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arXiv 2107.06404quant-ph
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大Trotter步绝热数值模拟的成功

Success of digital adiabatic simulation with large Trotter step

Changhao Yi

发表信息：
Phys. Rev. A 104, 052603 (2021)
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摘要：

绝热演化的模拟与绝热量子计算、量子近似优化算法和绝热态制备有着密切的联系。本文讨论了用Trotter公式对绝热过程进行量子模拟的误差分析问题。我们表明，与其他条件下，电路深度可以是线性的模拟时间$T$。改进来自于观察到这里的保真度误差不能通过进化算子之间的范数距离来估计。这种现象被称为数字绝热模拟中离散化的鲁棒性。它可以解释为三个步骤，从分析和数字证据：（1）。保真度误差应该通过应用绝热定理来估计有效哈密顿量。（二）、由于Riemann-Lebesgue引理的特殊性，大多数绝热过程对离散化具有很强的鲁棒性。（三）、随着Trotter步长的增大，有效哈密顿量的谱隙趋于闭合，导致绝热数值模拟失败。

英文摘要：

The simulation of adiabatic evolution has deep connections with Adiabatic Quantum Computation, the Quantum Approximate Optimization Algorithm and adiabatic state preparation. Here we address the error analysis problem in quantum simulation of adiabatic process using Trotter formulas. We show that with additional conditions, the circuit depth can be linear in simulation time $T$. The improvement comes from the observation that the fidelity error here can't be estimated by the norm distance between evolution operators. This phenomenon is termed as the robustness of discretization in digital adiabatic simulation. It can be explained in three steps, from analytical and numerical evidence: (1). The fidelity error should be estimated by applying adiabatic theorem on the effective Hamiltonian instead. (2). Because of the specialty of Riemann-Lebesgue lemma, most adiabatic process is naturally robust against discretization. (3). As the Trotter step gets larger, the spectral gap of effective Hamiltonian tends to close, which results in the failure of digital adiabatic simulation.

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