Scott Aaronson BQP und PH A tale of two strong-willed complexity classes… A 16-year-old quest to find an oracle that separates them… A solution at lastbut only for relational problems… The beast guarding the inner sanctum unmasked: the Generalized Linial-Nisan Conjecture… Where others flee in terror, a Braver Man attacks… A $200 bounty for slaughtering the wounded beast… 1

Quantum Computing: Where Does It Fit? PH P BPP AM NP PP P #P BQP 2 Factoring, discrete log, etc.: In BQP Not known to be in BPP But in NP coNP Could there be a problem in BQP\PH?

First question: can we at least find an oracle A such that BQP A PH A ? Essentially the same as finding a problem in quantum logarithmic time, but not AC 0 Why? Standard correspondence between relativized PH and AC 0 : replace s by OR gates, s by AND gates, and the oracle string by an input of size 2 n Relativization is just the obvious way to address the BQP vs. PH question, not some woo-woo thing People who claim they dont like oracle results really just dont understand them 3

BQP vs. PH: A Timeline Bernstein and Vazirani define BQP They construct an oracle problem, R ECURSIVE F OURIER S AMPLING, that has quantum query complexity n but classical query complexity n (log n) First example where quantum is superpolynomially better! A simple extension yields RFS MA Natural conjecture: RFS PH Alas, we cant even prove RFS AM!

Why do we care whether BQP PH? Does simulating quantum mechanics reduce to search or approximate counting? What other candidates for exponential quantum speedups are therebesides NP-intermediate problems like factoring? Could quantum computers provide exponential speedups even if P=NP? Would a fast quantum algorithm for NP-complete problems collapse the polynomial hierarchy? 5

This Talk 1.We achieve an oracle separation between the relational versions of BQP and PH (FBQP and FBPP PH ) 2.We study a new oracle problemF OURIER C HECKING thats in BQP, but not in BPP, MA, BPP path, SZK... 3.We conjecture that F OURIER C HECKING is not in PH, and prove that this would follow from the Generalized Linial- Nisan Conjecture Original Linial-Nisan Conjecture was proved by Braverman 2009, after being open for 20 years 6

Relational Problems FBPP: Class of relations, R {0,1}* {0,1}*, for which there exists a BPP machine that, given any x, outputs a y such that If we compared FBQP to FP PH, a separation would be trivial! Output an n-bit string with Kolmogorov complexity n/2 FBQP: Same but with quantum Well produce separations where the FBQP machine succeeds with probability 1-1/exp(n), while the FBPP PH machine succeeds with probability at most (say) 99% Note: Amplification not obvious; constant could actually matter! 7

Fourier Sampling Problem Given oracle access to a random Boolean function The Task: Output strings z 1,…,z n, at least 75% of which satisfy and at least 25% of which satisfy where 8

F OURIER S AMPLING Is In BQP Algorithm: H H H H H H f |0 Repeat n times; output whatever you see Distribution over Fourier coefficients Distribution over Fourier coefficients output by quantum algorithm 9

F OURIER S AMPLING Is Not In PH Key Idea: Show that, if we had a constant-depth 2 poly(n) -size circuit C for F OURIER S AMPLING, then we could violate a known AC 0 lower bound, by sneaking a M AJORITY problem into the estimation of some random Fourier coefficient Obvious problem: How do we know C will output the particular s were interested in, thereby revealing anything about ? We dont! (Indeed, theres only a ~1/2 n chance it will) But we have a long time to wait, since our reduction can be nondeterministic! That just adds more layers to the AC 0 circuit 10

Suppose each bit of an N-bit string is 1 with independent probability p. Then any depth-d circuit to decide whether p=½ or p=½+ (with constant bias) must have size If youre here, you can prove this Starting Point for Reduction 11 Well take a circuit that outputs slightly-larger-than-average Fourier coefficients of f, and get a circuit for detecting bias

Alice: Chooses s {0,1} n and b {0,1} uniformly at random 12 The Fourier Guessing Game For each x {0,1} n, sets Bob: Must output a z such that Sends truth table of f to Bob Keeps s,b secret Key Theorem: Regardless of Bobs strategy, In other words, if >1.1, Bob outputs the true s with probability noticeably more than 1/2 n … even if he tries to avoid it!

Finishing the Proof Let A be a random oracle View A as encoding a random Boolean function f n :{0,1} n {-1,1} for each n Let R be the relational problem where, on input 0 n, youre asked to output z 1,…,z n, at least 75% of which satisfy and at least 25% of which satisfy Clearly 13 On the other hand, standard diagonalization tricks imply

Decision Version: F OURIER C HECKING Given oracle access to two Boolean functions Decide whether (i) f,g are drawn from the uniform distribution U, or (ii) f,g are drawn from the following forrelated distribution F: pick a random unit vector then let 14

F OURIER C HECKING Is In BQP H H H H H H f |0 g H H H Probability of observing |0 n : 15

Intuition: F OURIER C HECKING Shouldnt Be In PH Why? For any individual s, computing the Fourier coefficient is a #P-complete problem f and g being forrelated is an extremely global property: conditioning on a polynomial number of f(x) and g(y) values should reveal almost nothing about it But how to formalize and prove that? 16

Crucial Definition: A distribution D is -almost k-wise independent if for all k-terms C, Theorem: For all k, the forrelated distribution F is O(k 2 /2 n/2 )-almost k-wise independent Proof: A few pages of Gaussian integrals, then a discretization step A k-term is a product of k literals of the form x i or 1-x i A distribution D over {0,1} N is k-wise independent if for all k-terms C, 17 Approximation is multiplicative, not additive … thats important!

Bazzi07 proved the depth-2 case 18 Linial-Nisan Conjecture (1990) with weaker parameters that suffice for us : Let f:{0,1} n {0,1} be computed by a circuit of size and depth O(1). Then for all n (1) -wise independent distributions D, Generalized Linial-Nisan Conjecture: Let f be computed by a circuit of size and depth O(1). Then for all 1/n (1) -almost n (1) -wise independent distributions D, Razborov08 dramatically simplified Bazzis proofFinally, Braverman09 proved the whole thingAlas, we need the…

Low-Fat Sandwich Conjecture: Let f:{0,1} n {0,1} be computed by a circuit of size and depth O(1). Then there exist polynomials p l,p u :R n R, of degree n o(1), such that 19 Theorem (Bazzi): Low-Fat Sandwich Conjecture Generalized Linial-Nisan Conjecture (Without the low-fat condition,Sandwich Conjecture Linial-Nisan Conjecture) (i) Sandwiching. (ii) Approximation. (iii) Low-Fat. p l,p u can be written as where

We know how to prove constant-depth lower bounds! So why is BQP A PH A so much harder than (say) PP A PH A ? Because known techniques for showing a function f has no small constant-depth circuits, also involve (directly or indirectly) showing that f isnt approximated by a low-degree polynomial And this is a problem because… Lemma (Beals et al. 1998): Every Boolean function f that has a T-query quantum algorithm, also has a degree-2T real polynomial p such that |p(x)-f(x)| for all x {0,1} n Example: The following degree-4 polynomial distinguishes the uniform distribution over f,g from the forrelated one: 20

But this polynomial solves F OURIER C HECKING only by exploiting massive cancellations between positive and negative terms (Not coincidentally, the central feature of quantum algorithms!) You might conjecture that if f AC 0, then f is approximated not merely by a low-degree polynomial, but by a reasonable, classical-looking onewith some bound on the coefficients that prevents massive cancellations And thats exactly what the Low-Fat Sandwich Conjecture says! Such a low-fat approximation of AC 0 circuits would be useful for independent reasons in learning theory 21

Open Problems Prove the Generalized Linial-Nisan Conjecture! Yields an oracle A such that BQP A PH A Prove Generalized L-N even for the special case of DNFs. Yields an oracle A such that BQP A AM A Is there a Boolean function f:{0,1} n {-1,1} thats well- approximated in L 2 -norm by a low-degree real polynomial, but not by a low-degree low-fat polynomial? Can we instantiate F OURIER C HECKING by an explicit (unrelativized) problem? More generally, evidence for/against BQP PH in the real world? 22 $100$200