The rapid advancement of quantum computing has led to an extensive demand for effective techniques to extract classical information from quantum systems, particularly in fields like quantum machine learning and quantum chemistry. However, quantum systems are inherently susceptible to noises, which adversely corrupt the information encoded in quantum systems. In this work, we introduce an efficient algorithm that can recover information from quantum states under Pauli noise. The core idea is to learn the necessary information of the unknown Pauli channel by post-processing the classical shadows of the channel. For a local and bounded-degree observable, only partial knowledge of the channel is required rather than its complete classical description to recover the ideal information, resulting in a polynomial-time algorithm. This contrasts with conventional methods such as probabilistic error cancellation, which requires the full information of the channel and exhibits exponential scaling with the number of qubits. We also prove that this scalable method is optimal on the sample complexity and generalise the algorithm to the weight contracting channel. Furthermore, we demonstrate the validity of the algorithm on the 1D anisotropic Heisenberg-type model via numerical simulations. As a notable application, our method can be severed as a sample-efficient error mitigation scheme for Clifford circuits.

翻译：量子计算的快速发展使得从量子系统中高效提取经典信息的技术需求日益增长，特别是在量子机器学习和量子化学等领域。然而，量子系统天生易受噪声影响，这会对编码在量子系统中的信息造成破坏。本文提出了一种高效算法，能够从泡利噪声下的量子态中恢复信息。其核心思想是通过后处理信道本身的经典阴影，学习未知泡利信道的必要信息。对于局部和有界度的可观测量，仅需部分信道知识而非其完整经典描述即可恢复理想信息，从而得到多项式时间算法。这与概率误差消除等传统方法形成对比，后者需要信道的全部信息，且复杂度随量子比特数呈指数增长。我们还证明该可扩展方法在样本复杂度上达到最优，并将算法推广至权重收缩信道。此外，通过一维各向异性海森堡模型的数值模拟验证了算法的有效性。作为重要应用，该方法可作为克利福德电路的高效样品误差缓解方案。