# 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 = $̱, we show that there is a classical algorithm that outputs a data structure forallowing sampling and querying to the entries, whereis such that- A^-1≤ϵA^-1. This output can be viewed as a classical analogue to the output of quantum linear solvers. The complexity of our algorithm isO(κ_F^6 κ^2/ϵ^2 ), whereκ_F = A_FA^-1andκ= AA^-1. This improves the previous best algorithm [Gilyén, Song and Tang, arXiv:2009.07268] of complexityO(κ_F^6 κ^6/ϵ^4). Our algorithm is based on the randomized Kaczmarz method, which is a particular case of stochastic gradient descent. We also find that whenAis row sparse, this method already returns an approximate solutionin timeO(κ_F^2), while the best quantum algorithm known returns|⟩in timeO(κ_F)whenAis stored in the QRAM data structure. As a result, assuming access to QRAM and ifAis row sparse, the speedup based on current quantum algorithms is quadratic.

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