day after the triumphant demonstration, a crowd cheered as Memeson boarded a train in Grand Central Station to return to his factory near Chicago, there to supervise the mass production of Unmistakables. Meanwhile Laurel and Yanny, now both in their thirties and helping to run the family firm, stood on the platform and beamed. It hadn’t been easy to grow up with such a singleminded father, one who seemed to love his radios a million times more than them, but at a moment like this, it almost felt worth it.  When Laurel and Yanny returned to the Fair to continue overseeing the Memeson Audio exhibition, they’d be the highest-ranking representatives of the company, and would bask in their old man’s reflected glory.

In biographies, Memeson is described as a pathological recluse, who’d hole himself up in his workshop for days at a time, with strict orders not to be disturbed by anyone.  But on this one occasion—as it turned out, the last time he’d ever be seen in public—Memeson was as hammy as could be.  As the train pulled out of Grand Central, he leaned out of an open window in his private car and grinned for the cameras, waving with one arm and holding up the Unmistakable with the other.

Every schoolchild knows what happened next: the train derailed an hour later.  Along with twenty other passengers, Memeson was killed, while all that remained of his Unmistakable was a mess of wires and splintered wood.

Famously, there was one last exchange. As the train began moving, a journalist waved his hat at Memeson and called out “safe travels, sir!”

Memeson smiled and tipped his hat.

Then, noticing Laurel and Yanny on the platform, the journalist yelled to Memeson, in jest (or so he thought): “if something happens, which of these two is next in line to run the business?”

The old man had never been known for his sense of humor, and seemed from his facial expression (or so witnesses would later say) to treat the question with utmost seriousness. As the train receded into the distance, he shouted—well, everyone agrees that it was two syllables. But which? With no written will to consult—one of Memeson’s many idiosyncrasies was his defiance of legal advice—it all came down to what people heard, or believed, or believed they heard.

On the one hand, it would of course be extremely unusual back then for a woman to lead a major technology firm. And Memeson had never shown the slightest interest in social causes: not women’s suffrage, not the New Deal, nothing. In court, Yanny’s lawyers would press these points, arguing that the old man couldn’t possibly have intended to pass on his empire to a daughter.

On the other hand, Laurel was his first-born child.  And some people said that, if Memeson had ever had a human connection with anyone, it was with her.  There were even employees who swore that, once in a while, Laurel was seen entering and leaving her dad’s workshop—a privilege the old man never extended to Yanny or anyone else. Years later, Laurel would go so far as to claim that, during these visits, she’d contributed crucial ideas to the design of the Unmistakable. Most commentators dismiss this claim as bluster: why would she wait to drop such a bombshell until she and Yanny had severed their last ties, until both siblings’ only passion in life was to destroy the other, to make the world unable to hear the other’s name?

At any rate, neither Laurel nor anyone else was ever able to build another Unmistakable, or to give a comprehensible account of how it worked.  But Laurel certainly has die-hard defenders to this day—and while I’ve tried to be evenhanded in this account, I confess to being one of them.

In the end, who people believed about this affair seemed to come down to where they stood—literally. Among the passengers in the train cars adjoining Memeson’s, the ones who heard him are generally adamant that they heard “Laurel”; while most who stood on the platform are equally insistent about “Yanny.”  Today, some Memeson scholars theorize that this discrepancy is due to a Doppler effect.  People on the platform would’ve heard a lower pitch than people comoving with Memeson, and modern reconstructions raise the possibility, however farfetched, that this alone could “morph” one name to the other.  If we accept this, then it suggests that Memeson himself would have intended “Laurel”—but pitch changing a word?  Really?

Today, Laurel and Yanny are both gone, like their father and his company, but their dispute is carried on by their children and grandchildren, with several claims still winding their way through the courts.

Are there any recordings from the platform?  There is one, which was lost for generations before it unexpectedly turned up again. Alas, any hopes that this recording would definitively resolve the matter were … well, just listen to the thing.  Maybe the audio quality isn’t good enough.  Maybe an Unmistakable recording, had it existed, would’ve revealed the observer-independent truth, given us a unique map from the sensory world to the world of meaning.