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.
翻译:我们在类似于量子线性求解器使用的QRAM模型中建立了一个改进的经典算法来解决线性系统,精确地说,对于线性系统$A\x = \b$,我们展示了一种经典算法,输出一个$\x$的数据结构,允许对条目进行抽样和查询,其中$\x$是满足$\|\x - A^{+}\b\| \leq \epsilon\|A^{+}\b\|$的解,该输出可以被视为量子线性求解器的输出的经典对应物。我们算法的复杂度为$\widetilde{O}(\kappa_F^4\kappa^2/\epsilon^2)$,其中$\kappa_F = \|A\|_F \|A^{+}\|$,$\kappa = \|A\|\|A^{+}\|$。这比以前最好的算法[Gily{\'e}n,Song和Tang,arXiv:2009.07268]的复杂度$\widetilde{O}(\kappa_F^6\kappa^6/\epsilon^4)$有所改善。我们的算法基于随机Kaczmarz方法,它是随机梯度下降的一种特殊情况。我们还发现当$A$是行稀疏矩阵时,该方法已经可以在时间$\widetilde{O}(\kappa_F^2)$内返回一个近似解$\x$,而已知的最佳量子算法在$A$存储在QRAM结构中时在时间$\widetilde{O}(\kappa_F)$返回$\ket{\x}$。因此,假设访问QRAM并且$A$是行稀疏的,基于当前已知的量子算法加速是二次的。