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Approximate Counting and Quantum Computation

Published online by Cambridge University Press:  11 October 2005

M. BORDEWICH
Affiliation:
Mathematical Institute, University of Oxford, 24-29 St. Giles', Oxford, OX1 (e-mail: magnusb@comp.leeds.ac.uk, dwelsh@maths.ox.ac.uk)
M. FREEDMAN
Affiliation:
Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA (e-mail: michaelf@microsoft.com, lovasz@microsoft.com)
L. LOVÁSZ
Affiliation:
Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA (e-mail: michaelf@microsoft.com, lovasz@microsoft.com)
D. WELSH
Affiliation:
Mathematical Institute, University of Oxford, 24-29 St. Giles', Oxford, OX1 (e-mail: magnusb@comp.leeds.ac.uk, dwelsh@maths.ox.ac.uk)

Abstract

Motivated by the result that an ‘approximate’ evaluation of the Jones polynomial of a braid at a 5th root of unity can be used to simulate the quantum part of any algorithm in the quantum complexity class BQP, and results relating BQP to the counting class GapP, we introduce a form of additive approximation which can be used to simulate a function in BQP. We show that all functions in the classes #P and GapP have such an approximation scheme under certain natural normalizations. However, we are unable to determine whether the particular functions we are motivated by, such as the above evaluation of the Jones polynomial, can be approximated in this way. We close with some open problems motivated by this work.

Type
Paper
Copyright
2005 Cambridge University Press

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