TY - GEN

T1 - Inverting well conditioned matrices in quantum logspace

AU - Ta-Shma, Amnon

PY - 2013

Y1 - 2013

N2 - 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]).

AB - 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]).

KW - Approximating matrix spectrum

KW - Matrix inversion

KW - Quan- tum phase estimation

KW - Quantum computation

KW - Quantum space complexity

KW - Quantum state tomography

UR - http://www.scopus.com/inward/record.url?scp=84879817452&partnerID=8YFLogxK

U2 - 10.1145/2488608.2488720

DO - 10.1145/2488608.2488720

M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.conference???

AN - SCOPUS:84879817452

SN - 9781450320290

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 881

EP - 890

BT - STOC 2013 - Proceedings of the 2013 ACM Symposium on Theory of Computing

Y2 - 1 June 2013 through 4 June 2013

ER -