发表机构
Center on Frontiers of Computing Studies, School of Computer Science, Peking University; School of Physics, Peking University; School of Artificial Intelligence, Beijing Normal University; Mathematical Institute, University of Oxford（北京大学计算机科学学院前沿计算研究中心; 北京大学物理学院; 北京师范大学人工智能学院; 牛津大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一种模拟几何局部 Lindblad 动力学的浅层量子电路算法，通过共享幺正膨胀实现可逆空间修正，达到近最优的深度和门数复杂度，并证明匹配的下界。
AI 中文摘要
开放系统动力学支配着量子多体系统中的退相干、输运和工程耗散。我们提出了一种量子算法，用于模拟在 $n$ 个格点的常维晶格上具有有界局部强度的、与时间无关的、有限范围的 Lindblad 动力学，允许非对易跳跃算符。对于时间 $T\ge1$ 且金刚石范数误差至多 $\epsilon$，该算法的深度为 $O(T\operatorname{polylog}(nT/\epsilon))$，门数为 $O(nT\operatorname{polylog}(nT/\epsilon))$。这些结果将 HHKL 空间分解扩展到开放系统。主要障碍在于用于幺正动力学的可逆重叠修正对耗散通道没有直接类比。我们通过将环境保留在共享的幺正膨胀中，使系统丢失的信息相干地可用，从而使空间修正可逆，从而克服了这一障碍。我们证明了由分解生成的环境状态的指数占据界，允许对出现的输入上的局部因子进行紧凑表示和相干模拟。该构造将每个短时演化简化为对数直径区域上的恒定数量的并行层，使用最近邻门和对数辅助密度。我们还使用传输量子信息的、与时间无关的、纯耗散过程证明了无条件的最坏情况下的下界。对于具有局部量子通信和经典通信的电路，在时间尺度达到晶格直径之前，期望门数的下界为 $\Omega(nT)$，深度的下界为 $\Omega(T)$。它们与上界的前导时间和体积依赖在多项式对数因子内匹配。在固定时间和足够小的固定误差下，我们进一步证明了深度下界为 $\Omega(\log n/\log\log n)$，表明全局精度可能需要随着晶格的增长而增加深度。
英文摘要
Open-system dynamics govern decoherence, transport, and engineered dissipation in quantum many-body systems. We propose a quantum algorithm for simulating time-independent, finite-range Lindblad dynamics with bounded local strength on a constant-dimensional lattice of $n$ sites, allowing noncommuting jump operators. For time $T\ge1$ and diamond-norm error at most $ε$, it has depth $O(T\operatorname{polylog}(nT/ε))$ and gate count $O(nT\operatorname{polylog}(nT/ε))$. These results extend the HHKL spatial decomposition to open systems. The main obstacle is that the reversible overlap corrections used for unitary dynamics have no direct analogue for dissipative channels. We overcome this by retaining the environment in a shared unitary dilation, keeping information lost from the system coherently available and making the spatial corrections reversible. We prove exponential occupation bounds for the environmental states generated by the decomposition, allowing compact representation and coherent simulation of local factors on the inputs that arise. The construction reduces each short-time evolution to a constant number of parallel layers on regions of logarithmic diameter, using nearest-neighbour gates and polylogarithmic auxiliary density. We also prove unconditional worst-case lower bounds using time-independent, purely dissipative processes that transport quantum information. For circuits with local quantum and classical communication, the lower bounds are $Ω(nT)$ for the expected gate count and $Ω(T)$ for depth on time scales up to the lattice diameter. They match the upper bounds' leading time and volume dependence up to polylogarithmic factors. At fixed time and sufficiently small fixed error, we further prove a depth lower bound of $Ω(\log n/\log\log n)$, showing that global accuracy can require increasing depth as the lattice grows.
Comments57 pages, 4 figures
