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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

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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?

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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

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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!

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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

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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

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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

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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

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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

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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

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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

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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!

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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

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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

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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

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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

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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!

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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…

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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

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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

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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

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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

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