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. [Subaşi et al., Phys. Rev. Lett. (2019)] proposed a randomized algorithm inspired by adiabatic quantum computing, based on a sequence of random Hamiltonian simulation steps, with suboptimal scaling in the condition number $κ$ of the linear system and the target error $ε$. Here we go beyond these results in several ways. Firstly, using filtering~[Lin et al., Quantum (2019)] and Poissonization techniques [Cunningham et al., arXiv:2406.03972 (2024)], the algorithm complexity is improved to the optimal scaling $O(κ\log(1/ε))$ -- an exponential improvement in $ε$, and a shaving of a $\log κ$ scaling factor in $κ$. Secondly, the algorithm is further modified to achieve constant factor improvements, which are vital as we progress towards hardware implementations on fault-tolerant devices. We introduce a cheaper randomized walk operator method replacing Hamiltonian simulation -- which also removes the need for potentially challenging classical precomputations; randomized routines are sampled over optimized random variables; circuit constructions are improved. We obtain a closed formula rigorously upper bounding the expected number of times one needs to apply a block-encoding of the linear system matrix to output a quantum state encoding the solution to the linear system. The upper bound is $837 κ$ at $ε=10^{-10}$ for Hermitian matrices.

翻译：求解线性方程组是科学各领域中具有广泛应用的基础问题，开发量子线性求解器算法的努力日益增加。[Subaşi等人，Phys. Rev. Lett. (2019)] 提出了一种受绝热量子计算启发的随机化算法，该算法基于一系列随机哈密顿量模拟步骤，但在线性系统的条件数 $κ$ 和目标误差 $ε$ 方面具有次优的标度。本文在多个方面超越了这些结果。首先，通过采用滤波技术~[Lin等人，Quantum (2019)] 和泊松化技术 [Cunningham等人，arXiv:2406.03972 (2024)]，算法复杂度被改进至最优标度 $O(κ\log(1/ε))$ —— 在 $ε$ 上实现了指数级改进，并在 $κ$ 上消除了一个 $\log κ$ 标度因子。其次，进一步修改算法以实现常数因子的改进，这对于我们向容错设备上的硬件实现迈进至关重要。我们引入了一种更廉价的随机游走算子方法以取代哈密顿量模拟——这也消除了对可能具有挑战性的经典预计算的需求；随机化例程通过对优化的随机变量进行采样；改进了电路构造。我们获得了一个闭合公式，严格上界了为输出编码线性方程组解量子态所需应用线性系统矩阵块编码的期望次数。对于厄米矩阵，在 $ε=10^{-10}$ 时，该上界为 $837 κ$。