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## Scott Aaronson (MIT)

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**BQPandPH**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 (MIT)**Quantum Computing: Where Does It Fit?**P#P Factoring, discrete log, etc.: In BQP Not known to be in BPP But in NPcoNP PH AM NP Could there be a problem in BQP\PH? BQP BPP P**First question: can we at least find an oracle A such that**BQPAPHA? Essentially the same as finding a problem in quantum logarithmic time, but not AC0 Why? Well-known correspondence between relativizedPH and AC0: interpret the ’s as OR gates, the ’s as AND gates, and the oracle string as an input of size 2n Oracles are just the “obvious” way to address the BQP vs. PH question, not some woo-woo thing Recall that the early evidence for BPP≠BQP (e.g. Simon’s alg) was also oracle evidence; then Shor found a similar oracle that could be “instantiated” by Factoring**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 RFSMA Natural conjecture: RFSPH Alas, we can’t even prove RFSAM!**Why do we care whether BQPPH?**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?**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**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**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**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 specific 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! Just adds more layers to theAC0circuit Challenge: Show that w.h.p., C is forced to estimate eventually, even if it tries to avoid it**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**Fourier Checking Is In BQP**|0 H H H |0 H f H g H |0 H H H Probability of observing |0n:**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?**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**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,**“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:RnR, 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)**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 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: Our conjecture says that if fAC0, 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 cancellations Such a “low-fat” approximation of AC0 circuits would be useful for independent reasons in learning theory But this polynomial solves Fourier Checking only by exploiting “massive cancellations” between positive and negative terms(Not coincidentally, a central feature of quantum algorithms!)**Open Problems**Prove the Generalized Linial-Nisan Conjecture!Yields an oracle A such that BQPAPHA Prove Generalized L-N even for the special case of DNFs.Yields an oracle A such that BQPAAMA 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 BQPPH in the real world? $100 $200