哈密顿模拟、微分方程与线性系统求解器的根稀疏度缩放
Root-sparsity scaling for Hamiltonian simulation, differential equations and linear-system solvers
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中文总结 AI 辅助
提出量子算法，实现哈密顿模拟、线性微分方程和线性系统的稀疏度平方根依赖，并通过联合下界证明最优性，同时提供门高效实现。
中文摘要 AI 辅助
我们给出了哈密顿模拟、线性微分方程和线性系统的量子算法，其稀疏度依赖为平方根。对于行和列欧几里得范数以$\eta$为界的$s$-稀疏埃尔米特矩阵，演化时间$t$至误差$\epsilon$所需的查询次数为$\mathcal{O}(\sqrt{s}\tau+\sqrt{s\tau\log(1/\epsilon)}+\log(1/\epsilon))$，其中$\tau=\eta|t|$。相同界限适用于具有相干时间索引稀疏访问和认证时间离散化的含时哈密顿量。我们的构造将精确逐项表示与凯莱换能器相结合。一个联合下界几乎匹配时间、稀疏度和精度依赖，排除了在$\sqrt{s}\tau$和$\log(1/\epsilon)$上的均匀加性界限。对于耗散线性微分方程，我们获得$\mathcal{O}(\bar g[\sqrt{s}(\tau+\sqrt{\tau})+1]\log(C\bar g/\epsilon))$次查询，其中$\tau$是演化时间乘以范数界限，$\bar g$限制组合输入和源范数与最终解范数的比率，$C$是绝对常数。我们通过分离哈密顿和耗散分量来锐化此界限，并处理一般常数生成器。下界捕获了含时方程的联合参数依赖和常数正半定耗散的精度难度。对于线性系统，凯莱游走和切比雪夫滤波器在$\\|A\\|\le1$且$\\|A^{-1}\\|\le\kappa$时给出$\mathcal{O}(\kappa\sqrt{s}\log(1/\epsilon))$次查询，并且我们给出了匹配联合下界的替代证明。我们还提供了含时无关哈密顿模拟和耗散ODE求解器的门高效实现，包括Lipschitz时间依赖，在给定高效条目解码、算术和所需源制备的情况下，以逆精度的多对数开销保留查询界限。
英文摘要
We give quantum algorithms for Hamiltonian simulation, linear differential equations, and linear systems with square-root dependence on sparsity. For an $s$-sparse Hermitian matrix with row and column Euclidean norms bounded by $η$, evolution for time $t$ to error $ε$ uses $\mathcal{O}(\sqrt{s}τ+\sqrt{sτ\log(1/ε)}+\log(1/ε))$ queries, where $τ=η|t|$. The same bound holds for time-dependent Hamiltonians with coherent time-indexed sparse access and certified time discretisation. Our construction combines an exact entrywise representation with Cayley transducers. A joint lower bound nearly matches the time, sparsity, and precision dependence, ruling out a uniformly additive bound in $\sqrt{s}τ$ and $\log(1/ε)$. For dissipative linear differential equations we obtain $\mathcal{O}(\bar g[\sqrt{s}(τ+\sqrtτ)+1]\log(C\bar g/ε))$ queries, where $τ$ is the evolution time multiplied by a norm bound, $\bar g$ bounds the ratio of combined input and source norms to the final-solution norm, and $C$ is an absolute constant. We sharpen this bound by separating Hamiltonian and dissipative components, and treat general constant generators. Lower bounds capture joint parameter dependence for time-dependent equations and precision hardness for constant positive semidefinite dissipation. For linear systems, a Cayley walk and Chebyshev filter give $\mathcal{O}(κ\sqrt{s}\log(1/ε))$ queries when $\|A\|\le1$ and $\|A^{-1}\|\leκ$, and we give an alternative proof of a matching joint lower bound. We also give gate-efficient implementations of time-independent Hamiltonian simulation and the dissipative ODE solver, including Lipschitz time dependence, retaining the query bounds with polylogarithmic overhead in inverse precision given efficient entry decoding, arithmetic, and required source preparations.
发表机构
- Macquarie University（麦考瑞大学）
- University of Queensland（昆士兰大学）
- University of Toronto（多伦多大学）
- Google Quantum AI（谷歌量子AI）
机构由 AI 辅助整理，请以论文原文为准。
