Archive for the ‘Quantum’ Category

Is “information is physical” contentful?

Thursday, July 20th, 2017

“Information is physical.”

This slogan seems to have originated around 1991 with Rolf Landauer.  It’s ricocheted around quantum information for the entire time I’ve been in the field, incanted in funding agency reports and popular articles and at the beginnings and ends of talks.

But what the hell does it mean?

There are many things it’s taken to mean, in my experience, that don’t make a lot of sense when you think about them—or else they’re vacuously true, or purely a matter of perspective, or not faithful readings of the slogan’s words.

For example, some people seem to use the slogan to mean something more like its converse: “physics is informational.”  That is, the laws of physics are ultimately not about mass or energy or pressure, but about bits and computations on them.  As I’ve often said, my problem with that view is less its audacity than its timidity!  It’s like, what would the universe have to do in order not to be informational in this sense?  “Information” is just a name we give to whatever picks out one element from a set of possibilities, with the “amount” of information given by the log of the set’s cardinality (and with suitable generalizations to infinite sets, nonuniform probability distributions, yadda yadda).  So, as long as the laws of physics take the form of telling us that some observations or configurations of the world are possible and others are not, or of giving us probabilities for each configuration, no duh they’re about information!

Other people use “information is physical” to pour scorn on the idea that “information” could mean anything without some actual physical instantiation of the abstract 0’s and 1’s, such as voltage differences in a loop of wire.  Here I certainly agree with the tautology that in order to exist physically—that is, be embodied in the physical world—a piece of information (like a song, video, or computer program) does need to be embodied in the physical world.  But my inner Platonist slumps in his armchair when people go on to assert that, for example, it’s meaningless to discuss the first prime number larger than 1010^125, because according to post-1998 cosmology, one couldn’t fit its digits inside the observable universe.

If the cosmologists revise their models next week, will this prime suddenly burst into existence, with all the mathematical properties that one could’ve predicted for it on general grounds—only to fade back into the netherworld if the cosmologists revise their models again?  Why would anyone want to use language in such a tortured way?

Yes, brains, computers, yellow books, and so on that encode mathematical knowledge comprise only a tiny sliver of the physical world.  But it’s equally true that the physical world we observe comprises only a tiny sliver of mathematical possibility-space.

Still other people use “information is physical” simply to express their enthusiasm for the modern merger of physical and information sciences, as exemplified by quantum computing.  Far be it from me to temper that enthusiasm: rock on, dudes!

Yet others use “information is physical” to mean that the rules governing information processing and transmission in the physical world aren’t knowable a priori, but can only be learned from physics.  This is clearest in the case of quantum information, which has its own internal logic that generalizes the logic of classical information.  But in some sense, we didn’t need quantum mechanics to tell us this!  Of course the laws of physics have ultimate jurisdiction over whatever occurs in the physical world, information processing included.

My biggest beef, with all these unpackings of the “information is physical” slogan, is that none of them really engage with any of the deep truths that we’ve learned about physics.  That is, we could’ve had more-or-less the same debates about any of them, even in a hypothetical world where the laws of physics were completely different.


So then what should we mean by “information is physical”?  In the rest of this post, I’d like to propose an answer to that question.

We get closer to the meat of the slogan if we consider some actual physical phenomena, say in quantum mechanics.  The double-slit experiment will do fine.

Recall: you shoot photons, one by one, at a screen with two slits, then examine the probability distribution over where the photons end up on a second screen.  You ask: does that distribution contain alternating “light” and “dark” regions, the signature of interference between positive and negative amplitudes?  And the answer, predicted by the math and confirmed by experiment, is: yes, but only if the information about which slit the photon went through failed to get recorded anywhere else in the universe, other than the photon location itself.

Here a skeptic interjects: but that has to be wrong!  The criterion for where a physical particle lands on a physical screen can’t possibly depend on anything as airy as whether “information” got “recorded” or not.  For what counts as “information,” anyway?  As an extreme example: what if God, unbeknownst to us mortals, took divine note of which slit the photon went through?  Would that destroy the interference pattern?  If so, then every time we do the experiment, are we collecting data about the existence or nonexistence of an all-knowing God?

It seems to me that the answer is: insofar as the mind of God can be modeled as a tensor factor in Hilbert space, yes, we are.  And crucially, if quantum mechanics is universally true, then the mind of God would have to be such a tensor factor, in order for its state to play any role in the prediction of observed phenomena.

To say this another way: it’s obvious and unexceptionable that, by observing a physical system, you can often learn something about what information must be in it.  For example, you need never have heard of DNA to deduce that chickens must somehow contain information about making more chickens.  What’s much more surprising is that, in quantum mechanics, you can often deduce things about what information can’t be present, anywhere in the physical world—because if such information existed, even a billion light-years away, it would necessarily have a physical effect that you don’t see.

Another famous example here concerns identical particles.  You may have heard the slogan that “if you’ve seen one electron, you’ve seen them all”: that is, apart from position, momentum, and spin, every two electrons have exactly the same mass, same charge, same every other property, including even any properties yet to be discovered.  Again the skeptic interjects: but that has to be wrong.  Logically, you could only ever confirm that two electrons were different, by observing a difference in their behavior.  Even if the electrons had behaved identically for a billion years, you couldn’t rule out the possibility that they were actually different, for example because of tiny nametags (“Hi, I’m Emily the Electron!” “Hi, I’m Ernie!”) that had no effect on any experiment you’d thought to perform, but were visible to God.

You can probably guess where this is going.  Quantum mechanics says that, no, you can verify that two particles are perfectly identical by doing an experiment where you swap them and see what happens.  If the particles are identical in all respects, then you’ll see quantum interference between the swapped and un-swapped states.  If they aren’t, you won’t.  The kind of interference you’ll see is different for fermions (like electrons) than for bosons (like photons), but the basic principle is the same in both cases.  Once again, quantum mechanics lets you verify that a specific type of information—in this case, information that distinguishes one particle from another—was not present anywhere in the physical world, because if it were, it would’ve destroyed an interference effect that you in fact saw.

This, I think, already provides a meatier sense in which “information is physical” than any of the senses discussed previously.


But we haven’t gotten to the filet mignon yet.  The late, great Jacob Bekenstein will forever be associated with the discovery that information, wherever and whenever it occurs in the physical world, takes up a minimum amount of space.  The most precise form of this statement, called the covariant entropy bound, was worked out in detail by Raphael Bousso.  Here I’ll be discussing a looser version of the bound, which holds in “non-pathological” cases, and which states that a bounded physical system can store at most A/(4 ln 2) bits of information, where A is the area in Planck units of any surface that encloses the system—so, about 1069 bits per square meter.  (Actually it’s 1069 qubits per square meter, but because of Holevo’s theorem, an upper bound on the number of qubits is also an upper bound on the number of classical bits that can be reliably stored in a system and then retrieved later.)

You might have heard of the famous way Nature enforces this bound.  Namely, if you tried to create a hard drive that stored more than 1069 bits per square meter of surface area, the hard drive would necessarily collapse to a black hole.  And from that point on, the information storage capacity would scale “only” with the area of the black hole’s event horizon—a black hole itself being the densest possible hard drive allowed by physics.

Let’s hear once more from our skeptic.  “Nonsense!  Matter can take up space.  Energy can take up space.  But information?  Bah!  That’s just a category mistake.  For a proof, suppose God took one of your black holes, with a 1-square-meter event horizon, which already had its supposed maximum of ~1069 bits of information.  And suppose She then created a bunch of new fundamental fields, which didn’t interact with gravity, electromagnetism, or any of the other fields that we know from observation, but which had the effect of encoding 10300 new bits in the region of the black hole.  Presto!  An unlimited amount of additional information, exactly where Bekenstein said it couldn’t exist.”

We’d like to pinpoint what’s wrong with the skeptic’s argument—and do so in a self-contained, non-question-begging way, a way that doesn’t pull any rabbits out of hats, other than the general principles of relativity and quantum mechanics.  I was confused myself about how to do this, until a month ago, when Daniel Harlow helped set me straight (any remaining howlers in my exposition are 100% mine, not his).

I believe the logic goes like this:

  1. Relativity—even just Galilean relativity—demands that, in flat space, the laws of physics must have the same form for all inertial observers (i.e., all observers who move through space at constant speed).
  2. Anything in the physical world that varies in space—say, a field that encodes different bits of information at different locations—also varies in time, from the perspective of an observer who moves through the field at a constant speed.
  3. Combining 1 and 2, we conclude that anything that can vary in space can also vary in time.  Or to say it better, there’s only one kind of varying: varying in spacetime.
  4. More strongly, special relativity tells us that there’s a specific numerical conversion factor between units of space and units of time: namely the speed of light, c.  Loosely speaking, this means that if we know the rate at which a field varies across space, we can also calculate the rate at which it varies across time, and vice versa.
  5. Anything that varies across time carries energy.  Why?  Because this is essentially the definition of energy in quantum mechanics!  Up to a constant multiple (namely, Planck’s constant), energy is the expected speed of rotation of the global phase of the wavefunction, when you apply your Hamiltonian.  If the global phase rotates at the slowest possible speed, then we take the energy to be zero, and say you’re in a vacuum state.  If it rotates at the next highest speed, we say you’re in a first excited state, and so on.  Indeed, assuming a time-independent Hamiltonian, the evolution of any quantum system can be fully described by simply decomposing the wavefunction into a superposition of energy eigenstates, then tracking of the phase of each eigenstate’s amplitude as it loops around and around the unit circle.  No energy means no looping around means nothing ever changes.
  6. Combining 3 and 5, any field that varies across space carries energy.
  7. More strongly, combining 4 and 5, if we know how quickly a field varies across space, we can lower-bound how much energy it has to contain.
  8. In general relativity, anything that carries energy couples to the gravitational field.  This means that anything that carries energy necessarily has an observable effect: if nothing else, its effect on the warping of spacetime.  (This is dramatically illustrated by dark matter, which is currently observable via its spacetime warping effect and nothing else.)
  9. Combining 6 and 8, any field that varies across space couples to the gravitational field.
  10. More strongly, combining 7 and 8, if we know how quickly a field varies across space, then we can lower-bound by how much it has to warp spacetime.  This is so because of another famous (and distinctive) feature of gravity: namely, the fact that it’s universally attractive, so all the warping contributions add up.
  11. But in GR, spacetime can only be warped by so much before we create a black hole: this is the famous Schwarzschild bound.
  12. Combining 10 and 11, the information contained in a physical field can only vary so quickly across space, before it causes spacetime to collapse to a black hole.

Summarizing where we’ve gotten, we could say: any information that’s spatially localized at all, can only be localized so precisely.  In our world, the more densely you try to pack 1’s and 0’s, the more energy you need, therefore the more you warp spacetime, until all you’ve gotten for your trouble is a black hole.  Furthermore, if we rewrote the above conceptual argument in math—keeping track of all the G’s, c’s, h’s, and so on—we could derive a quantitative bound on how much information there can be in a bounded region of space.  And if we were careful enough, that bound would be precisely the holographic entropy bound, which says that the number of (qu)bits is at most A/(4 ln 2), where A is the area of a bounding surface in Planck units.

Let’s pause to point out some interesting features of this argument.

Firstly, we pretty much needed the whole kitchen sink of basic physical principles: special relativity (both the equivalence of inertial frames and the finiteness of the speed of light), quantum mechanics (in the form of the universal relation between energy and frequency), and finally general relativity and gravity.  All three of the fundamental constants G, c, and h made appearances, which is why all three show up in the detailed statement of the holographic bound.

But secondly, gravity only appeared from step 8 onwards.  Up till then, everything could be said solely in the language of quantum field theory: that is, quantum mechanics plus special relativity.  The result would be the so-called Bekenstein bound, which upper-bounds the number of bits in any spatial region by the product of the region’s radius and its energy content.  I learned that there’s an interesting history here: Bekenstein originally deduced this bound using ingenious thought experiments involving black holes.  Only later did people realize that the Bekenstein bound can be derived purely within QFT (see here and here for example)—in contrast to the holographic bound, which really is a statement about quantum gravity.  (An early hint of this was that, while the holographic bound involves Newton’s gravitational constant G, the Bekenstein bound doesn’t.)

Thirdly, speaking of QFT, some readers might be struck by the fact that at no point in our 12-step program did we ever seem to need QFT machinery.  Which is fortunate, because if we had needed it, I wouldn’t have been able to explain any of this!  But here I have to confess that I cheated slightly.  Recall step 4, which said that “if you know the rate at which a field varies across space, you can calculate the rate at which it varies across time.”  It turns out that, in order to give that sentence a definite meaning, one uses the fact that in QFT, space and time derivatives in the Hamiltonian need to be related by a factor of c, since otherwise the Hamiltonian wouldn’t be Lorentz-invariant.

Fourthly, eagle-eyed readers might notice a loophole in the argument.  Namely, we never upper-bounded how much information God could add to the world, via fields that are constant across all of spacetime.  For example, there’s nothing to stop Her from creating a new scalar field that takes the same value everywhere in the universe—with that value, in suitable units, encoding 1050000 separate divine thoughts in its binary expansion.  But OK, being constant, such a field would interact with nothing and affect no observations—so Occam’s Razor itches to slice it off, by rewriting the laws of physics in a simpler form where that field is absent.  If you like, such a field would at most be a comment in the source code of the universe: it could be as long as the Great Programmer wanted it to be, but would have no observable effect on those of us living inside the program’s execution.


Of course, even before relativity and quantum mechanics, information had already been playing a surprisingly fleshy role in physics, through its appearance as entropy in 19th-century thermodynamics.  Which leads to another puzzle.  To a computer scientist, the concept of entropy, as the log of the number of microstates compatible with a given macrostate, seems clear enough, as does the intuition for why it should increase monotonically with time.  Or at least, to whatever extent we’re confused about these matters, we’re no more confused than the physicists are!

But then why should this information-theoretic concept be so closely connected to tangible quantities like temperature, and pressure, and energy?  From the mere assumption that a black hole has a nonzero entropy—that is, that it takes many bits to describe—how could Bekenstein and Hawking have possibly deduced that it also has a nonzero temperature?  Or: if you put your finger into a tub of hot water, does the heat that you feel somehow reflect how many bits are needed to describe the water’s microstate?

Once again our skeptic pipes up: “but surely God could stuff as many additional bits as She wanted into the microstate of the hot water—for example, in degrees of freedom that are still unknown to physics—without the new bits having any effect on the water’s temperature.”

But we should’ve learned by now to doubt this sort of argument.  There’s no general principle, in our universe, saying that you can hide as many bits as you want in a physical object, without those bits influencing the object’s observable properties.  On the contrary, in case after case, our laws of physics seem to be intolerant of “wallflower bits,” which hide in a corner without talking to anyone.  If a bit is there, the laws of physics want it to affect other nearby bits and be affected by them in turn.

In the case of thermodynamics, the assumption that does all the real work here is that of equidistribution.  That is, whatever degrees of freedom might be available to your thermal system, your gas in a box or whatever, we assume that they’re all already “as randomized as they could possibly be,” subject to a few observed properties like temperature and volume and pressure.  (At least, we assume that in classical thermodynamics.  Non-equilibrium thermodynamics is a whole different can of worms, worms that don’t stay in equilibrium.)  Crucially, we assume this despite the fact that we might not even know all the relevant degrees of freedom.

Why is this assumption justified?  “Because experiment bears it out,” the physics teacher explains—but we can do better.  The assumption is justified because, as long as the degrees of freedom that we’re talking about all interact with each other, they’ve already had plenty of time to equilibrate.  And conversely, if a degree of freedom doesn’t interact with the stuff we’re observing—or with anything that interacts with the stuff we’re observing, etc.—well then, who cares about it anyway?

But now, because the microscopic laws of physics have the fundamental property of reversibility—that is, they never destroy information—a new bit has to go somewhere, and it can’t overwrite degrees of freedom that are already fully randomized.  This is why, if you pump more bits of information into a tub of hot water, while keeping it at the same volume, the new bits have nowhere to go except into pushing up the energy.  Now, there are often ways to push up the energy other than by raising the temperature—the concept of specific heat, in chemistry, is precisely about this—but if you need to stuff more bits into a substance, at the cost of raising its energy, certainly one of the obvious ways to do it is to describe a greater range of possible speeds for the water molecules.  So since that can happen, by equidistribution it typically does happen, which means that the molecules move faster on average, and your finger feels the water get hotter.


In summary, our laws of physics are structured in such a way that even pure information often has “nowhere to hide”: if the bits are there at all in the abstract machinery of the world, then they’re forced to pipe up and have a measurable effect.  And this is not a tautology, but comes about only because of nontrivial facts about special and general relativity, quantum mechanics, quantum field theory, and thermodynamics.  And this is what I think people should mean when they say “information is physical.”

Anyway, if this was all obvious to you, I apologize for having wasted your time!  But in my defense, it was never explained to me quite this way, nor was it sorted out in my head until recently—even though it seems like one of the most basic and general things one can possibly say about physics.


Endnotes. Thanks again to Daniel Harlow, not only for explaining the logic of the holographic bound to me but for several suggestions that improved this post.

Some readers might suspect circularity in the arguments we’ve made: are we merely saying that “any information that has observable physical consequences, has observable physical consequences”?  No, it’s more than that.  In all the examples I discussed, the magic was that we inserted certain information into our abstract mathematical description of the world, taking no care to ensure that the information’s presence would have any observable consequences whatsoever.  But then the principles of quantum mechanics, quantum gravity, or thermodynamics forced the information to be detectable in very specific ways (namely, via the destruction of quantum interference, the warping of spacetime, or the generation of heat respectively).

Three things

Monday, July 17th, 2017

I was shocked and horrified to learn of the loss of Maryam Mirzakhani at age 40, after a battle with cancer (see here or here).  Mirzakhani was a renowned mathematician at Stanford and the world’s first and so far only female Fields Medalist.  I never had the privilege of meeting her, but everything I’ve read about her fills me with admiration.  I wish to offer condolences to her friends and family, including her husband Jan Vondrák, also a professor at Stanford and a member of the CS theory community.


In other depressing news, discussion continues to rage on social media about “The Uninhabitable Earth,” the New York magazine article by David Wallace-Wells arguing that the dangers of climate change have been systematically understated even by climate scientists; that sea level rise is the least of the problems; and that if we stay the current course, much of the earth’s landmass has a good chance of being uninhabitable by the year 2100.  In an unusual turn of events, the Wallace-Wells piece has been getting slammed by climate scientists, including Michael Mann (see here and also this interview)—people who are usually in the news to refute the claims of deniers.

Some of the critics’ arguments seem cogent to me: for example, that Wallace-Wells misunderstood some satellite data, and more broadly, that the piece misleadingly presents its scenario as overwhelmingly probable by 2100 if we do nothing, rather than as “only” 10% likely or whatever—i.e., a mere Trump-becoming-president level of risk.  Other objections to the article impressed me less: for example, that doom-and-gloom is a bad way to motivate people about climate change; that the masses need a more optimistic takeaway.  That obviously has no bearing on the truth of what’s going to happen—but even if we did agree to entertain such arguments, well, it’s not as if mainstream messaging on climate change has been an unmitigated success.  What if everyone should be sweating-in-the-night terrified?

As far as I understand it, the question of the plausibility of Wallace-Wells’s catastrophe scenario mostly just comes down to a single scientific unknown: namely, will the melting permafrost belch huge amounts of methane into the atmosphere?  If it does, then “Armageddon” is probably a fair description of what awaits us in the next century, and if not, not.  Alas, our understanding of permafrost doesn’t seem especially reliable, and it strikes me that models of such feedbacks have a long history of erring on the side of conservatism (for example, researchers were astonished by how quickly glaciers and ice shelves fell apart).

So, while I wish the article was written with more caveats, I submit that runaway warming scenarios deserve more attention rather than less.  And we should be putting discussion of those scenarios in exactly the broader context that Wallace-Wells does: namely, that of the Permian-Triassic extinction event, the Fermi paradox, and the conditions for a technological civilization to survive past its infancy.

Certainly we spend much more time on risks to civilization (e.g., nuclear terrorism, bioengineered pandemics) that strike me as less probable than this one.  And certainly this tail, in the distribution of possible outcomes, deserves at least as much attention as its more popular opposite, the tail where climate change turns out not to be much of a problem at all.  For the grim truth about climate change is that history won’t end in 2100: only the projections do.  And the mere addition of 50 more years could easily suffice to turn a tail risk into a body risk.

Of course, that the worst will happen is a clear prediction of reverse Hollywoodism theory—besides being the “natural, default” prediction for a computer scientist used to worst-case analysis.  This is one prediction that I hope turns out to be as wrong as possible.


OK, now for something to cheer us all up.  Yesterday the group of Misha Lukin, at Harvard, put a paper on the arXiv reporting the creation of a 51-qubit quantum simulator using cold atoms.  The paper doesn’t directly address the question of quantum supremacy, or indeed of performance comparisons between the new device and classical simulations at all.  But this is clearly a big step forward, while the world waits for the fully-programmable 50-qubit superconducting QCs that have been promised by the groups at Google and IBM.

Indeed, this strikes me as the most exciting news in experimental quantum information since last month, when Jian-Wei Pan’s group in Shanghai reported the first transmission of entangled photons from a satellite to earth—thereby allowing violations of the Bell inequality over 1200 kilometers, teleportation of a qubit from earth to space, and other major firsts.  These are breakthroughs that we knew were in the works ever since the Chinese government launched the QUESS satellite devoted to quantum communications.  I should’ve blogged about them in June.  Then again, regular readers of Shtetl-Optimized, familiar as they already are with the universal reach of quantum mechanics and with the general state of quantum information technology, shouldn’t find anything here that fundamentally surprises them, should they?

Amsterdam art museums plagiarizing my blog?

Wednesday, July 12th, 2017

This past week I had the pleasure of attending COLT (Conference on Learning Theory) 2017 in Amsterdam, and of giving an invited talk on “PAC-Learning and Reconstruction of Quantum States.”  You can see the PowerPoint slides here; videos were also made, but don’t seem to be available yet.

This was my first COLT, but almost certainly not the last.  I learned lots of cool new tidbits, from the expressive power of small-depth neural networks, to a modern theoretical computer science definition of “non-discriminatory” (namely, your learning algorithm’s output should be independent of protected categories like race, sex, etc. after conditioning on the truth you’re trying to predict), to the inapproximability of VC dimension (assuming the Exponential Time Hypothesis).  You can see the full schedule here.  Thanks so much to the PC chairs, Ohad Shamir and Satyen Kale, for inviting me and for putting on a great conference.

And one more thing: I’m not normally big on art museums, but Amsterdam turns out to have two in close proximity to each other—the Rijksmuseum and the Stedelijk—each containing something that Shtetl-Optimized readers might recognize.

Photo credits: Ronald de Wolf and Marijn Heule.

Yet more errors in papers

Wednesday, May 24th, 2017

Following up on my posts PostBQP Postscripts and More Wrong Things I Said In Papers, it felt like time for another post in which I publicly flog myself for mistakes in my research papers.  [Warning: The rest of this post is kinda, sorta technical.  Read at your own risk.]


(1) In my 2006 paper “Oracles are subtle but not malicious,” I claimed to show that if PP is contained in BQP/qpoly, then the counting hierarchy collapses to QMA (Theorem 5).  But on further reflection, I only know how to show a collapse of the counting hierarchy under the stronger assumption that PP is in BQP/poly.  If PP is in BQP/qpoly, then certainly P#P=PP=QMA, but I don’t know how to collapse any further levels of the counting hierarchy.  The issue is this: in QMA, we can indeed nondeterministically guess an (amplified) quantum advice state for a BQP/qpoly algorithm.  We can then verify that the advice state works to solve PP problems, by using (for example) the interactive protocol for the permanent, or some other #P-complete problem.  But having done that, how do we then unravel the higher levels of the counting hierarchy?  For example, how do we simulate PPPP in PPBQP=PP?  We don’t have any mechanism to pass the quantum advice up to the oracle PP machine, since queries to a PP oracle are by definition classical strings.  We could try to use tools from my later paper with Andy Drucker, passing a classical description of the quantum advice up to the oracle and then using the description to reconstruct the advice for ourselves.  But doing so just doesn’t seem to give us a complexity class that’s low for PP, which is what would be needed to unravel the counting hierarchy.  I still think this result might be recoverable, but a new idea is needed.


(2) In my 2008 algebrization paper with Avi Wigderson, one of the most surprising things we showed was a general connection between communication complexity lower bounds and algebraic query complexity lower bounds.  Specifically, given a Boolean oracle A:{0,1}n→{0,1}, let ~A be a low-degree extension of A over a finite field F (that is, ~A(x)=A(x) whenever x∈{0,1}n).  Then suppose we have an algorithm that’s able to learn some property of A, by making k black-box queries to ~A.  We observed that, in such a case, if Alice is given the top half of the truth table of A, and Bob is given the bottom half of the truth table, then there’s necessarily a communication protocol by which Alice and Bob can learn the same property of A, by exchanging at most O(kn log|F|) bits.  This connection is extremely model-independent: a randomized query algorithm gives rise to a randomized communication protocol, a quantum query algorithm gives rise to a quantum communication protocol, etc. etc.  The upshot is that, if you want to lower-bound the number of queries that an algorithm needs to make to the algebraic extension oracle ~A, in order to learn something about A, then it suffices to prove a suitable communication complexity lower bound.  And the latter, unlike algebraic query complexity, is a well-studied subject with countless results that one can take off the shelf.  We illustrated how one could use this connection to prove, for example, that there exists an oracle A such that NPA ⊄ BQP~A, for any low-degree extension ~A of A—a separation that we didn’t and don’t know how to prove any other way. Likewise, there exists an oracle B such that BQPB ⊄ BPP~B for any low-degree extension ~B of B.

The trouble is, our “proof sketches” for these separations (in Theorem 5.11) are inadequate, even for “sketches.”  They can often be fixed, but only by appealing to special properties of the communication complexity separations in question, properties that don’t necessarily hold for an arbitrary communication separation between two arbitrary models.

The issue is this: while it’s true, as we claimed, that a communication complexity lower bound implies an algebraic query complexity lower bound, it’s not true in general that a communication complexity upper bound implies an algebraic query complexity upper bound!  So, from a communication separation between models C and D, we certainly obtain a query complexity problem that’s not in D~A, but then the problem might not be in CA.  What tripped us up was that, in the cases we had in mind (e.g. Disjointness), it’s obvious that the query problem is in CA.  In other cases, however, such as Raz’s separation between quantum and randomized communication complexity, it probably isn’t even true.  In the latter case, to recover the desired conclusion about algebraic query complexity (namely, the existence of an oracle B such that BQPB ⊄ BPP~B), what seems to be needed is to start from a later quantum vs. classical communication complexity separation due to Klartag and Regev, and then convert their communication problem into a query problem using a recent approach by myself and Shalev Ben-David (see Section 4).  Unfortunately, my and Shalev’s approach only tells us nonconstructively that there exists a query problem with the desired separation, with no upper bound on the gate complexity of the quantum algorithm.  So strictly speaking, I still don’t know how to get a separation between the relativized complexity classes BQPB and BPP~B defined in terms of Turing machines.

In any case, I of course should have realized this issue with the algebrization paper the moment Shalev and I encountered the same issue when writing our later paper.  Let me acknowledge Shalev, as well as Robin Kothari, for helping to spur my realization of the issue.


In case it wasn’t clear, the mistakes I’ve detailed here have no effect on the main results of the papers in question (e.g., the existence of an oracle relative to which PP has linear-sized circuits; the existence and pervasiveness of the algebrization barrier).  The effect is entirely on various “bonus” results—results that, because they’re bonus, were gone over much less carefully by authors and reviewers alike.

Nevertheless, I’ve always felt like in science, the louder you are about your own mistakes, the better.  Hence this post.

This Week’s BS

Friday, May 5th, 2017

There are two pieces of BosonSampling-related news that people have asked me about this week.

First, a group in Shanghai, led by Chaoyang Lu and Jianwei Pan, has reported in Nature Photonics that they can do BosonSampling with a coincidence rate that’s higher than in previous experiments by a factor of several thousand.  This, in particular, lets them do BosonSampling with 5 photons.  Now, 5 might not sound like that much, especially since the group in Bristol previously did 6-photon BosonSampling.  But to make their experiment work, the Bristol group needed to start its photons in the initial state |3,3〉: that is, two modes with 3 photons each.  This gives rise to matrices with repeated rows, whose permanents are much easier to calculate than the permanents of arbitrary matrices.  By contrast, the Shangai group starts its photons in the “true BosonSampling initial state” |1,1,1,1,1〉: that is, five modes with 1 photon each.  That’s the kind of initial state we ultimately want.

The second piece of news is that on Monday, a group at Bristol—overlapping with the group we mentioned before—submitted a preprint to the arXiv with the provocative title “No imminent quantum supremacy by boson sampling.”  In this paper, they give numerical evidence that BosonSampling, with n photons and m modes, can be approximately simulated by a classical computer in “merely” about n2n time (that is, the time needed to calculate a single n×n permanent), as opposed to the roughly mn time that one would need if one had to calculate permanents corresponding to all the possible outcomes of the experiment.  As a consequence of that, they argue that achieving quantum supremacy via BosonSampling would probably require at least ~50 photons—which would in turn require a “step change” in technology, as they put it.

I completely agree with the Bristol group’s view of the asymptotics.  In fact, Alex Arkhipov and I ourselves repeatedly told experimentalists, in our papers and talks about BosonSampling (the question came up often…), that the classical complexity of the problem should only be taken to scale like 2n, rather than like mn.  Despite not having a general proof that the problem could actually be solved in ~2n time in the worst case, we said that for two main reasons:

  1. Even under the most optimistic assumptions, our hardness reductions, from Gaussian permanent estimation and so forth, only yielded ~2n hardness, not ~mn hardness.  (Hardness reductions giving us important clues about the real world?  Whuda thunk??)
  2. If our BosonSampling matrix is Haar-random—or otherwise not too skewed to produce outcomes with huge probabilities—then it’s not hard to see that we can do approximate BosonSampling in O(n2n) time classically, by using rejection sampling.

Indeed, Alex and I insisted on these points despite some pushback from experimentalists, who were understandably hoping that they could get to quantum supremacy just by upping m, the number of modes, without needing to do anything heroic with n, the number of photons!  So I’m happy to see that a more careful analysis supports the guess that Alex and I made.

On the other hand, what does this mean for the number of photons needed for “quantum supremacy”: is it 20? 30? 50?  I confess that that sort of question interests me much less, since it all depends on the details of how you define the comparison (are we comparing against ENIAC? a laptop? a server farm? how many cores? etc etc).  As I’ve often said, my real hope with quantum supremacy is to see a quantum advantage that’s so overwhelming—so duh-obvious to the naked eye—that we don’t have to squint or argue about the definitions.

Insert D-Wave Post Here

Thursday, March 16th, 2017

In the two months since I last blogged, the US has continued its descent into madness.  Yet even while so many certainties have proven ephemeral as the morning dew—the US’s autonomy from Russia, the sanity of our nuclear chain of command, the outcome of our Civil War, the constraints on rulers that supposedly set us apart from the world’s dictator-run hellholes—I’ve learned that certain facts of life remain constant.

The moon still waxes and wanes.  Electrons remain bound to their nuclei.  P≠NP proofs still fill my inbox.  Squirrels still gather acorns.  And—of course!—people continue to claim big quantum speedups using D-Wave devices, and those claims still require careful scrutiny.

With that preamble, I hereby offer you eight quantum computing news items.


Cathy McGeoch Episode II: The Selby Comparison

On January 17, a group from D-Wave—including Cathy McGeoch, who now works directly for D-Wave—put out a preprint claiming a factor-of-2500 speedup for the D-Wave machine (the new, 2000-qubit one) compared to the best classical algorithms.  Notably, they wrote that the speedup persisted when they compared against simulated annealing, quantum Monte Carlo, and even the so-called Hamze-de Freitas-Selby (HFS) algorithm, which was often the classical victor in previous performance comparisons against the D-Wave machine.

Reading this, I was happy to see how far the discussion has advanced since 2013, when McGeoch and Cong Wang reported a factor-of-3600 speedup for the D-Wave machine, but then it turned out that they’d compared only against classical exact solvers rather than heuristics—a choice for which they were heavily criticized on this blog and elsewhere.  (And indeed, that particular speedup disappeared once the classical computer’s shackles were removed.)

So, when people asked me this January about the new speedup claim—the one even against the HFS algorithm—I replied that, even though we’ve by now been around this carousel several times, I felt like the ball was now firmly in the D-Wave skeptics’ court, to reproduce the observed performance classically.  And if, after a year or so, no one could, that would be a good time to start taking seriously that a D-Wave speedup might finally be here to stay—and to move on to the next question, of whether this speedup had anything to do with quantum computation, or only with the building of a piece of special-purpose optimization hardware.


A&M: Annealing and Matching

As it happened, it only took one month.  On March 2, Salvatore Mandrà, Helmut Katzgraber, and Creighton Thomas put up a response preprint, pointing out that the instances studied by the D-Wave group in their most recent comparison are actually reducible to the minimum-weight perfect matching problem—and for that reason, are solvable in polynomial time on a classical computer.   Much of Mandrà et al.’s paper just consists of graphs, wherein they plot the running times of the D-Wave machine and of a classical heuristic on the relevant instances—clearly all different flavors of exponential—and then Edmonds’ matching algorithm from the 1960s, which breaks away from the pack into polynomiality.

But let me bend over backwards to tell you the full story.  Last week, I had the privilege of visiting Texas A&M to give a talk.  While there, I got to meet Helmut Katzgraber, a condensed-matter physicist who’s one of the world experts on quantum annealing experiments, to talk to him about their new response paper.  Helmut was clear in his prediction that, with only small modifications to the instances considered, one could see similar performance by the D-Wave machine while avoiding the reduction to perfect matching.  With those future modifications, it’s possible that one really might see a D-Wave speedup that survived serious attempts by skeptics to make it go away.

But Helmut was equally clear in saying that, even in such a case, he sees no evidence at present that the speedup would be asymptotic or quantum-computational in nature.  In other words, he thinks the existing data is well explained by the observation that we’re comparing D-Wave against classical algorithms for Ising spin minimization problems on Chimera graphs, and D-Wave has heroically engineered an expensive piece of hardware specifically for Ising spin minimization problems on Chimera graphs and basically nothing else.  If so, then the prediction would be that such speedups as can be found are unlikely to extend either to more “practical” optimization problems—which need to be embedded into the Chimera graph with considerable losses—or to better scaling behavior on large instances.  (As usual, as long as the comparison is against the best classical algorithms, and as long as we grant the classical algorithm the same non-quantum advantages that the D-Wave machine enjoys, such as classical parallelism—as Rønnow et al advocated.)

Incidentally, my visit to Texas A&M was partly an “apology tour.”  When I announced on this blog that I was moving from MIT to UT Austin, I talked about the challenge and excitement of setting up a quantum computing research center in a place that currently had little quantum computing for hundreds of miles around.  This thoughtless remark inexcusably left out not only my friends at Louisiana State (like Jon Dowling and Mark Wilde), but even closer to home, Katzgraber and the others at Texas A&M.  I felt terrible about this for months.  So it gives me special satisfaction to have the opportunity to call out Katzgraber’s new work in this post.  In football, UT and A&M were longtime arch-rivals, but when it comes to the appropriate level of skepticism to apply to quantum supremacy claims, the Texas Republic seems remarkably unified.


When 15 MilliKelvin is Toasty

In other D-Wave-related scientific news, on Monday night Tameem Albash, Victor Martin-Mayer, and Itay Hen put out a preprint arguing that, in order for quantum annealing to have any real chance of yielding a speedup over classical optimization methods, the temperature of the annealer should decrease at least like 1/log(n), where n is the instance size, and more likely like 1/nβ (i.e., as an inverse power law).

If this is correct, then cold as the D-Wave machine is, at 0.015 degrees or whatever above absolute zero, it still wouldn’t be cold enough to see a scalable speedup, at least not without quantum fault-tolerance, something that D-Wave has so far eschewed.  With no error-correction, any constant temperature that’s above zero would cause dangerous level-crossings up to excited states when the instances get large enough.  Only a temperature that actually converged to zero as the problems got larger would suffice.

Over the last few years, I’ve heard many experts make this exact same point in conversation, but this is the first time I’ve seen the argument spelled out in a paper, with explicit calculations (modulo assumptions) of the rate at which the temperature would need to go to zero for uncorrected quantum annealing to be a viable path to a speedup.  I lack the expertise to evaluate the calculations myself, but any experts who’d like to share their insight in the comments section are “warmly” (har har) invited.


“Their Current Numbers Are Still To Be Checked”

As some of you will have seen, The Economist now has a sprawling 10-page cover story about quantum computing and other quantum technologies.  I had some contact with the author while the story was in the works.

The piece covers a lot of ground and contains many true statements.  It could be much worse.

But I take issue with two things.

First, The Economist claims: “What is notable about the effort [to build scalable QCs] now is that the challenges are no longer scientific but have become matters of engineering.”  As John Preskill and others pointed out, this is pretty far from true, at least if we interpret the claim in the way most engineers and businesspeople would.

Yes, we know the rules of quantum mechanics, and the theory of quantum fault-tolerance, and a few promising applications; and the basic building blocks of QC have already been demonstrated in several platforms.  But if (let’s say) someone were to pony up $100 billion, asking only for a universal quantum computer as soon as possible, I think the rational thing to do would be to spend initially on a frenzy of basic research: should we bet on superconducting qubits, trapped ions, nonabelian anyons, photonics, a combination thereof, or something else?  (Even that is far from settled.)  Can we invent better error-correcting codes and magic state distillation schemes, in order to push the resource requirements for universal QC down by three or four orders of magnitude?  Which decoherence mechanisms will be relevant when we try to do this stuff at scale?  And of course, which new quantum algorithms can we discover, and which new cryptographic codes resistant to quantum attack?

The second statement I take issue with is this:

“For years experts questioned whether the [D-Wave] devices were actually exploiting quantum mechanics and whether they worked better than traditional computers.  Those questions have since been conclusively answered—yes, and sometimes”

I would instead say that the answers are:

  1. depends on what you mean by “exploit” (yes, there are quantum tunneling effects, but do they help you solve problems faster?), and
  2. no, the evidence remains weak to nonexistent that the D-Wave machine solves anything faster than a traditional computer—certainly if, by “traditional computer,” we mean a device that gets all the advantages of the D-Wave machine (e.g., classical parallelism, hardware heroically specialized to the one type of problem we’re testing on), but no quantum effects.

Shortly afterward, when discussing the race to achieve “quantum supremacy” (i.e., a clear quantum computing speedup for some task, not necessarily a useful one), the Economist piece hedges: “D-Wave has hinted it has already [achieved quantum supremacy], but has made similar claims in the past; their current numbers are still to be checked.”

To me, “their current numbers are still to be checked” deserves its place alongside “mistakes were made” among the great understatements of the English language—perhaps a fitting honor for The Economist.


Defeat Device

Some of you might also have seen that D-Wave announced a deal with Volkswagen, to use D-Wave machines for traffic flow.  I had some advance warning of this deal, when reporters called asking me to comment on it.  At least in the materials I saw, no evidence is discussed that the D-Wave machine actually solves whatever problem VW is interested in faster than it could be solved with a classical computer.  Indeed, in a pattern we’ve seen repeatedly for the past decade, the question of such evidence is never even directly confronted or acknowledged.

So I guess I’ll say the same thing here that I said to the journalists.  Namely, until there’s a paper or some other technical information, obviously there’s not much I can say about this D-Wave/Volkswagen collaboration.  But it would be astonishing if quantum supremacy were to be achieved on an application problem of interest to a carmaker, even as scientists struggle to achieve that milestone on contrived and artificial benchmarks, even as the milestone seems repeatedly to elude D-Wave itself on contrived and artificial benchmarks.  In the previous such partnerships—such as that with Lockheed Martin—we can reasonably guess that no convincing evidence for quantum supremacy was found, because if it had been, it would’ve been trumpeted from the rooftops.

Anyway, I confess that I couldn’t resist adding a tiny snark—something about how, if these claims of amazing performance were found not to withstand an examination of the details, it would not be the first time in Volkswagen’s recent history.


Farewell to a Visionary Leader—One Who Was Trash-Talking Critics on Social Media A Decade Before President Trump

This isn’t really news, but since it happened since my last D-Wave post, I figured I should share.  Apparently D-Wave’s outspoken and inimitable founder, Geordie Rose, left D-Wave to form a machine-learning startup (see D-Wave’s leadership page, where Rose is absent).  I wish Geordie the best with his new venture.


Martinis Visits UT Austin

On Feb. 22, we were privileged to have John Martinis of Google visit UT Austin for a day and give the physics colloquium.  Martinis concentrated on the quest to achieve quantum supremacy, in the near future, using sampling problems inspired by theoretical proposals such as BosonSampling and IQP, but tailored to Google’s architecture.  He elaborated on Google’s plan to build a 49-qubit device within the next few years: basically, a 7×7 square array of superconducting qubits with controllable nearest-neighbor couplings.  To a layperson, 49 qubits might sound unimpressive compared to D-Wave’s 2000—but the point is that these qubits will hopefully maintain coherence times thousands of times longer than the D-Wave qubits, and will also support arbitrary quantum computations (rather than only annealing).  Obviously I don’t know whether Google will succeed in its announced plan, but if it does, I’m very optimistic about a convincing quantum supremacy demonstration being possible with this sort of device.

Perhaps most memorably, Martinis unveiled some spectacular data, which showed near-perfect agreement between Google’s earlier 9-qubit quantum computer and the theoretical predictions for a simulation of the Hofstadter butterfly (incidentally invented by Douglas Hofstadter, of Gödel, Escher, Bach fame, when he was still a physics graduate student).  My colleague Andrew Potter explained to me that the Hofstadter butterfly can’t be used to show quantum supremacy, because it’s mathematically equivalent to a system of non-interacting fermions, and can therefore be simulated in classical polynomial time.  But it’s certainly an impressive calibration test for Google’s device.


2000 Qubits Are Easy, 50 Qubits Are Hard

Just like the Google group, IBM has also publicly set itself the ambitious goal of building a 50-qubit superconducting quantum computer in the near future (i.e., the next few years).  Here in Austin, IBM held a quantum computing session at South by Southwest, so I went—my first exposure of any kind to SXSW.  There were 10 or 15 people in the audience; the purpose of the presentation was to walk through the use of the IBM Quantum Experience in designing 5-qubit quantum circuits and submitting them first to a simulator and then to IBM’s actual superconducting device.  (To the end user, of course, the real machine differs from the simulation only in that with the former, you can see the exact effects of decoherence.)  Afterward, I chatted with the presenters, who were extremely friendly and knowledgeable, and relieved (they said) that I found nothing substantial to criticize in their summary of quantum computing.

Hope everyone had a great Pi Day and Ides of March.

“THE TALK”: My quantum computing cartoon with Zach Weinersmith

Wednesday, December 14th, 2016

OK, here’s the big entrée that I promised you yesterday:

“THE TALK”: My joint cartoon about quantum comgputing with Zach Weinersmith of SMBC Comics.

Just to whet your appetite:

In case you’re wondering how this came about: after our mutual friend Sean Carroll introduced me and Zach for a different reason, the idea of a joint quantum computing comic just seemed too good to pass up.  The basic premise—“The Talk”—was all Zach.  I dutifully drafted some dialogue for him, which he then improved and illustrated.  I.e., he did almost all the work (despite having a newborn competing for his attention!).  Still, it was an honor for me to collaborate with one of the great visual artists of our time, and I hope you like the result.  Beyond that, I’ll let the work speak for itself.

Quantum computing news (98% Trump-free)

Thursday, November 24th, 2016

(1) Apparently Microsoft has decided to make a major investment in building topological quantum computers, which will include hiring Charles Marcus and Matthias Troyer among others.  See here for their blog post, and here for the New York Times piece.  In the race to implement QC among the established corporate labs, Microsoft thus joins the Martinis group at Google, as well as the IBM group at T. J. Watson—though both Google and IBM are focused on superconducting qubits, rather than the more exotic nonabelian anyon approach that Microsoft has long favored and is now doubling down on.  I don’t really know more about this new initiative beyond what’s in the articles, but I know many of the people involved, they’re some of the most serious in the business, and Microsoft intensifying its commitment to QC can only be good for the field.  I wish the new effort every success, despite being personally agnostic between superconducting qubits, trapped ions, photonics, nonabelian anyons, and other QC hardware proposals—whichever one gets there first is fine with me!


(2) For me, though, perhaps the most exciting QC development of the last month was a new preprint by my longtime friend Dorit Aharonov and her colleague Yosi Atia, entitled Fast-Forwarding of Hamiltonians and Exponentially Precise Measurements.  In this work, Dorit and Yosi wield the clarifying sword of computational complexity at one of the most historically confusing issues in quantum mechanics: namely, the so-called “time-energy uncertainty principle” (TEUP).

The TEUP says that, just as position and momentum are conjugate in quantum mechanics, so too are energy and time—with greater precision in energy corresponding to lesser precision in time and vice versa.  The trouble is, it was never 100% clear what the TEUP even meant—after all, time isn’t even an observable in quantum mechanics, just an external parameter—and, to whatever extent the TEUP did have a definite meaning, it wasn’t clear that it was true.  Indeed, as Dorit and Yosi’s paper discusses in detail, in 1961 Dorit’s uncle Yakir Aharonov, together with David Bohm, gave a counterexample to a natural interpretation of the TEUP.  But, despite this and other counterexamples, the general feeling among physicists—who, after all, are physicists!—seems to have been that some corrected version of the TEUP should hold “in all reasonable circumstances.”

But, OK, what do we mean by a “reasonable circumstance”?  This is where Dorit and Yosi come in.   In the new work, they present a compelling case that the TEUP should really be formulated as a tradeoff between the precision of energy measurements and circuit complexity (that is, the minimum number of gates needed to implement the energy measurement)—and in that amended form, the TEUP holds for exactly those Hamiltonians H that can’t be “computationally fast-forwarded.”  In other words, it holds whenever applying the unitary transformation e-iHt requires close to t computation steps, when there’s no magical shortcut that lets you simulate t steps of time evolution with only (say) log(t) steps.  And, just as the physicists handwavingly thought, that should indeed hold for “generic” Hamiltonians H (assuming BQP≠PSPACE), although it’s possible to use Shor’s algorithm, for finding the order of an element in a multiplicative group, to devise a counterexample to it.

Anyway, there’s lots of other stuff in the paper, including a connection to the stuff Lenny Susskind and I have been doing about the “generic” growth of circuit complexity, in the CFT dual of an expanding wormhole (where we also needed to assume BQP≠PSPACE and closely related complexity separations, for much the same reasons).  Congratulations to Dorit and Yosi for once again illustrating the long reach of computational complexity in physics, and for giving me a reason to be happy this month!


(3) As many of you will have seen, my former MIT colleagues, Lior Eldar and Peter Shor, recently posted an arXiv preprint claiming a bombshell result: namely, a polynomial-time quantum algorithm to solve a variant of the Closest Vector Problem in lattices.  Their claimed algorithm wouldn’t yet break lattice-based cryptography, but if the approximation factors could be improved, it would be well on the way to doing so.  This has been one of the most tempting targets for quantum algorithms research for more than twenty years—ever since Shor’s “original” algorithm laid waste to RSA, Diffie-Hellman, elliptic-curve cryptography, and more in a world with scalable quantum computers, leaving lattice-based cryptography as one of the few public-key crypto proposals still standing.

Unfortunately, Lior tells me that Oded Regev has discovered a flaw in the algorithm, which he and Peter don’t currently know how to fix.  So for now, they’re withdrawing the paper (because of the Thanksgiving holiday, the withdrawal won’t take effect on the arXiv until Monday).  It’s still a worthy attempt on a great problem—here’s hoping that they or someone else manage to, as Lior put it to me, “make the algorithm great again.”

My 5-minute quantum computing talk at the White House

Tuesday, October 25th, 2016

(OK, technically it was in the Eisenhower Executive Office Building, which is not exactly the White House itself, but is adjacent to the West Wing in the White House complex.  And President Obama wasn’t there—maybe, like Justin Trudeau, he already knows everything about quantum computing?  But lots of people from the Office of Science and Technology Policy were!  And some of us talked with Valerie Jarrett, Obama’s adviser, when she passed us on her way to the West Wing.

The occasion was a Quantum Information Science policy workshop that OSTP held, and which the White House explicitly gave us permission to discuss on social media.  Indeed, John Preskill already tweeted photos from the event.  Besides me and Preskill, others in attendance included Umesh Vazirani, Seth Lloyd, Yaoyun Shi, Rob Schoelkopf, Krysta Svore, Hartmut Neven, Stephen Jordan…

I don’t know whether this is the first time that the polynomial hierarchy, or the notion of variation distance, were ever invoked in a speech at the White House.  But in any case, I was proud to receive a box of Hershey Kisses bearing the presidential seal.  I thought of not eating them, but then I got hungry, and realized that I can simply refill the box later if desired.

For regular readers of Shtetl-Optimized, my talk won’t have all that much that’s new, but in any case it’s short.

Incidentally, during the workshop, a guy from OSTP told me that, when he and others at the White House were asked to prepare materials about quantum computing, posts on Shtetl-Optimized (such as Shor I’ll Do It) were a huge help.  Honored though I was to have “served my country,” I winced, thinking about all the puerile doofosities I might’ve self-censored had I had any idea who might read them.  I didn’t dare ask whether anyone at the White House also reads the comment sections!

Thanks so much to all the other participants and to the organizers for a great workshop.  –SA)


Quantum Supremacy

by Scott Aaronson (UT Austin)

October 18, 2016

Thank you; it’s great to be here.  There are lots of directions that excite me enormously right now in quantum computing theory, which is what I work on.  For example, there’s the use of quantum computing to get new insight into classical computation, into condensed matter physics, and recently, even into the black hole information problem.

But since I have five minutes, I wanted to talk here about one particular direction—one that, like nothing else that I know of, bridges theory and experiment in the service of what we hope will be a spectacular result in the near future.  This direction is what’s known as “Quantum Supremacy”—John [Preskill], did you help popularize that term?  [John nods yes]—although some people have been backing away from the term recently, because of the campaign of one of the possible future occupants of this here complex.

But what quantum supremacy means to me, is demonstrating a quantum speedup for some task as confidently as possible.  Notice that I didn’t say a useful task!  I like to say that for me, the #1 application of quantum computing—more than codebreaking, machine learning, or even quantum simulation—is just disproving the people who say quantum computing is impossible!  So, quantum supremacy targets that application.

What is important for quantum supremacy is that we solve a clearly defined problem, with some relationship between inputs and outputs that’s independent of whatever hardware we’re using to solve the problem.  That’s part of why it doesn’t cut it to point to some complicated, hard-to-simulate molecule and say “aha!  quantum supremacy!”

One discovery, which I and others stumbled on 7 or 8 years ago, is that quantum supremacy seems to become much easier to demonstrate if we switch from problems with a single valid output to sampling problems: that is, problems of sampling exactly or approximately from some specified probability distribution.

Doing this has two advantages.  First, we no longer need a full, fault-tolerant quantum computer—in fact, very rudimentary types of quantum hardware appear to suffice.  Second, we can design sampling problems for which we can arguably be more confident that they really are hard for a classical computer, than we are that (say) factoring is classically hard.  I like to say that a fast classical factoring algorithm might collapse the world’s electronic commerce, but as far as we know, it wouldn’t collapse the polynomial hierarchy!  But with sampling problems, at least with exact sampling, we can often show the latter implication, which is about the best evidence you can possibly get for such a problem being hard in the present state of mathematics.

One example of these sampling tasks that we think are classically hard is BosonSampling, which Alex Arkhipov and I proposed in 2011.  BosonSampling uses a bunch of identical photons that are sent through a network of beamsplitters, then measured to count the number of photons in each output mode.  Over the past few years, this proposal has been experimentally demonstrated by quantum optics groups around the world, with the current record being a 6-photon demonstration by the O’Brien group in Bristol, UK.  A second example is the IQP (“Instantaneous Quantum Polynomial-Time”) or Commuting Hamiltonians model of Bremner, Jozsa, and Shepherd.

A third example—no doubt the simplest—is just to sample from the output distribution of a random quantum circuit, let’s say on a 2D square lattice of qubits with nearest-neighbor interactions.  Notably, this last task is one that the Martinis group at Google is working toward achieving right now, with 40-50 qubits.  They say that they’ll achieve it in as little as one or two years, which translated from experimental jargon, means maybe five years?  But not infinity years.

The challenges on the experimental side are clear: get enough qubits with long enough coherence times to achieve this.  But there are also some huge theoretical challenges remaining.

A first is, can we still solve classically hard sampling problems even in the presence of realistic experimental imperfections?  Arkhipov and I already thought about that problem—in particular, about sampling from a distribution that’s merely close in variation distance to the BosonSampling one—and got results that admittedly weren’t as satisfactory as the results for exact sampling.  But I’m delighted to say that, just within the last month or two, there have been some excellent new papers on the arXiv that tackle exactly this question, with both positive and negative results.

A second theoretical challenge is, how do we verify the results of a quantum supremacy experiment?  Note that, as far as we know today, verification could itself require classical exponential time.  But that’s not the showstopper that some people think, since we could target the “sweet spot” of 40-50 qubits, where classical verification is difficult (and in particular, clearly “costlier” than running the experiment itself), but also far from impossible with cluster computing resources.

If I have any policy advice, it’s this: recognize that a clear demonstration of quantum supremacy is at least as big a deal as (say) the discovery of the Higgs boson.  After this scientific milestone is achieved, I predict that the whole discussion of commercial applications of quantum computing will shift to a new plane, much like the Manhattan Project shifted to a new plane after Fermi built his pile under the Chicago stadium in 1942.  In other words: at this point, the most “applied” thing to do might be to set applications aside temporarily, and just achieve this quantum supremacy milestone—i.e., build the quantum computing Fermi pile—and thereby show the world that quantum computing speedups are a reality.  Thank you.

Avi Wigderson’s “Permanent” Impact on Me

Wednesday, October 12th, 2016

The following is the lightly-edited transcript of a talk that I gave a week ago, on Wednesday October 5, at Avi Wigderson’s 60th birthday conference at the Institute for Advanced Study in Princeton.  Videos of all the talks (including mine) are now available here.

Thanks so much to Sanjeev Arora, Boaz Barak, Ran Raz, Peter Sarnak, and Amir Shpilka for organizing the conference and for inviting me to speak; to all the other participants and speakers for a great conference; and of course to Avi himself for being Avi. –SA


I apologize that I wasn’t able to prepare slides for today’s talk. But the good news is that, ever since I moved to Texas two months ago, I now carry concealed chalk everywhere I go. [Pull chalk out of pocket]

My history with Avi goes back literally half my life. I spent a semester with him at Hebrew University, and then a year with him as a postdoc here at IAS. Avi has played a bigger role in my career than just about anyone—he helped me professionally, he helped me intellectually, and once I dated and then married an Israeli theoretical computer scientist (who was also a postdoc in Avi’s group), Avi even helped me learn Hebrew. Just today, Avi taught me the Hebrew word for the permanent of a matrix. It turns out that it’s [throaty noises] pehhrmahnent.

But it all started with a talk that Avi gave in Princeton in 2000, which I attended as a prospective graduate student. That talk was about the following function of an n×n matrix A∈Rn×n, the permanent:

$$ \text{Per}(A) = \sum_{\sigma \in S_n} \prod_{i=1}^n a_{i,\sigma(i)}. $$

Avi contrasted that function with a superficially similar function, the determinant:

$$ \text{Det}(A) = \sum_{\sigma \in S_n} (-1)^{\text{sgn}(\sigma)} \prod_{i=1}^n a_{i,\sigma(i)}. $$

In this talk, I want to share a few of the amazing things Avi said about these two functions, and how the things he said then reverberated through my entire career.

Firstly, like we all learn in kindergarten or whatever, the determinant is computable in polynomial time, for example by using Gaussian elimination. By contrast, Valiant proved in 1979 that computing the permanent is #P-complete—which means, at least as hard as any combinatorial counting problem, a class believed to be even harder than NP-complete.

So, despite differing from each other only by some innocent-looking -1 factors, which the determinant has but the permanent lacks, these two functions effectively engage different regions of mathematics. The determinant is linear-algebraic and geometric; it’s the product of the eigenvalues and the volume of the parallelipiped defined by the row vectors. But the permanent is much more stubbornly combinatorial. It’s not quite as pervasive in mathematics as the determinant is, though it does show up. As an example, if you have a bipartite graph G, then the permanent of G’s adjacency matrix counts the number of perfect matchings in G.

When n=2, computing the permanent doesn’t look too different from computing the determinant: indeed, we have

$$
\text{Per}\left(
\begin{array}
[c]{cc}%
a & b\\
c & d
\end{array}
\right) =\text{Det}\left(
\begin{array}
[c]{cc}%
a & -b\\
c & d
\end{array}
\right).
$$

But as n gets larger, the fact that the permanent is #P-complete means that it must get exponentially harder to compute than the determinant, unless basic complexity classes collapse. And indeed, to this day, the fastest known algorithm to compute an n×n permanent, Ryser’s algorithm, takes O(n2n) time, which is only modestly better than the brute-force algorithm that just sums all n! terms.

Yet interestingly, not everything about the permanent is hard. So for example, if A is nonnegative, then in 1997, Jerrum, Sinclair, and Vigoda famously gave a polynomial-time randomized algorithm to approximate Per(A) to within a 1+ε multiplicative factor, for ε>0 as small as you like. As an even simpler example, if A is nonnegative and you just want to know whether its permanent is zero or nonzero, that’s equivalent to deciding whether a bipartite graph has at least one perfect matching. And we all know that that can be done in polynomial time.


OK, but the usual algorithm from the textbooks that puts the matching problem in the class P is already a slightly nontrivial one—maybe first grade rather than kindergarten! It involves repeatedly using depth-first search to construct augmenting paths, then modifying the graph, etc. etc.

Sixteen years ago in Princeton, the first thing Avi said that blew my mind is that there’s a vastly simpler polynomial-time algorithm to decide whether a bipartite graph has a perfect matching—or equivalently, to decide whether a nonnegative matrix has a zero or nonzero permanent. This algorithm is not quite as efficient as the textbook one, but it makes up for it by being more magical.

So here’s what you do: you start with the 0/1 adjacency matrix of your graph. I’ll do a 2×2 example, since that’s all I’ll be able to compute on the fly:

$$ \left(
\begin{array}
[c]{cc}%
1 & 1\\
0 & 1
\end{array}
\right). $$

Then you normalize each row so it sums to 1. In the above example, this would give

$$ \left(
\begin{array}
[c]{cc}%
\frac{1}{2} & \frac{1}{2} \\
0 & 1
\end{array}
\right). $$

Next you normalize each column so it sums to 1:

$$ \left(
\begin{array}
[c]{cc}%
1 & \frac{1}{3} \\
0 & \frac{2}{3}
\end{array}
\right). $$

OK, but now the problem is that the rows are no longer normalized, so you normalize them again:

$$ \left(
\begin{array}
[c]{cc}%
\frac{3}{4} & \frac{1}{4} \\
0 & 1
\end{array}
\right). $$

Then you normalize the columns again, and so on. You repeat something like n3log(n) times. If, after that time, all the row sums and column sums have become within ±1/n of 1, then you conclude that the permanent was nonzero and the graph had a perfect matching. Otherwise, the permanent was zero and the graph had no perfect matching.

What gives? Well, it’s a nice exercise to prove why this works. I’ll just give you a sketch: first, when you multiply any row or column of a matrix by a scalar, you multiply the permanent by that same scalar. By using that fact, together with the arithmetic-geometric mean inequality, it’s possible to prove that, in every iteration but the first, when you rebalance all the rows or all the columns to sum to 1, you can’t be decreasing the permanent. The permanent increases monotonically.

Second, if the permanent is nonzero, then after the first iteration it’s at least 1/nn, simply because we started with a matrix of 0’s and 1’s.

Third, the permanent is always at most the product of the row sums or the product of the column sums, which after the first iteration is 1.

Fourth, at any iteration where there’s some row sum or column sum that’s far from 1, rescaling must not only increase the permanent, but increase it by an appreciable amount—like, a 1+1/n2 factor or so.

Putting these four observations together, we find that if the permanent is nonzero, then our scaling procedure must terminate after a polynomial number of steps, with every row sum and every column sum close to 1—since otherwise, the permanent would just keep on increasing past its upper bound of 1.

But a converse statement is also true. Suppose the matrix can be scaled so that every row sum and every column sum gets within ±1/n of 1. Then the rescaled entries define a flow through the bipartite graph, with roughly the same amount of flow through each of the 2n vertices. But if such a flow exists, then Hall’s Theorem tells you that there must be a perfect matching (and hence the permanent must be nonzero)—since if a matching didn’t exist, then there would necessarily be a bottleneck preventing the flow.

Together with Nati Linial and Alex Samorodnitsky, Avi generalized this to show that scaling the rows and columns gives you a polynomial-time algorithm to approximate the permanent of a nonnegative matrix. This basically follows from the so-called Egorychev-Falikman Theorem, which says that the permanent of a doubly stochastic matrix is at least n!/nn. The approximation ratio that you get this way, ~en, isn’t nearly as good as Jerrum-Sinclair-Vigoda’s, but the advantage is that the algorithm is deterministic (and also ridiculously simple).

For myself, though, I just filed away this idea of matrix scaling for whenever I might need it. It didn’t take long. A year after Avi’s lecture, when I was a beginning grad student at Berkeley, I was obsessing about the foundations of quantum mechanics. Specifically, I was obsessing about the fact that, when you measure a quantum state, the rules of quantum mechanics tell you how to calculate the probability that you’ll see a particular outcome. But the rules are silent about so-called multiple-time or transition probabilities. In other words: suppose we adopt Everett’s Many-Worlds view, according to which quantum mechanics needs to be applied consistently to every system, regardless of scale, so in particular, the state of the entire universe (including us) is a quantum superposition state. We perceive just one branch, but there are also these other branches where we made different choices or where different things happened to us, etc.

OK, fine: for me, that’s not the troubling part! The troubling part is that quantum mechanics rejects as meaningless questions like the following: given that you’re in this branch of the superposition at time t1, what’s the probability that you’ll be in that branch at time t2, after some unitary transformation is applied? Orthodox quantum mechanics would say: well, either someone measured you at time t1, in which case their act of measuring collapsed the superposition and created a whole new situation. Or else no one measured at t1—but in that case, your state at time t1 was the superposition state, full stop. It’s sheer metaphysics to imagine a “real you” that jumps around from one branch of the superposition to another, having a sequence of definite experiences.

Granted, in practice, branches of the universe’s superposition that split from each other tend never to rejoin, for the same thermodynamic reasons why eggs tend never to unscramble themselves. And as long as the history of the Everettian multiverse has the structure of a tree, we can sensibly define transition probabilities. But if, with some technology of the remote future, we were able to do quantum interference experiments on human brains (or other conscious entities), the rules of quantum mechanics would no longer predict what those beings should see—not even probabilistically.

I was interested in the question: suppose we just wanted to postulate transition probabilities, with the transitions taking place in some fixed orthogonal basis. What would be a mathematically reasonable way to do that? And it occurred to me that one thing you could do is the following. Suppose for simplicity that you have a pure quantum state, which is just a unit vector of n complex numbers called amplitudes:

$$ \left(
\begin{array}
[c]{c}%
\alpha_{1}\\
\vdots\\
\alpha_{n}%
\end{array}
\right) $$

Then the first rule of quantum mechanics says that you can apply any unitary transformation U (that is, any norm-preserving linear transformation) to map this state to a new one:

$$ \left(
\begin{array}
[c]{c}%
\beta_{1}\\
\vdots\\
\beta_{n}%
\end{array}
\right) =\left(
\begin{array}
[c]{ccc}%
u_{11} & \cdots & u_{1n}\\
\vdots & \ddots & \vdots\\
u_{n1} & \cdots & u_{nn}%
\end{array}
\right) \left(
\begin{array}
[c]{c}%
\alpha_{1}\\
\vdots\\
\alpha_{n}%
\end{array}
\right). $$

The second rule of quantum mechanics, the famous Born Rule, says that if you measure in the standard basis before applying U, then the probability that you’ll find youself in state i equals |αi|2. Likewise, if you measure in the standard basis after applying U, the probability that you’ll find youself in state j equals |βj|2.

OK, but what’s the probability that you’re in state i at the initial time, and then state j at the final time? These joint probabilities, call them pij, had better add up to |αi|2 and |βj|2, if we sum the rows and columns respectively. And ideally, they should be “derived” in some way from the unitary U—so that for example, if uij=0 then pij=0 as well.

So here’s something you could do: start by replacing each uij by its absolute value, to get a nonnegative matrix. Then, normalize the ith row so that it sums to |αi|2, for each i. Then normalize the jth column so that it sums to |βj|2, for each j. Then normalize the rows, then the columns, and keep iterating until hopefully you end up with all the rows and columns having the right sums.

So the first question I faced was, does this process converge? And I remembered what Avi taught me about the permanent. In this case, because of the nonuniform row and column scalings, the permanent no longer works as a progress measure, but there’s something else that does work. Namely, as a first step, we can use the Max-Flow/Min-Cut Theorem to show that there exists a nonnegative matrix F=(fij) such that fij=0 whenever uij=0, and also

$$ \sum_j f_{ij} = \left|\alpha_i\right|^2 \forall i,\ \ \ \ \ \sum_i f_{ij} = \left|\beta_j\right|^2 \forall j. $$

Then, letting M=(mij) be our current rescaled matrix (so that initially mij:=|uij|), we use

$$ \prod_{i,j : u_{ij}\ne 0} m_{ij}^{f_{ij}} $$

as our progress measure. By using the nonnegativity of the Kullback-Leibler divergence, one can prove that this quantity never decreases. So then, just like with 0/1 matrices and the permanent, we get eventual convergence, and indeed convergence after a number of iterations that’s polynomial in n.

I was pretty stoked about this until I went to the library, and discovered that Erwin Schrödinger had proposed the same matrix scaling process in 1931! And Masao Nagasawa and others then rigorously analyzed it. OK, but their motivations were somewhat different, and for some reason they never talked about finite-dimensional matrices, only infinite-dimensional ones.

I can’t resist telling you my favorite open problem about this matrix scaling process: namely, is it stable under small perturbations? In other words, if I change one of the αi‘s or uij‘s by some small ε, then do the final pij‘s also change by at most some small δ? To clarify, several people have shown me how to prove that the mapping to the pij‘s is continuous. But for computer science applications, one needs something stronger: namely that when the matrix M, and the row and column scalings, actually arise from a unitary matrix in the way above, we get strong uniform continuity, with a 1/nO(1) change to the inputs producing only a 1/nO(1) change to the outputs (and hopefully even better than that).

The more general idea that I was groping toward or reinventing here is called a hidden-variable theory, of which the most famous example is Bohmian mechanics. Again, though, Bohmian mechanics has the defect that it’s only formulated for some exotic state space that the physicists care about for some reason—a space involving pointlike objects called “particles” that move around in 3 Euclidean dimensions (why 3? why not 17?).

Anyway, this whole thing led me to wonder: under the Schrödinger scaling process, or something like it, what’s the computational complexity of sampling an entire history of the hidden variable through a quantum computation? (“If, at the moment of your death, your whole life history flashes before you in an instant, what can you then efficiently compute?”)

Clearly the complexity is at least BQP (i.e., quantum polynomial time), because even sampling where the hidden variable is at a single time is equivalent to sampling the output distribution of a quantum computer. But could the complexity be even more than BQP, because of the correlations between the hidden variable values at different times? I noticed that, indeed, sampling a hidden variable history would let you do some crazy-seeming things, like solve the Graph Isomorphism problem in polynomial time (OK, fine, that seemed more impressive at the time than it does after Babai’s breakthrough), or find collisions in arbitrary cryptographic hash functions, or more generally, solve any problem in the complexity class SZK (Statistical Zero Knowledge).

But you might ask: what evidence do we have that any these problems are hard even for garden-variety quantum computers? As many of you know, it’s widely conjectured today that NP⊄BQP—i.e., that quantum computers can’t solve NP-complete problems in polynomial time. And in the “black box” setting, where all you know how to do is query candidate solutions to your NP-complete problem to check whether they’re valid, it’s been proven that quantum computers don’t give you an exponential speedup: the best they can give is the square-root speedup of Grover’s algorithm.

But for these SZK problems, like finding collisions in hash functions, who the hell knows? So, this is the line of thought that led me to probably the most important thing I did in grad school, the so-called quantum lower bound for collision-finding. That result says that, if (again) your hash function is only accessible as a black box, then a quantum computer can provide at most a polynomial speedup over a classical computer for finding collisions in it (in this case, it turns out, at most a two-thirds power speedup). There are several reasons you might care about that, such as showing that one of the basic building blocks of modern cryptography could still be secure in a world with quantum computers, or proving an oracle separation between SZK and BQP. But my original motivation was just to understand how transition probabilities would change quantum computation.


The permanent has also shown up in a much more direct way in my work on quantum computation. If we go back to Avi’s lecture from 2000, a second thing he said that blew my mind was that apparently, or so he had heard, even the fundamental particles of the universe know something about the determinant and the permanent. In particular, he said, fermions—the matter particles, like the quarks and electrons in this stage—have transition amplitudes that are determinants of matrices. Meanwhile, bosons—the force-carrying particles, like the photons coming from the ceiling that let you see this talk—have transition amplitudes that are permanents of matrices.

Or as Steven Weinberg, one of the great physicists on earth, memorably put it in the first edition of his recent quantum mechanics textbook: “in the case of bosons, it is also a determinant, except without minus signs.” I’ve had the pleasure of getting to know Weinberg at Austin, so recently I asked him about that line. He told me that of course he knew that the determinant without minus signs is called a permanent, but he thought no one else would know! As far as he knew, the permanent was just some esoteric function used by a few quantum field theorists who needed to calculate boson amplitudes.

Briefly, the reason why the permanent and determinant turn up here is the following: whenever you have n particles that are identical, to calculate the amplitude for them to do something, you need to sum over all n! possible permutations of the particles. Furthermore, each contribution to the sum is a product of n complex numbers, one uij for each particle that hops from i to j. But there’s a difference: when you swap two identical bosons, nothing happens, and that’s why bosons give rise to the permanent (of an n×n complex matrix, if there are n bosons). By contrast, when you swap two identical fermions, the amplitude for that state of the universe gets multiplied by -1, and that’s why fermions give rise to the determinant.

Anyway, Avi ended his talk with a quip about how unfair it seemed to the bosons that they should have to work so much harder than the fermions just to calculate where they should be!

And then that one joke of Avi—that way of looking at things—rattled around in my head for a decade, like a song I couldn’t get rid of. It raised the question: wait a minute, bosons—particles that occur in Nature—are governed by a #P-complete function? Does that mean we could actually use bosons to solve #P-complete problems in polynomial time? That seems ridiculous, like the kind of nonsense I’m fighting every few weeks on my blog! As I said before, most of us don’t even expect quantum computers to be able to solve NP-complete problems in polynomial time, let alone #P-complete ones.

As it happens, Troyansky and Tishby had already taken up that puzzle in 1996. (Indeed Avi, being the social butterfly and hub node of our field that he is, had learned about the role of permaments and determinants in quantum mechanics from them.) What Troyansky and Tishby said was, it’s true that if you have a system of n identical, non-interacting bosons, their transition amplitudes are given by permanents of n×n matrices. OK, but amplitudes in quantum mechanics are not directly observable. They’re just what you use to calculate the probability that you’ll see this or that measurement outcome. But if you try to encode a hard instance of a #P-complete problem into a bosonic system, the relevant amplitudes will in general be exponentially small. And that means that, if you want a decent estimate of the permanent, you’ll need to repeat the experiment an exponential number of times. So OK, they said, nice try, but this doesn’t give you a computational advantage after all in calculating the permanent compared to classical brute force.

In our 2011 work on BosonSampling, my student Alex Arkhipov and I reopened the question. We said, not so fast. It’s true that bosons don’t seem to help you in estimating the permanent of a specific matrix of your choice. But what if your goal was just to sample a random n×n matrix A∈Cn×n, in a way that’s somehow biased toward matrices with larger permanents? Now, why would that be your goal? I have no idea! But this sampling is something that a bosonic system would easily let you do.

So, what Arkhipov and I proved was that this gives rise to a class of probability distributions that can be sampled in quantum polynomial time (indeed, by a very rudimentary type of quantum computer), but that can’t be sampled in classical polynomial time unless the polynomial hierarchy collapses to the third level. And even though you’re not solving a #P-complete problem, the #P-completeness of the permanent still plays a crucial role in explaining why the sampling problem is hard. (Basically, one proves that the probabilities are #P-hard even to approximate, but that if there were a fast classical sampling algorithm, then the probabilities could be approximated in the class BPPNP. So if a fast classical sampling algorithm existed, then P#P would equal BPPNP, which would collapse the polynomial hierarchy by Toda’s Theorem.)

When we started on this, Arkhipov and I thought about it as just pure complexity theory—conceptually clarifying what role the #P-completeness of the permanent plays in physics. But then at some point it occurred to us: bosons (such as photons) actually exist, and experimentalists in quantum optics like to play with them, so maybe they could demonstrate some of this stuff in the lab. And as it turned out, the quantum optics people were looking for something to do at the time, and they ate it up.

Over the past five years, a trend has arisen in experimental physics that goes by the name “Quantum Supremacy,” although some people are now backing away from the name because of Trump. The idea is: without yet having a universal quantum computer, can we use the hardware that we’re able to build today to demonstrate the reality of a quantum-computational speedup as clearly as possible? Not necessarily for a useful problem, but just for some problem? Of course, no experiment can prove that something is scaling polynomially rather than exponentially, since that’s an asymptotic statement. But an experiment could certainly raise the stakes for the people who deny such a statement—for example, by solving something a trillion times faster than we know how to solve it otherwise, using methods for which we don’t know a reason for them not to scale.

I like to say that for me, the #1 application of quantum computing, more than breaking RSA or even simulating physics and chemistry, is simply disproving the people who say that quantum computing is impossible! So, quantum supremacy targets that application.

Experimental BosonSampling has become a major part of the race to demonstrate quantum supremacy. By now, at least a half-dozen groups around the world have reported small-scale implementations—the record, so far, being an experiment at Bristol that used 6 photons, and experimentally confirmed that, yes, their transition amplitudes are given by permanents of 6×6 complex matrices. The challenge now is to build single-photon sources that are good enough that you could scale up to (let’s say) 30 photons, which is where you’d really start seeing a quantum advantage over the best known classical algorithms. And again, this whole quest really started with Avi’s joke.

A year after my and Arkhipov’s work, I noticed that one also can run the connection between quantum optics and the permanent in the “reverse” direction. In other words: with BosonSampling, we used the famous theorem of Valiant, that the permanent is #P-complete, to help us argue that bosons can solve hard sampling problems. But if we know by some other means that quantum optics lets us encode #P-complete problems, then we can use that to give an independent, “quantum” proof that the permanent is #P-complete in the first place! As it happens, there is another way to see why quantum optics lets us encode #P-complete problems. Namely, we can use celebrated work by Knill, Laflamme, and Milburn (KLM) from 2001, which showed how to perform universal quantum computation using quantum optics with the one additional resource of “feed-forward measurements.” With minor modifications, the construction by KLM also lets us encode a #P-complete problem into a bosonic amplitude, which we know is a permanent—thereby proving that the permanent is #P-complete, in what I personally regard as a much more intuitive way than Valiant’s original approach based on cycle covers. This illustrates a theme that we’ve seen over and over in the last 13 years or so, which is the use of quantum methods and arguments to gain insight even about classical computation.

Admittedly, I wasn’t proving anything here in classical complexity theory that wasn’t already known, just giving a different proof for an old result! Extremely recently, however, my students Daniel Grier and Luke Schaeffer have extended my argument based on quantum optics, to show that computing the permanent of a unitary or orthogonal matrix is #P-complete. (Indeed, even over finite fields of characteristic k, computing the permanent of an orthogonal matrix is a ModkP-complete problem, as long as k is not 2 or 3—which turns out to be the tight answer.) This is not a result that we previously knew by any means, whether quantum or classical.

I can’t resist telling you the biggest theoretical open problem that arose from my and Arkhipov’s work. We would like to say: even if you had a polynomial-time algorithm that sampled a probability distribution that was merely close, in variation distance, to the BosonSampling distribution, that would already imply a collapse of the polynomial hierarchy. But we’re only able to prove that assuming a certain problem is #P-complete, which no one has been able to prove #P-complete. That problem is the following:

Given an n×n matrix A, each of whose entries is an i.i.d. complex Gaussian with mean 0 and variance 1 (that is, drawn from N(0,1)C), estimate |Per(A)|2, to within additive error ±ε·n!, with probability at least 1-δ over the choice of A. Do this in time polynomial in n, 1/ε, and 1/δ.

Note that, if you care about exactly computing the permanent of a Gaussian random matrix, or about approximating the permanent of an arbitrary matrix, we know how to prove both of those problems #P-complete. The difficulty “only” arises when we combine approximation and average-case in the same problem.

At the moment, we don’t even know something more basic, which is: what’s the distribution over |Per(A)|2, when A is an n×n matrix of i.i.d. N(0,1)C Gaussians? Based on numerical evidence, we conjecture that the distribution converges to lognormal as n gets large. By using the interpretation of the determinant as the volume of a parallelipiped, we can prove that the distribution over |Det(A)|2 converges to lognormal. And the distribution over |Per(A)|2 looks almost the same when you plot it. But not surprisingly, the permanent is harder to analyze.


This brings me to my final vignette. Why would anyone even suspect that approximating the permanent of a Gaussian random matrix would be a #P-hard problem? Well, because if you look at the permanent of an n×n matrix over a large enough finite field, say Fp, that function famously has the property of random self-reducibility. This means: the ability to calculate such a permanent in polynomial time, on 90% all matrices in Fpn×n, or even for that matter on only 1% of them, implies the ability to calculate it in polynomial time on every such matrix.

The reason for this is simply that the permanent is a low-degree polynomial, and low-degree polynomials have extremely useful error-correcting properties. In particular, if you can compute such a polynomial on any large fraction of points, then you can do noisy polynomial interpolation (e.g., the Berlekamp-Welch algorithm, or list decoding), in order to get the value of the polynomial on an arbitrary point.

I don’t specifically remember Avi talking about the random self-reducibility of the permanent in his 2000 lecture, but he obviously would have talked about it! And it was really knowing about the random self-reducibility of the permanent, and how powerful it was, that let me and Alex Arkhipov to the study of BosonSampling in the first place.

In complexity theory, the random self-reducibility of the permanent is hugely important because it was sort of the spark for some of our most convincing examples of non-relativizing results—that is, results that fail relative to a suitable oracle. The most famous such result is that #P, and for that matter even PSPACE, admit interactive protocols (the IP=PSPACE theorem). In the 1970s, Baker, Gill, and Solovay pointed out that non-relativizing methods would be needed to resolve P vs. NP and many of the other great problems of the field.

In 2007, Avi and I wrote our only joint paper so far. In that paper, we decided to take a closer look at the non-relativizing results based on interactive proofs. We said: while it’s true that these results don’t relativize—that is, there are oracles relative to which they fail—nevertheless, these results hold relative to all oracles that themselves encode low-degree polynomials over finite fields (such as the permanent). So, introducing a term, Avi and I said that results like IP=PSPACE algebrize.

But then we also showed that, if you want to prove P≠NP—or for that matter, even prove circuit lower bounds that go “slightly” beyond what’s already known (such as NEXPP/poly)—you’ll need techniques that are not only non-relativizing, but also non-algebrizing. So in some sense, the properties of the permanent that are used (for example) in proving that it has an interactive protocol, just “aren’t prying the black box open wide enough.”

I have a more recent result, from 2011 or so, that I never got around to finishing a paper about. In this newer work, I decided to take another look at the question: what is it about the permanent that actually fails to relativize? And I prove the following result: relative to an arbitrary oracle A, the class #P has complete problems that are both random self-reducible and downward self-reducible (that is, reducible to smaller instances of the same problem). So, contrary to what certainly I and maybe others had often thought, it’s not the random self-reducibility of the permanent that’s the crucial thing about it. What’s important, instead, is a further property that the permanent has, of being self-checkable and self-correctible.

In other words: given (say) a noisy circuit for the permanent, it’s not just that you can correct that circuit to compute whichever low-degree polynomial it was close to computing. Rather, it’s that you can confirm that the polynomial is in fact the permanent, and nothing else.

I like the way Ketan Mulmuley thinks about this phenomenon in his Geometric Complexity Theory, which is a speculative, audacious program to try to prove that the permanent is harder than the determinant, and to tackle the other great separation questions of complexity theory (including P vs. NP), by using algebraic geometry and representation theory. Mulmuley says: the permanent is a polynomial in the entries of an n×n matrix that not only satisfies certain symmetries (e.g., under interchanging rows or columns), but is uniquely characterized by those symmetries. In other words, if you find a polynomial that passes certain tests—for example, if it behaves in the right way under rescaling and interchanging rows and columns—then that polynomial must be the permanent, or a scalar multiple of the permanent. Similarly, if you find a polynomial that passes the usual interactive proof for the permanent, that polynomial must be the permanent. I think this goes a long way toward explaining why the permanent is so special: it’s not just any hard-to-compute, low-degree polynomial; it’s one that you can recognize when you come across it.


I’ve now told you about the eventual impact of one single survey talk that Avi gave 16 years ago—not even a particularly major or important one. So you can only imagine what Avi’s impact must have been on all of us, if you integrate over all the talks he’s given and papers he’s written and young people he’s mentored and connections he’s made his entire career. May that impact be permanent.