As we progress towards physical implementation of quantum algorithms it is vital to determine the explicit resource costs needed to run them. Solving linear systems of equations is a fundamental problem with a wide variety of applications across many fields of science, and there is increasing effort to develop quantum linear solver algorithms. Here we introduce a quantum linear solver algorithm combining ideas from adiabatic quantum computing with filtering techniques based on quantum signal processing. We give a closed formula for the non-asymptotic query complexity $Q$ -- the exact number of calls to a block-encoding of the linear system matrix -- as a function of condition number $\kappa$, error tolerance $\epsilon$ and block-encoding scaling factor $\alpha$. Our protocol reduces the cost of quantum linear solvers over state-of-the-art close to an order of magnitude for early implementations. The asymptotic scaling is $O(\kappa \log(\kappa/\epsilon))$, slightly looser than the $O(\kappa \log(1/\epsilon))$ scaling of the asymptotically optimal algorithm of Costa et al. However, our algorithm outperforms the latter for all condition numbers up to $\kappa \approx 10^{32}$, at which point $Q$ is comparably large, and both algorithms are anyway practically unfeasible. The present optimized analysis is both problem-agnostic and architecture-agnostic, and hence can be deployed in any quantum algorithm that uses linear solvers as a subroutine.

翻译：随着我们向量子算法的物理实现迈进，确定运行这些算法所需的显式资源成本至关重要。求解线性方程组是一个基础性问题，在众多科学领域有着广泛的应用，目前开发量子线性求解器算法的努力日益增加。本文提出了一种量子线性求解器算法，该算法将绝热量子计算的思想与基于量子信号处理的滤波技术相结合。我们给出了非渐近查询复杂度 $Q$ 的封闭公式——即对线性系统矩阵块编码的精确调用次数——作为条件数 $\kappa$、误差容限 $\epsilon$ 和块编码缩放因子 $\alpha$ 的函数。我们的协议将量子线性求解器的成本较现有最优方法降低了近一个数量级，适用于早期实现。其渐近缩放比例为 $O(\kappa \log(\kappa/\epsilon))$，略逊于 Costa 等人提出的渐近最优算法的 $O(\kappa \log(1/\epsilon))$。然而，对于条件数高达 $\kappa \approx 10^{32}$ 的情况，我们的算法优于后者，此时 $Q$ 相当大，且两种算法实际上均不可行。本文优化的分析既与问题无关，也与架构无关，因此可部署在任何使用线性求解器作为子程序的量子算法中。