Inverting well conditioned matrices in quantum logspace

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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 in- termediate measurements. This should be compared with the best known classical algorithm for the problem that re- quires (log2 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]).

Original languageEnglish
Title of host publicationSTOC 2013 - Proceedings of the 2013 ACM Symposium on Theory of Computing
Pages881-890
Number of pages10
DOIs
StatePublished - 2013
Event45th Annual ACM Symposium on Theory of Computing, STOC 2013 - Palo Alto, CA, United States
Duration: 1 Jun 20134 Jun 2013

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Conference

Conference45th Annual ACM Symposium on Theory of Computing, STOC 2013
Country/TerritoryUnited States
CityPalo Alto, CA
Period1/06/134/06/13

Keywords

  • Approximating matrix spectrum
  • Matrix inversion
  • Quan- tum phase estimation
  • Quantum computation
  • Quantum space complexity
  • Quantum state tomography

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