微扰参数化量子演化导数的“真”移位规则
"Proper" Shift Rules for Derivatives of Perturbed-Parametric Quantum Evolutions
AI总结：
本文针对含微扰项的参数化量子演化期望值导数估计问题，提出仅需移位参数的真移位规则，实现精确无偏估计且最坏情况方差与Banchi-Crooks方法相同，并通过傅里叶分析刻画其理论性质。
摘要：
Banchi与Crooks（Quantum, 2021）给出了估计期望值导数的方法，这些期望值依赖于一个通过我们称之为“微扰”量子演化 $x\mapsto e^{i(x A + B)/\hbar}$ 进入的参数。他们的方法需要对出现的酉算子进行修改，而不仅仅是改变参数。此外，在 $B$ 项不可避免的情况下，似乎尚无已知的精确方法（无偏估计器）来求导数：Banchi与Crooks的方法给出的是近似。本文针对估计此类参数化期望值的导数，提出了一种只需移位参数、无需对量子演化做其他任何修改的方法（一种“真”移位规则）。我们的方法是精确的（即给出解析导数、无偏估计器），并且具有与Banchi-Crooks方法相同的最坏情况方差。此外，我们讨论了围绕真移位规则的理论，基于微扰参数化量子演化的傅里叶分析，得到了用傅里叶变换刻画真移位规则的结果，这进而导出了移位具有指数集中性的真移位规则的不存在性结果。我们推导出具有近似误差的截断方法，并基于初步数值模拟与Banchi-Crooks方法进行了比较。
英文摘要：
Banchi & Crooks (Quantum, 2021) have given methods to estimate derivatives of expectation values depending on a parameter that enters via what we call a "perturbed" quantum evolution $x\mapsto e^{i(x A + B)/\hbar}$. Their methods require modifications, beyond merely changing parameters, to the unitaries that appear. Moreover, in the case when the $B$-term is unavoidable, no exact method (unbiased estimator) for the derivative seems to be known: Banchi & Crooks's method gives an approximation. In this paper, for estimating the derivatives of parameterized expectation values of this type, we present a method that only requires shifting parameters, no other modifications of the quantum evolutions (a "proper" shift rule). Our method is exact (i.e., it gives analytic derivatives, unbiased estimators), and it has the same worst-case variance as Banchi-Crooks's. Moreover, we discuss the theory surrounding proper shift rules, based on Fourier analysis of perturbed-parametric quantum evolutions, resulting in a characterization of the proper shift rules in terms of their Fourier transforms, which in turn leads us to non-existence results of proper shift rules with exponential concentration of the shifts. We derive truncated methods that exhibit approximation errors, and compare to Banchi-Crooks's based on preliminary numerical simulations.
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