论文标题
“适当”的驱动 - 参数量子演变的衍生物的偏移规则
"Proper" Shift Rules for Derivatives of Perturbed-Parametric Quantum Evolutions
论文作者
论文摘要
Banchi＆Crooks（Quantum，2021）提供了根据我们所谓的“扰动”量子进化$ x \ mapSto e^{i（x a + b）/\ hbar} $的参数来估计期望值的衍生物的方法。他们的方法需要修改，而不是仅将参数更改为出现的单位。此外，在不可避免的$ b $ term的情况下，似乎已知衍生物的确切方法（无偏估计器）：Banchi＆Crooks的方法给出了近似值。 在本文中，为了估算此类型的参数化期望值的导数，我们提出了一种仅需要转移参数的方法，而没有其他修改量子发展的方法（“适当”的移位规则）。我们的方法是准确的（即，它提供了分析衍生物，公正的估计器），并且与Banchi-Crooks的差异相同。 此外，我们基于对扰动参数量子演变的傅立叶分析，讨论围绕适当转移规则的理论，从而从其傅立叶变换方面表征了适当的转移规则的表征，这又导致我们不存在转移规则的不存在结果，并以指数的转移浓度来表征。我们得出表现出近似误差的截短方法，并与基于初步数值模拟的Banchi-Crooks进行比较。
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.
