How to Solve Longstanding Open Problems In Quantum Computing Using Only Fourier Analysis

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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. Problem 1: BQP  PH?.

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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

### Problem 1: BQP  PH?

“Natural” conjecture would be that BQPPH. But we don’t even have an oracle separation

In fact, we don’t even have an oracle A such that BQPAAMA. (Best is BQPAMAA)

Open since Bernstein-Vazirani 1993

Furthermore, until recently our only candidate problem was a monstrosity (“Recursive Fourier Sampling”)

### New Candidate Problem: “Fourier Checking”

Promised: Either

All f(x) and g(x) values are drawn independently from the Gaussian distribution N(0,1), or

The f(x)’s are drawn independently from N(0,1), and g=FT(f) is the Fourier transform of f over Z2n

Problem: Decide which, with constant bias.

Claim: Fourier Checking is in BQP

Conjecture: Fourier Checking is not in PH

As usual, the problem boils down to showing Fourier Checking has no AC0 circuit of size 2poly(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 AC0[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 AC0

### Problem 2: The Need for Structure in Quantum Speedups

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 Shor’s factoring algorithm)

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?

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 Infi1/poly(d).

Conjecture 2: If P=P#P, then PA=BQPA with probability 1 for a random oracle A. [insert avg, i.o. to taste]

Conjecture: Let Q be a T-query quantum algorithm. Then a classical randomized algorithm that makes TO(1) queries can approximate Q’s acceptance probability on most inputs x{0,1}n.

### What We Know

Dinur, Friedgut, Kindler, O’Donnell 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.)

O’Donnell, Saks, Schramm, Servedio 2005: Every classical decision tree of depth d has a variable with influence (1/d).

### Problem 3: Quantum Algorithm for a #P-complete Problem?!?

Then a simple quantum algorithm outputs each y{0,1}n with probability

Let f:{0,1}n{0,1} be efficiently computable.

Can we find a fixed f (depending only on the input length n), such that computinggiven y as input is #P-complete?

If even estimating is #P-complete on average, then FBPP=FBQP  P#P=AM.