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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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