Superpolynomial speedups based on almost any quantum circuit
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.
Cite
@article{arxiv.0805.0007,
title = {Superpolynomial speedups based on almost any quantum circuit},
author = {Sean Hallgren and Aram W. Harrow},
journal= {arXiv preprint arXiv:0805.0007},
year = {2008}
}
Comments
16 pages, 1 figure, to appear in ICALP '08. v2 includes references and acknowledgments