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arXiv 2111.04773quant-ph
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随机输入的哈密顿模拟

Hamiltonian simulation with random inputs

Qi Zhao, You Zhou, Alexander F. Shaw, Tongyang Li, Andrew M. Childs

发表信息：
Physical Review Letters 129, 270502 (2022)
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摘要：

数字量子模拟的算法误差通常是根据实际和理想演化算子之间的谱范数距离来探索的。在实践中，这种最坏情况下的错误分析可能是不必要的悲观。为了解决这个问题，我们开发了一个理论的平均情况下的随机初始状态的哈密顿模拟性能。我们将平均情况下的误差与乘性误差的Frobenius范数联系起来，并给出了乘积公式（PF）和截断泰勒级数方法的上界。作为应用，我们估计了一般格点哈密顿量和$k$-局部哈密顿量的数字哈密顿模拟的平均情况误差。特别地，对于具有$n$自旋的最近邻海森堡链，对于PF方法和泰勒级数方法，误差从最坏情况下的$\mathcal O(n)$二次减少到平均$\mathcal O(\sqrt{n})$。数值证据表明，该理论准确地描述了具体模型的平均误差。我们也将我们的结果应用于量子置乱模拟中的误差分析。

英文摘要：

The algorithmic error of digital quantum simulations is usually explored in terms of the spectral norm distance between the actual and ideal evolution operators. In practice, this worst-case error analysis may be unnecessarily pessimistic. To address this, we develop a theory of average-case performance of Hamiltonian simulation with random initial states. We relate the average-case error to the Frobenius norm of the multiplicative error and give upper bounds for the product formula (PF) and truncated Taylor series methods. As applications, we estimate average-case error for digital Hamiltonian simulation of general lattice Hamiltonians and $k$-local Hamiltonians. In particular, for the nearest-neighbor Heisenberg chain with $n$ spins, the error is quadratically reduced from $\mathcal O(n)$ in the worst case to $\mathcal O(\sqrt{n})$ on average for both the PF method and the Taylor series method. Numerical evidence suggests that this theory accurately characterizes the average error for concrete models. We also apply our results to error analysis in the simulation of quantum scrambling.

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