1. 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

2. 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?

3. 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

4. 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! 19901995200020052010 4

5. 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

6. 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

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 7

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 8

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 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

10. 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

11. 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

12. 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

13. 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!

14. 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…

15. 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

16. 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

17. 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