通过交换子分解实现 Trotter 误差时间标度分离
Trotter error time scaling separation via commutant decomposition
AI总结：
针对动态量子模拟中Trotter误差抑制需深电路的问题，提出交换子分解框架分离不同时间标度的误差分量，改进误差估计并通过解析与数值验证，还应用于Floquet动力学相关可观测量误差分析。
摘要：
在动态量子模拟中抑制 Trotter 误差通常需要运行更深的电路，这对有噪声的近期量子设备构成了巨大挑战。研究表明，经验误差通常远小于现有界所暗示的误差，这意味着实际所需的电路成本远低于基于这些界的成本。在此，我们通过引入交换子分解的通用框架改进了对现有界的 Trotter 误差估计，该框架分离了与时间具有根本不同标度的不相交误差分量。特别地，我们确定了两个误差分量，对于演化到时间 t 的 p 阶乘积公式，每个分量分别标度为 O(τ^p t) 和 O(τ^p)，其中 τ 是固定步长。我们证明，这种形式主义不仅直接重现了先前的结果，而且为高阶乘积公式提供了更好的误差估计。我们通过解析和数值方法证明了这种改进。我们还将分析应用于与 Floquet 动力学中的加热和热化相关的可观测量误差，这具有独立的研究意义。
英文摘要：
Suppressing the Trotter error in dynamical quantum simulation typically requires running deeper circuits, posing a great challenge for noisy near-term quantum devices. Studies have shown that the empirical error is usually much smaller than the one suggested by existing bounds, implying the actual circuit cost required is much less than the ones based on those bounds. Here, we improve the estimate of the Trotter error over existing bounds, by introducing a general framework of commutant decomposition that separates disjoint error components that have fundamentally different scaling with time. In particular we identify two error components that each scale as $O(τ^pt)$ and $O(τ^p)$ for a $p$th-order product formula evolving to time $t$ using a fixed step size $τ$, it implies one would scale linearly with time $t$ and the other would be constant of $t$. We show that this formalism not only straightforwardly reproduces previous results but also provides a better error estimate for higher-order product formulas. We demonstrate the improvement both analytically and numerically. We also apply the analysis to observable error relating to the heating in Floquet dynamics and thermalization, which is of independent interest.
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