提出新方法，实现对任意量子通道在一般产品分布下的高效预测。
Predicting quantum channels over general product distributions
- 采用类经典偏置傅里叶分析的量子保罗正交基方法
- 在非经典产品分布下，误差ε对应时间复杂度为n^{O(log(1/ε))}
- 适用于广义产品态输入，突破此前仅限于克利福德不变分布的限制
我们研究未知量子通道输出行为的预测问题。给定对n比特量子通道E和可观测量O的查询访问，目标是在分布D采样的大多数输入ρ上，以小误差ε学习映射ρ↦Tr(O E[ρ])。此前Huang、Chen和Preskill证明，即使通道任意，该任务也可在约n^{O(log(1/ε))}时间内完成，但其保证仅适用于对所有单比特克利福德门不变的输入分布。当输入为一般产品态分布时，其算法失效。本文提出新方法，在几乎任意产品分布下实现准确预测，前提是分布不为“经典”——否则存在平凡的指数下界。方法基于“有偏保罗分析”，类比经典偏置傅里叶分析。实现中需克服量子设置下缺乏合适正交基等挑战，所发展技术可能具有更广泛量子信息应用价值。
原文摘要 · Abstract (English)
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*} ρ\mapsto \mathrm{Tr}(O E[ρ]) \end{equation*} to within a small error for most $ρ$ 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 $n^{O(\log(1/ε))}$, where $ε$ 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 $ρ$. 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.
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