Extracting information efficiently from quantum systems is a major component of quantum information processing tasks. Randomized measurements, or classical shadows, enable predicting many properties of arbitrary quantum states using few measurements. While random single qubit measurements are experimentally friendly and suitable for learning low-weight Pauli observables, they perform poorly for nonlocal observables. Prepending a shallow random quantum circuit before measurements maintains this experimental friendliness, but also has favorable sample complexities for observables beyond low-weight Paulis, including high-weight Paulis and global low-rank properties such as fidelity. However, in realistic scenarios, quantum noise accumulated with each additional layer of the shallow circuit biases the results. To address these challenges, we propose the robust shallow shadows protocol. Our protocol uses Bayesian inference to learn the experimentally relevant noise model and mitigate it in postprocessing. This mitigation introduces a bias-variance trade-off: correcting for noise-induced bias comes at the cost of a larger estimator variance. Despite this increased variance, as we demonstrate on a superconducting quantum processor, our protocol correctly recovers state properties such as expectation values, fidelity, and entanglement entropy, while maintaining a lower sample complexity compared to the random single qubit measurement scheme. We also theoretically analyze the effects of noise on sample complexity and show how the optimal choice of the shallow shadow depth varies with noise strength. This combined theoretical and experimental analysis positions the robust shallow shadow protocol as a scalable, robust, and sample-efficient protocol for characterizing quantum states on current quantum computing platforms.

翻译：从量子系统中高效提取信息是量子信息处理任务的重要组成部分。随机测量（或称经典阴影）方法能够利用少量测量预测任意量子态的多种属性。虽然随机单量子比特测量具有实验友好性且适用于学习低权重泡利可观测量，但其对非局域可观测量的表现较差。在测量前添加浅层随机量子电路既能保持这种实验友好性，又能对超出低权重泡利算符的可观测量（包括高权重泡利算符和保真度等全局低秩属性）获得优越的样本复杂度。然而在现实场景中，浅层电路每增加一层累积的量子噪声会引入结果偏差。针对这些挑战，我们提出鲁棒浅层阴影协议。该协议通过贝叶斯推断学习实验相关噪声模型并在后处理中对其进行抑制。这种抑制方法引入了偏差-方差权衡：校正噪声引起的偏差以增大估计器方差为代价。尽管方差增大，但如我们在超导量子处理器上所示范的，该协议能正确恢复期望值、保真度和纠缠熵等状态属性，同时相较随机单量子比特测量方案保持更低的样本复杂度。我们还从理论上分析了噪声对样本复杂度的影响，揭示了浅层阴影深度的最优选择如何随噪声强度变化。这种理论与实验相结合的分析表明，鲁棒浅层阴影协议是当前量子计算平台上可扩展、鲁棒且样本高效的量子态表征方案。