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arXiv · 0805.0007

Superpolynomial speedups based on almost any quantum circuit

Sean Hallgren·Aram W. Harrow
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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.

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Sean Hallgren, Aram W. Harrow. 2008-05-02. Superpolynomial speedups based on almost any quantum circuit. https://doi.org/10.1007/978-3-540-70575-8_64

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