论文标题
归一化schatten规范和学习应用的量子近似
Quantum Approximation of Normalized Schatten Norms and Applications to Learning
论文作者
论文摘要
已广泛研究了确定量子状态（例如保真度度量）相似性的有效度量。在本文中，我们解决了可以定义可以\ textit {有效估计}的量子操作的相似性度量的问题。给定两个量子操作，即$ u_1 $和$ u_2 $，以其电路表格表示，我们首先开发一个量子采样电路，以估算其差异的归一化schatten 2-norm（$ \ | | U_1-U_2 \ | _ {s_2} $）使用精确的$ε$，仅使用一个清洁的随机变量和一个classical callibal and classical andial。我们证明了一个poly $（\ frac {1}ε）$上限在样品复杂性上，该限制与量子系统的大小无关。然后，我们表明，这种相似性度量与使用量子状态的常规保真度度量（$ f $）直接相关。 u_1-u_2 \ | _ {s_2} $很小（例如$ \ leq \ leq \fracε{1+ \ sqrt {2（1/δ-1）} $），是通过处理相同的随机和均匀的纯净状态而获得的状态的忠诚度，$ | （$ f（{u} _1 |ψ\ rangle，{u} _2 |ψ\ rangle）\ geq1-ε$），概率超过$ 1-δ$。我们为量子电路学习任务提供了这种有效的相似性度量估计框架的示例应用，例如找到给定统一操作的平方根。
Efficient measures to determine similarity of quantum states, such as the fidelity metric, have been widely studied. In this paper, we address the problem of defining a similarity measure for quantum operations that can be \textit{efficiently estimated}. Given two quantum operations, $U_1$ and $U_2$, represented in their circuit forms, we first develop a quantum sampling circuit to estimate the normalized Schatten 2-norm of their difference ($\| U_1-U_2 \|_{S_2}$) with precision $ε$, using only one clean qubit and one classical random variable. We prove a Poly$(\frac{1}ε)$ upper bound on the sample complexity, which is independent of the size of the quantum system. We then show that such a similarity metric is directly related to a functional definition of similarity of unitary operations using the conventional fidelity metric of quantum states ($F$): If $\| U_1-U_2 \|_{S_2}$ is sufficiently small (e.g. $ \leq \fracε{1+\sqrt{2(1/δ- 1)}}$) then the fidelity of states obtained by processing the same randomly and uniformly picked pure state, $|ψ\rangle$, is as high as needed ($F({U}_1 |ψ\rangle, {U}_2 |ψ\rangle)\geq 1-ε$) with probability exceeding $1-δ$. We provide example applications of this efficient similarity metric estimation framework to quantum circuit learning tasks, such as finding the square root of a given unitary operation.
