超越条件数的更快量子线性系统求解器
Faster quantum linear system solver beyond the condition number
查看机构详情
- AWS Center for Quantum Computing（亚马逊量子计算中心）
- Rice University（莱斯大学）
- The University of Texas at Dallas（德克萨斯大学达拉斯分校）
机构由 AI 辅助整理，请以论文原文为准。
浏览论文内容
中文总结 AI 辅助
研究超越条件数的更快量子线性系统求解，提出两种量子算法，基于截断的求解器对|b⟩和A进行特定次数查询，证明有效条件数上界，基于滤波的求解器简单且运行时前置因子良好，还提出解范数估计器，改进了Li的算法。
中文摘要 AI 辅助
光谱条件数是量子线性系统求解器最坏情况成本的广泛采用度量。但它可能显著高估典型问题实例的实际运行时间。我们提出两种量子算法，能以与条件数κ无关的复杂度，将线性系统Ax = |b⟩的归一化解|x⟩求解到精度ϵ。我们专注于标准输入模型，还引入联合编码A和|b⟩的仿射扩张模型。基于截断的求解器对|b⟩进行最优次数查询，对A进行O(κeff polylog(κeff/ϵ))次查询。我们证明了有效条件数的一系列上界。基于滤波的求解器极其简单且运行时前置因子良好。我们还提出了类似简单的解范数估计器。我们的量子线性系统求解器大幅改进了Li的近期算法，实现了超越条件数的更快量子线性系统求解。
英文摘要
The spectral condition number is a widely adopted measure of worst-case cost for quantum linear system solvers. Yet it can significantly overestimate the actual runtime for a typical problem instance. We present two quantum algorithms that produce the normalized solution $|x\rangle$ of linear system $Ax=| b \rangle$ to accuracy $ε$ with complexity independent of the condition number $κ=\lVert A^{-1}\rVert$. We focus on the standard input model where $A$ is accessed through a block encoding and $| b \rangle$ is prepared by a unitary. But we also introduce an affine dilation model that encodes $A$ and $| b \rangle$ jointly, allowing further refinements of the query complexity. Our truncation-based solver makes an optimal number of queries to $| b \rangle$ and $\operatorname{\mathbf{O}}\left(κ_{\mathrm{eff}}\operatorname{polylog}\left(\frac{κ_{\mathrm{eff}}}ε\right)\right)$ queries to $A$. We prove a family of upper bounds on the effective condition number, including $κ_{\mathrm{eff}}\leq\frac{\lVert(A^\dagger A)^{-t/2}|x\rangle\rVert^{1/t}}{ε^{1/t}}$ for positive even integer $t$ and $κ_{\mathrm{eff}}\leq\frac{\lVert A^{-1\dagger}(A^\dagger A)^{-(t-1)/2}|x\rangle\rVert^{1/t}}{ε^{1/t}}$ for positive odd $t$, overcoming the $κ$-barrier. Our filtering-based solver is extremely simple with a favorable runtime prefactor. In particular, the solver has query complexity $3\frac{\lVert A^{-1\dagger}|x\rangle\rVert}ε$ to leading order when the solution norm is known. We then present a similarly simple solution norm estimator with the same asymptotic cost up to logarithmic factors. Our quantum linear system solvers thus substantially improve a recent algorithm of Li, enabling faster quantum linear system solving beyond the condition number.
