TY - GEN
T1 - Quantum-inspired algorithms for solving low-rank linear equation systems with logarithmic dependence on the dimension
AU - Chia, Nai Hui
AU - Gilyén, András
AU - Lin, Han Hsuan
AU - Lloyd, Seth
AU - Tang, Ewin
AU - Wang, Chunhao
N1 - Publisher Copyright:
© Nai-Hui Chia, András Gilyén, Han-Hsuan Lin, Seth Lloyd, Ewin Tang, and Chunhao Wang.
PY - 2020/12
Y1 - 2020/12
N2 - We present two efficient classical analogues of the quantum matrix inversion algorithm [16] for low-rank matrices. Inspired by recent work of Tang [27], assuming length-square sampling access to input data, we implement the pseudoinverse of a low-rank matrix allowing us to sample from the solution to the problem Ax = b using fast sampling techniques. We construct implicit descriptions of the pseudo-inverse by finding approximate singular value decomposition of A via subsampling, then inverting the singular values. In principle, our approaches can also be used to apply any desired “smooth” function to the singular values. Since many quantum algorithms can be expressed as a singular value transformation problem [15], our results indicate that more low-rank quantum algorithms can be effectively “dequantised” into classical length-square sampling algorithms.
AB - We present two efficient classical analogues of the quantum matrix inversion algorithm [16] for low-rank matrices. Inspired by recent work of Tang [27], assuming length-square sampling access to input data, we implement the pseudoinverse of a low-rank matrix allowing us to sample from the solution to the problem Ax = b using fast sampling techniques. We construct implicit descriptions of the pseudo-inverse by finding approximate singular value decomposition of A via subsampling, then inverting the singular values. In principle, our approaches can also be used to apply any desired “smooth” function to the singular values. Since many quantum algorithms can be expressed as a singular value transformation problem [15], our results indicate that more low-rank quantum algorithms can be effectively “dequantised” into classical length-square sampling algorithms.
UR - http://www.scopus.com/inward/record.url?scp=85100924131&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85100924131&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.ISAAC.2020.47
DO - 10.4230/LIPIcs.ISAAC.2020.47
M3 - Conference contribution
AN - SCOPUS:85100924131
T3 - Leibniz International Proceedings in Informatics, LIPIcs
SP - 471
EP - 4717
BT - 31st International Symposium on Algorithms and Computation, ISAAC 2020
A2 - Cao, Yixin
A2 - Cheng, Siu-Wing
A2 - Li, Minming
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 31st International Symposium on Algorithms and Computation, ISAAC 2020
Y2 - 14 December 2020 through 18 December 2020
ER -