NDLI: Inverting well conditioned matrices in quantum logspace

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Inverting well conditioned matrices in quantum logspace

Content Provider	ACM Digital Library
Author	Ta-Shma, Amnon
Abstract	We show that quantum computers improve on the best known classical algorithms for matrix inversion (and singular value decomposition) as far as space is concerned. This adds to the (still short) list of important problems where quantum computers are of help. Specifically, we show that the inverse of a well conditioned matrix can be approximated in quantum logspace with intermediate measurements. This should be compared with the best known classical algorithm for the problem that requires $Ω(log^{2}$ n) space. We also show how to approximate the spectrum of a normal matrix, or the singular values of an arbitrary matrix, with ε additive accuracy, and how to approximate the singular value decomposition (SVD) of a matrix whose singular values are well separated. The technique builds on ideas from several previous works, including simulating Hamiltonians in small quantum space (building on [2] and [10]), treating a Hermitian matrix as a Hamiltonian and running the quantum phase estimation procedure on it (building on [5]) and making small space probabilistic (and quantum) computation consistent through the use of offline randomness and the shift and truncate method (building on [8]).
Starting Page	881
Ending Page	890
Page Count	10
File Format	PDF
ISBN	9781450320290
DOI	10.1145/2488608.2488720
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2013-06-01
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Quantum computation Matrix inversion Quantum phase estimation Approximating matrix spectrum Quantum space complexity Quantum state tomography
Content Type	Text
Resource Type	Article

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