Faster quantum-inspired algorithms for solving linear systems

Abstract

We establish an improved classical algorithm for solving linear systems in a model analogous to the QRAM that is used by quantum linear solvers. Precisely, for the linear system $A\x = \b$, we show that there is a classical algorithm that outputs a data structure for $\x$ allowing sampling and querying to the entries, where $\x$ is such that $\|\x - A^{+}\b\|\leq ε\|A^{+}\b\|$. This output can be viewed as a classical analogue to the output of quantum linear solvers. The complexity of our algorithm is $\tilde{O}({\kappa }_F^4{\kappa }^2/{\epsilon }^2)$, where ${\kappa }_F=\Vert A{\Vert }_F\Vert A^{+}\Vert$ and $\kappa =\Vert A\Vert \Vert A^{+}\Vert$. This improves the previous best algorithm [Gily{é}n, Song and Tang, arXiv:2009.07268] of complexity $\tilde{O}({\kappa }_F^6{\kappa }^6/{\epsilon }^4)$. Our algorithm is based on the randomized Kaczmarz method, which is a particular case of stochastic gradient descent. We also find that when $A$ is row sparse, this method already returns an approximate solution $\x$ in time $\tilde{O}({\kappa }_F^2)$, while the best quantum algorithm known returns $\ket{\x}$ in time $\tilde{O}({\kappa }_F)$ when $A$ is stored in the QRAM data structure. As a result, assuming access to QRAM and if $A$ is row sparse, the speedup based on current quantum algorithms is quadratic.

24 pages. The main algorithm (Theorem 13) was improved via a better complexity analysis

Cited by in corpus (2)

• Block-encoding dense and full-rank kernels using hierarchical matrices: applications in quantum numerical linear algebra
• Linear Regression by Quantum Amplitude Estimation and its Extension to Convex Optimization