Scott Aaronson

1. BQPundPH A tale of two strong-willed complexity classes… A 16-year-old quest to find an oracle that separates them… A solution at last—but 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… Scott Aaronson

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

3. First question: can we at least find an oracle A such that BQPAPHA? Essentially the same as finding a problem in quantum logarithmic time, but not AC0 Why? Standard correspondence between relativized PH and AC0: replace ’s by OR gates, ’s by AND gates, and the oracle string by an input of size 2n Relativization is just the “obvious” way to address the BQP vs. PH question, not some woo-woo thing People who claim they don’t like oracle results really just don’t understand them

4. BQP vs. PH: A Timeline 1990 1995 2000 2005 2010 Bernstein and Vazirani define BQP They construct an oracle problem, Recursive Fourier Sampling, 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 can’t even prove RFSAM!

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 there—besides 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?

6. This Talk • We achieve an oracle separation between the relational versions of BQP and PH (FBQP and FBPPPH) • We study a new oracle problem—Fourier Checking—that’s in BQP, but not in BPP, MA, BPPpath, SZK... • We conjecture that Fourier Checking is not in PH, and prove that this would follow from the Generalized Linial-Nisan ConjectureOriginal Linial-Nisan Conjecture was proved by Braverman 2009, after being open for 20 years

7. 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 FBQP: Same but with quantum We’ll produce separations where the FBQP machine succeeds with probability 1-1/exp(n), while the FBPPPH machine succeeds with probability at most (say) 99%Note: Amplification not obvious; constant could actually matter! If we compared FBQP to FPPH, a separation would be trivial! “Output an n-bit string with Kolmogorov complexity  n/2”

8. Fourier Sampling Problem Given oracle access to a random Boolean function The Task: Output strings z1,…,zn, at least 75% of which satisfy and at least 25% of which satisfy where

9. Fourier Sampling Is In BQP |0 H H Repeat n times; output whatever you see Algorithm: |0 H f H |0 H H Distribution over Fourier coefficients Distribution over Fourier coefficients output by quantum algorithm

10. Fourier Sampling Is Not In PH Key Idea: Show that, if we had a constant-depth 2poly(n)-size circuit C for Fourier Sampling, then we could violate a known AC0 lower bound, by “sneaking a Majority problem” into the estimation of some random Fourier coefficient Obvious problem: How do we know C will output the particular s we’re interested in, thereby revealing anything about ? We don’t! (Indeed, there’s only a ~1/2n chance it will) But we have a long time to wait, since our reduction can be nondeterministic!That just adds more layers to the AC0 circuit

11. Starting Point for Reduction 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 you’re here, you can prove this We’ll take a circuit that outputs slightly-larger-than-average Fourier coefficients of f, and get a circuit for detecting  bias

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

13. Finishing the Proof Let A be a random oracle View A as encoding a random Boolean function fn:{0,1}n{-1,1} for each n Let R be the relational problem where, on input 0n, you’re asked to output z1,…,zn, at least 75% of which satisfyand at least 25% of which satisfy Clearly On the other hand, standard diagonalization tricks imply

14. Decision Version: Fourier Checking Given oracle access to two Boolean functions • Decide whether • f,g are drawn from the uniform distribution U, or • f,g are drawn from the following “forrelated” distribution F: pick a random unit vector then let

15. Fourier Checking Is In BQP |0 H H H |0 H f H g H |0 H H H Probability of observing |0n:

16. Intuition: Fourier Checking Shouldn’t 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?

17. A k-term is a product of k literals of the form xi or 1-xi A distribution D over {0,1}N is k-wise independent if for all k-terms C, Crucial Definition: A distribution D is -almost k-wise independent if for all k-terms C, Approximation is multiplicative, not additive … that’s important! Theorem: For all k, the forrelated distribution F is O(k2/2n/2)-almost k-wise independent Proof: A few pages of Gaussian integrals, then a discretization step

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, Razborov’08 dramatically simplified Bazzi’s proof Finally, Braverman’09 proved the whole thing Bazzi’07 proved the depth-2 case Alas, we need the… “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,

19. “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 pl,pu:RnR, of degree no(1), such that (i) Sandwiching. (ii) Approximation. (iii) Low-Fat. pl,pu can be written as where Theorem (Bazzi): Low-Fat Sandwich Conjecture Generalized Linial-Nisan Conjecture (Without the low-fat condition, Sandwich Conjecture Linial-Nisan Conjecture)

20. We know how to prove constant-depth lower bounds! So why is BQPAPHA so much harder than (say) PPAPHA? Because known techniques for showing a function f has no small constant-depth circuits, also involve (directly or indirectly) showing that f isn’t 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:

21. But this polynomial solves Fourier Checking only by exploiting “massive cancellations” between positive and negative terms(Not coincidentally, the central feature of quantum algorithms!) You might conjecture that if fAC0, then f is approximated not merely by a low-degree polynomial, but by a “reasonable,” “classical-looking” one—with some bound on the coefficients that prevents massive cancellationsAnd that’s exactly what the Low-Fat Sandwich Conjecture says! Such a “low-fat” approximation of AC0 circuits would be useful for independent reasons in learning theory

22. Open Problems Prove the Generalized Linial-Nisan Conjecture!Yields an oracle A such that BQPAPHA Prove Generalized L-N even for the special case of DNFs.Yields an oracle A such that BQPAAMA Is there a Boolean function f:{0,1}n{-1,1} that’s well-approximated in L2-norm by a low-degree real polynomial, but not by a low-degree low-fat polynomial? Can we “instantiate” Fourier Checking by an explicit (unrelativized) problem? More generally, evidence for/against BQPPH in the real world? $100 $200