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?

Open since Bernstein-Vazirani 1993

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

In fact, we don’t even have an oracle A such that BQPAAMA. (Best is BQPAMAA)

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

New Candidate Problem: “Fourier Checking”

Given: Oracle access to functions f,g:{-1,1}nPromised: 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 Z2

n

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

11

11

2

Pr11

N

C

N CD

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 Shor’s factoring algorithm)

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.

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 Infi1/poly(d).

.i

xi xpxpEInf

.1,

ypxpEyx

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

What We KnowDinur, 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).

Can we find a fixed f (depending only on the input length n), such that computinggiven 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.

2ˆ yf

2ˆ yf

2ˆ yf

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.

Open Problems