Scott Aaronson (MIT) BQP and 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 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? Well-known correspondence between relativized PH and AC 0 : interpret the s as OR gates, the s as AND gates, and the oracle string as an input of size 2 n Oracles are just the obvious way to address the BQP vs. PH question, not some woo-woo thing Recall that the early evidence for BPPBQP (e.g. Simons alg) was also oracle evidence; then Shor found a similar oracle that could be instantiated by F ACTORING 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

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 7

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 8

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 specific 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! Just adds more layers to the AC 0 circuit Challenge: Show that w.h.p., C is forced to estimate eventually, even if it tries to avoid it 9

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 10

F OURIER C HECKING Is In BQP H H H H H H f |0 g H H H Probability of observing |0 n : 11

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? 12

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, 13 Approximation is multiplicative, not additive … thats important!

Bazzi07 proved the depth-2 case 14 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 15 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

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 But every function with a T-query quantum algorithm, is approximated by a degree-2T real polynomial! [Beals et al. 98] Example: The following degree-4 polynomial distinguishes the uniform distribution over f,g from the forrelated one: 16 But this polynomial solves F OURIER C HECKING only by exploiting massive cancellations between positive and negative terms (Not coincidentally, a central feature of quantum algorithms!) Our conjecture says 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 Such a low-fat approximation of AC 0 circuits would be useful for independent reasons in learning theory

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? 17 $100$200