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 \( A{\bf x}= {\bf b} \) , we show that there is a classical algorithm that outputs a data structure for \( {\bf x} \) allowing sampling and querying to the entries, where \( {\bf x} \) is such that \( \Vert {\bf x}- A^{+}{\bf b}\Vert \le \epsilon \Vert A^{+}{\bf b}\Vert \) . This output can be viewed as a classical analogue to the output of quantum linear solvers. The complexity of our algorithm is \( \widetilde{O}(\kappa _F^6 \kappa ^2/\epsilon ^2) \) , where \( \kappa _F = \Vert A\Vert _F\Vert A^{+}\Vert \) and \( \kappa = \Vert A\Vert \Vert A^{+}\Vert \) . This improves the previous best algorithm [Gilyén, 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 \( {\bf x} \) in time \( \widetilde{O}(\kappa _F^2) \) , while the best quantum algorithm known returns \( | {\bf x} \rangle \) 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.