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How to Solve Longstanding Open Problems In Quantum Computing Using Only Fourier Analysis Scott Aaronson (MIT) For those who hate quantum: The open problems will be in off-white boxes like this one

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Problem 1: BQP PH? Open since Bernstein-Vazirani 1993 Natural conjecture would be that BQP PH. But we dont even have an oracle separation In fact, we dont even have an oracle A such that BQP A AM A. (Best is BQP A MA A ) Furthermore, until recently our only candidate problem was a monstrosity (Recursive Fourier Sampling)

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New Candidate Problem: Fourier Checking Given: Oracle access to functions f,g:{-1,1} n Promised: Either (i)All f(x) and g(x) values are drawn independently from the Gaussian distribution N(0,1), or (ii)The f(x)s are drawn independently from N(0,1), and g=FT(f) is the Fourier transform of f over Z 2 n Problem: Decide which, with constant bias. Claim: Fourier Checking is in BQP Conjecture: Fourier Checking is not in PH

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As usual, the problem boils down to showing Fourier Checking has no AC 0 circuit of size 2 poly(n) Alas, all known techniques for constant-depth circuit lower bounds (random restriction, Razborov-Smolensky, Nisan-Wigderson…) fail for interesting reasons! Conjecture (Linial-Nisan 1989): Polylog-wise independence fools AC 0 [recently proved by Bazzi for DNFs!] What I want: The Generalized Linial-Nisan Conjecture. Namely, no distribution D over {0,1} N such that for all conjunctions C of polylog(N) literals, can be distinguished from uniform (with (1) bias) in AC 0

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Problem 2: The Need for Structure in Quantum Speedups Beals et al 1998: Quantum and classical decision tree complexities are polynomially related for all total Boolean functions f: D(f)=O(Q(f) 6 ) But could a quantum computer evaluate an almost- total function with exponentially fewer queries? Suggests that if you want an exponential quantum speedup, then you need to exploit some structure in the oracle being queried (e.g. periodicity in the case of Shors factoring algorithm)

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Conjecture: Let Q be a T-query quantum algorithm. Then a classical randomized algorithm that makes T O(1) queries can approximate Qs acceptance probability on most inputs x {0,1} n. Would suffice to prove that every low-degree bounded polynomial has an influential variable: Let p:{-1,1} n [-1,1] be a real polynomial of degree d. Suppose Let Then there exists an i such that Inf i 1/poly(d). Conjecture 2: If P=P #P, then P A =BQP A with probability 1 for a random oracle A. [insert avg, i.o. to taste]

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What We Know Dinur, Friedgut, Kindler, ODonnell 2006: Every degree-d polynomial p:{-1,1} n [-1,1] with (1) variance has a variable with influence at least 1/exp(d). (Indeed, p is close to an exp(d)-junta.) ODonnell, Saks, Schramm, Servedio 2005: Every classical decision tree of depth d has a variable with influence (1/d).

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Can we find a fixed f (depending only on the input length n), such that computing given y as input is #P-complete? Problem 3: Quantum Algorithm for a #P-complete Problem?!? Let f:{0,1} n {0,1} be efficiently computable. Then a simple quantum algorithm outputs each y {0,1} n with probability If even estimating is #P-complete on average, then FBPP=FBQP P #P =AM.

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Open Problems

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