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Quantum Computation and BQP

Richard J. Lipton, Kenneth W. Regan

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

This chapter contains sections titled: 16.1 The Class BQP, 16.2 Equations, Solutions, and Complexity, 16.3 A Circuit Labeling Algorithm, 16.4 Sum-Over-Paths and Polynomial Roots, 16.5 The Additive Polynomial Simulation, 16.6 Bounding BQP, 16.7 Problems, 16.8 Summary and Notes

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Quantum computers are widely believed have an advantage over classical computers, and some have even published some empirical evidence that this is the case. However, these publications do not include a rigorous proof of this advantage, which would have to minimally state that the class of problems decidable by a quantum computer in polynomial time, BQP, contains problems that are not in the class of problems decidable by a classical computer with similar time bounds, P. Here, I provide the proof of a stronger result that implies this result: BQP contains problems that lie beyond the much larger classical computing class NP. This proves that quantum computation is able to efficiently solve problems which are far beyond the capabilities of classical computers.
A restricted path integral method is proposed to simulate a type of quantum system or Hamiltonian called a sum of controlled few-fermions on a classical computer using Monte Carlo without a numerical sign problem. Then a universality is proven to assert that any bounded-error quantum polynomial time (BQP) algorithm can be encoded into a sum of controlled few-fermions and simulated efficiently using classical Monte Carlo. Therefore, BQP is precisely the same as the class of bounded-error probabilistic polynomial time (BPP), namely, BPP = BQP.