发表机构
Xiangtan University; Hunan Research Center of the Basic Discipline Fundamental Algorithmic Theory and Novel Computational Methods; National Center for Applied Mathematics in Hunan; East China Normal University（湘潭大学; 湖南省基础学科基本算法理论与新型计算方法研究中心; 湖南应用数学中心; 华东师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对含时非幺正动力学量子算法研究不足的问题，提出基于时钟变量表述的自主化框架，将原系统提升为扩展空间的自主输运方程，结合已有量子ODE求解器，优化了精度依赖标度并经数值验证。
AI 中文摘要
用于模拟线性微分方程的量子算法受到了越来越多的关注，其应用范围从哈密顿动力学到一般非幺正动力学。虽然含时不变的情况已得到充分研究，但含时非幺正动力学的探索仍相当不足，目前尚不清楚如何系统地将现有含时不变系统的求解器适配到这类问题中。在本研究中，我们通过引入基于时钟变量表述的自主化框架来填补这一空白，该技术最初是在文献~\noindent\ref{CJL23TimeSchr}中为含时哈密顿系统开发的。通过将原始非自主系统提升为扩展空间上的自主输运型方程，并在时钟变量中应用傅里叶谱离散化，我们得到了一个显式的含时不变线性系统，以及合适的初始态和目标解的恢复映射。至关重要的是，该表述将含时性的处理与量子常微分方程求解器的选择解耦，从而使为含时不变系统设计的现有求解器能够直接应用于由此产生的自主问题。我们将该框架与薛定谔化以及基于泰勒展开的量子常微分方程求解器相结合。在基于薛定谔化的组合中，我们的复杂度分析表明，精度依赖标度可达到$\log^{5/4}(1/\varepsilon)$，优于现有方法中的$\log^2(1/\varepsilon)$标度。数值实验验证了自主化表述的正确性，并证实了目标解的成功恢复。
英文摘要
We present an improved autonomization method for quantum simulation of time-dependent homogeneous dissipative linear systems, combining a clock-variable reformulation with Schrödingerization to obtain a time-independent Hamiltonian system. To control both discretization error and recovery probability, we construct the clock profile from a compactly supported window function and a normalized Dirichlet kernel: the former imposes endpoint regularity, while the latter concentrates the sampled mass and keeps the discrete normalization bounded. For the subsequent Schrödingerization step, we use an exactly periodic version of an error-function initial profile, whose explicit Fourier coefficients allow direct error estimates and simplify the analysis of smooth initialization. By combining Fourier projection with weighted recovery, we retain a single logarithmic precision factor in the full algorithm. Under finite-order derivative bounds, sampled matrix access, and exact state-preparation access, we prepare the normalized solution at time $T$ to accuracy $\varepsilon$ with success probability $Θ(1)$ using $\mathcal O\!\bigl(gα_AT\log(gα_AT/\varepsilon)\bigr)$ matrix queries and $\mathcal O(g)$ queries to each state-preparation oracle, where $g=\|\boldsymbol{x}_0\|/\|\boldsymbol{x}(T)\|$ and $α_A$ is the matrix-oracle normalization. We illustrate the construction and its recovery probabilities through a numerical experiment on a time-dependent two-cavity system.
