面向几何局域哈密顿量模拟的最优电路深度
Toward Optimal Circuit Depth for Geometrically Local Hamiltonian Simulation
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中文总结 AI 辅助
本文针对几何局域哈密顿量模拟，构造了深度为O(T log(nT/ε))的最近邻量子电路，并通过下界证明该深度在双对数因子内最优，阐明了局域性和精度对电路复杂性的影响。
中文摘要 AI 辅助
理解相互作用量子系统的动力学是量子模拟的核心目标。对于几何局域哈密顿量，局域性表明数字模拟应保留底层动力学的大部分并行性。针对目标精度 $\epsilon$，我们构造了一个最近邻量子电路，用于模拟固定维度晶格上具有归一化局域相互作用强度的 $n$ 量子比特、时间无关、有限程哈密顿量在时间 $T\ge1$ 内的动力学，电路深度为 $O(T\log(nT/\epsilon))$，门数为 $O(nT\log(nT/\epsilon))$。这将 Haah、Hastings、Kothari 和 Low 的多对数开销减少为单一对数。在 Lieb-Robinson 分解的对数尺度块上，浅层但高精度的实现对于降低电路深度至关重要。我们的关键观察是，空间局域化也控制着高精度的成本。在每个块中，常数阶乘积公式的误差通过修正得到补偿，从而产生指数级精度。该修正可以以与块直径成比例的深度实现，在空间局域化所需的渐近深度尺度之外不引入额外的渐近深度尺度。此外，我们证明了均匀 XXZ 哈密顿量的无条件电路深度下界：在所述范围内固定演化时间 $T$ 且足够小的固定误差 $\epsilon$ 下，任何模拟这些动力学的最近邻量子电路都需要深度 $\Omega(\log n/\log\log n)$。这与我们的 $O(\log n)$ 上界匹配，相差一个双重对数因子，从而在该范围内建立了晶格哈密顿量模拟的近乎最优电路深度。这些结果共同阐明了局域性和精度在决定量子动力学电路复杂性中的作用，并为空间受限的量子计算提出了更广泛的原则。
英文摘要
Understanding the dynamics of interacting quantum systems is a central goal of quantum simulation. For geometrically local Hamiltonians, locality suggests that digital simulation should retain much of the parallelism of the underlying dynamics. For a target accuracy $ε$, we construct a nearest-neighbor quantum circuit that simulates the dynamics of an $n$-qubit, time-independent, finite-range Hamiltonian with normalized local interaction strength on a fixed-dimensional lattice for time $T\ge1$, with circuit depth $O(T\log(nT/ε))$ and gate count $O(nT\log(nT/ε))$. This reduces the polylogarithmic overhead of Haah, Hastings, Kothari, and Low to a single logarithm. On logarithmic-scale blocks from the Lieb--Robinson decomposition, a shallow but highly accurate implementation is essential for reducing the circuit depth. Our key observation is that spatial localization also controls the cost of high precision. In each block, the error of a constant-order product formula is compensated for by a correction, yielding exponential accuracy. The correction can be implemented with depth proportional to the block diameter, introducing no additional asymptotic depth scale beyond that required by spatial localization. Furthermore, we prove an unconditional circuit-depth lower bound for uniform XXZ Hamiltonians: at fixed evolution time $T$ in the stated regime and sufficiently small fixed error $ε$, any nearest-neighbor quantum circuit simulating these dynamics requires depth $Ω(\log n/\log\log n)$. This matches our $O(\log n)$ upper bound up to a doubly logarithmic factor, establishing near-optimal circuit depth for lattice Hamiltonian simulation in this regime. Together, these results clarify the role of locality and precision in determining the circuit complexity of quantum dynamics and suggest broader principles for spatially constrained quantum computation.
发表机构
- Center on Frontiers of Computing Studies, School of Computer Science, Peking University（北京大学计算机科学学院前沿计算研究中心）
- School of Physics, Peking University（北京大学物理学院）
- School of Artificial Intelligence, Beijing Normal University（北京师范大学人工智能学院）
- State Key Laboratory for Novel Software Technology, Nanjing University（南京大学软件新技术国家重点实验室）
- Hefei National Laboratory（合肥国家实验室）
- Mathematical Institute, University of Oxford（牛津大学数学研究所）
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