Lindblad动力学的因果查询压缩：最优查询与近线性局部模拟
Causal Query Compression for Lindblad Dynamics: Optimal Queries and Nearly Linear Local Simulation
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中文总结 AI 辅助
针对时间相关Lindblad动力学，提出因果查询编译器，在多种输入模型下实现最优查询复杂度和近线性局部模拟门限界，适用于非对易跳变。
中文摘要 AI 辅助
我们为时间相关的Lindblad动力学构建了一个因果查询编译器，并在不同输入模型下建立了查询和门限界。在哈密顿量和单个跳变算子的相干块编码下，一个Lipschitz生成元可以被模拟到钻石范数误差$\varepsilon$，使用$O(1+\tau+\log(1/\varepsilon)/\log(e+\log(1/\varepsilon)/\tau))$次受控查询，其中$\tau=T(\alpha_H+\sum_\mu\alpha_\mu^2)$。该界限与时间网格大小无关，并且在$\tau\ge1$时是最坏情况最优的。弱活动进入和干净返回产生阶乘历史尾部；结合两种重用长度可去除辅助输入。三次调用的极坐标校正提供Lindblad单元。该编译器还通过量子查询复杂度和一般对手界限（在常数因子内）刻画了有界自适应马尔可夫协议的最小预言机动作。对于$n$个格点上的有限范围动力学，局部矩阵和固定局部参数的高效相干求值给出$O(n(T+1)\operatorname{polylog}X)$个基本门，其中$X=\max\{e,n(T+1)(1+K_t+B)/\varepsilon\}$，适用于具有固定指数、局部变差界$K_t$且每项至多$B$个断点的分段Hölder生成元。跳变不需要对易。该门限界包括求值、算术、选择、路由和环境存储。其证明结合了公共浴池上的空间分解、占用输入上的编译和压缩记录上的路由。在短段之间重用浴池存储给出深度$(T+1)\operatorname{polylog}X$和空间$n\operatorname{polylog}X$。
英文摘要
We construct a causal query compiler for time-dependent Lindblad dynamics and establish query and gate bounds under distinct input models. With coherent block encodings of the Hamiltonian and individual jump operators, a Lipschitz generator can be simulated to diamond-norm error $\varepsilon$ using $O(1+τ+\log(1/\varepsilon)/\log(e+\log(1/\varepsilon)/τ))$ controlled queries, where $τ=T(α_H+\sum_μα_μ^2)$. This bound is independent of the time-grid size and is worst-case optimal for $τ\ge1$. Weak active entry and clean return yield a factorial history tail; combining two reuse lengths removes the auxiliary input. A three-call polar correction supplies the Lindblad cells. The compiler also characterizes the minimum oracle action of bounded adaptive Markovian protocols by quantum query complexity and the general adversary bound, up to constant factors. For finite-range dynamics on $n$ lattice sites, efficient coherent evaluation of the local matrices and fixed local parameters give $O(n(T+1)\operatorname{polylog}X)$ elementary gates, where $X=\max\{e,n(T+1)(1+K_t+B)/\varepsilon\}$, for piecewise Hölder generators with fixed exponent, local variation bound $K_t$, and at most $B$ breakpoints per term. The jumps need not commute. This gate bound includes evaluation, arithmetic, selection, routing, and environment storage. Its proof combines a spatial decomposition on a common bath, compilation on occupied inputs, and routing on compressed records. Reusing bath storage between short segments gives depth $(T+1)\operatorname{polylog}X$ and space $n\operatorname{polylog}X$.
