Quantum Linear System Algorithm for Dense Matrices

Physical Review Letters
Leonard WossnigAnupam Prakash

Abstract

Solving linear systems of equations is a frequently encountered problem in machine learning and optimization. Given a matrix A and a vector b the task is to find the vector x such that Ax=b. We describe a quantum algorithm that achieves a sparsity-independent runtime scaling of O(κ^{2}sqrt[n]polylog(n)/ε) for an n×n dimensional A with bounded spectral norm, where κ denotes the condition number of A, and ε is the desired precision parameter. This amounts to a polynomial improvement over known quantum linear system algorithms when applied to dense matrices, and poses a new state of the art for solving dense linear systems on a quantum computer. Furthermore, an exponential improvement is achievable if the rank of A is polylogarithmic in the matrix dimension. Our algorithm is built upon a singular value estimation subroutine, which makes use of a memory architecture that allows for efficient preparation of quantum states that correspond to the rows of A and the vector of Euclidean norms of the rows of A.

References

Jun 4, 2008·Physical Review Letters·Vittorio GiovannettiLorenzo Maccone
Nov 13, 2009·Physical Review Letters·Aram W HarrowSeth Lloyd
Sep 26, 2012·Physical Review Letters·Nathan WiebeSeth Lloyd
Jul 9, 2013·Physical Review Letters·B D CladerC R Sprouse
Oct 11, 2014·Physical Review Letters·Patrick RebentrostSeth Lloyd

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Citations

Feb 13, 2018·Proceedings. Mathematical, Physical, and Engineering Sciences·Carlo CilibertoLeonard Wossnig
Nov 9, 2019·Scientific Reports·Chih-Chieh ChenYuh-Renn Wu
Jul 18, 2020·Physical Review Letters·Suguru EndoXiao Yuan

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