@MISC{Høyer_efficientquantum, author = {Peter Høyer}, title = {Efficient Quantum Transforms}, year = {} }

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Abstract

Quantum mechanics requires the operation of quantum computers to be unitary, and thus makes it important to have general techniques for developing fast quantum algorithms for computing unitary transforms. A quantum routine for computing a generalized Kronecker product is given. Applications include re-development of the networks for computing the Walsh-Hadamard and the quantum Fourier transform. New networks for two wavelet transforms are given. Quantum computation of Fourier transforms for non-Abelian groups is defined. A slightly relaxed definition is shown to simplify the analysis and the networks that computes the transforms. Efficient networks for computing such transforms for a class of metacyclic groups are introduced. A novel network for computing a Fourier transform for a group used in quantum errorcorrection is also given. 1