标题： 在一般乘积分布上预测量子通道 显示英文标题

标题： Predicting quantum channels over general product distributions

摘要： 我们研究预测未知量子通道输出行为的问题。 给定对一个$n$-比特通道$E$和一个可观测量$O$的查询访问，我们的目标是学习映射\begin{equation*} \rho \mapsto \mathrm{Tr}(O E[\rho]) \end{equation*}，使其在从分布$D$中采样的大多数$\rho$上的误差很小。 先前，黄、陈和Preskill证明了一个令人惊讶的结果，即使$E$是任意的，这个任务也可以在大约$n^{O(\log(1/\epsilon))}$的时间内解决，其中$\epsilon$是目标预测误差。 然而，他们的保证仅适用于在所有单量子比特Clifford门下不变的输入分布$D$，并且他们的算法在重要的情况下会失败，例如在乘积状态上的通用乘积分布$\rho$。 在这项工作中，我们提出了一种新方法，在本质上任何乘积分布$D$上都能实现精确预测，前提是它不是“经典”的，在这种情况下存在一个平凡的指数下界。 我们的方法使用了“有偏的Pauli分析”，类似于经典的有偏傅里叶分析。 实施这种方法需要克服一些独特的量子设置中的挑战，包括缺乏具有适当正交性质的基。 我们为解决这些问题开发的技术可能在量子信息中有更广泛的应用。 显示英文摘要

摘要： We investigate the problem of predicting the output behavior of unknown quantum channels. Given query access to an $n$-qubit channel $E$ and an observable $O$, we aim to learn the mapping \begin{equation*} \rho \mapsto \mathrm{Tr}(O E[\rho]) \end{equation*} to within a small error for most $\rho$ sampled from a distribution $D$. Previously, Huang, Chen, and Preskill proved a surprising result that even if $E$ is arbitrary, this task can be solved in time roughly $n^{O(\log(1/\epsilon))}$, where $\epsilon$ is the target prediction error. However, their guarantee applied only to input distributions $D$ invariant under all single-qubit Clifford gates, and their algorithm fails for important cases such as general product distributions over product states $\rho$. In this work, we propose a new approach that achieves accurate prediction over essentially any product distribution $D$, provided it is not "classical" in which case there is a trivial exponential lower bound. Our method employs a "biased Pauli analysis," analogous to classical biased Fourier analysis. Implementing this approach requires overcoming several challenges unique to the quantum setting, including the lack of a basis with appropriate orthogonality properties. The techniques we develop to address these issues may have broader applications in quantum information.