Superpolynomial speedups based on almost any quantum circuit

TL;DR

This paper demonstrates that a broad class of quantum circuits, called dispersing circuits, can achieve superpolynomial speedups over classical algorithms, generalizing previous quantum-classical separation results.

Contribution

It introduces dispersing circuits as a general class for quantum speedups and provides a method to amplify these separations to superpolynomial levels.

Findings

Dispersing circuits include quantum Fourier transforms and many random circuits.

A general amplification method achieves n^{O(1)} vs. n^{Omega(log n)} separations.

Broad applicability of quantum speedups beyond Hadamard-based circuits.

Abstract

The first separation between quantum polynomial time and classical bounded-error polynomial time was due to Bernstein and Vazirani in 1993. They first showed a O(1) vs. Omega(n) quantum-classical oracle separation based on the quantum Hadamard transform, and then showed how to amplify this into a n^{O(1)} time quantum algorithm and a n^{Omega(log n)} classical query lower bound. We generalize both aspects of this speedup. We show that a wide class of unitary circuits (which we call dispersing circuits) can be used in place of Hadamards to obtain a O(1) vs. Omega(n) separation. The class of dispersing circuits includes all quantum Fourier transforms (including over nonabelian groups) as well as nearly all sufficiently long random circuits. Second, we give a general method for amplifying quantum-classical separations that allows us to achieve a n^{O(1)} vs. n^{Omega(log n)} separation from any dispersing circuit.