Faster quantum-inspired algorithms for solving linear systems

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 $\|\x - A^{+}\b\|\leq \epsilon \|A^{+}\b\|$. This output can be viewed as a classical analogue to the output of quantum linear solvers. The complexity of our algorithm is $\widetilde{O}(\kappa_F^4 \kappa^2/\epsilon^2 )$, where $\kappa_F = \|A\|_F\|A^{+}\|$ and $\kappa = \|A\|\|A^{+}\|$. This improves the previous best algorithm [Gily{\'e}n, Song and Tang, arXiv:2009.07268] of complexity $\widetilde{O}(\kappa_F^6 \kappa^6/\epsilon^4)$. 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 $\widetilde{O}(\kappa_F^2)$, while the best quantum algorithm known returns $\ket{\x}$ in time $\widetilde{O}(\kappa_F)$ 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.