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 - https://doi.org/10.1145/2488608.2488720
DO - https://doi.org/10.1145/2488608.2488720
M3 - منشور من مؤتمر
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
T2 - 45th Annual ACM Symposium on Theory of Computing, STOC 2013
Y2 - 1 June 2013 through 4 June 2013
ER -