Quantum Error Mitigated Classical Shadows
Quantum Error Mitigated Classical Shadows
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DOI：
10.1103/prxquantum.5.010324
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发表时间：
2023-05
期刊：
影响因子：
9.7
通讯作者：
Hamza Jnane;Jonathan Steinberg;Z. Cai;H. C. Nguyen;Bálint Koczor
中科院分区：
文献类型：
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作者：
Hamza Jnane;Jonathan Steinberg;Z. Cai;H. C. Nguyen;Bálint Koczor
经典阴影使我们能够通过很少的测量来了解量子态 $\rho$ 的许多属性。然而，近期和早期的容错量子计算机只能准备有噪声的量子态 $\rho$，因此有效学习理想的无噪声状态 $\rho_{id}$ 的属性是一个相当大的挑战。我们考虑误差减轻技术，例如概率误差消除（PEC）、零噪声外推（ZNE）和对称验证（SV），这些技术是为减轻单一期望值测量中的误差而开发的，并将它们推广到减轻经典阴影中的误差。我们发现 PEC 是最自然的候选者，因此为 PEC 影子开发了一个完整的理论框架，并具有以下严格的理论保证： PEC 影子是理想量子态 $\rho_{id}$ 的无偏估计器；同时预测 $\rho_{id}$ 的许多线性属性的样本复杂度与传统阴影方法的样本复杂度相同，乘法因子是由于误差缓解而产生的样本开销。由于阴影的高效后处理，这种开销并不直接取决于量子位的数量，而是随着噪声门的数量呈指数增长。这项工作中引入的广泛工具可能有助于开发近期和早期的容错量子计算机：我们在详细的数值模拟中演示了量子计算机的一系列实际应用，这些应用将大大受益于我们的技术。
Classical shadows enable us to learn many properties of a quantum state $\rho$ with very few measurements. However, near-term and early fault-tolerant quantum computers will only be able to prepare noisy quantum states $\rho$ and it is thus a considerable challenge to efficiently learn properties of an ideal, noise free state $\rho_{id}$. We consider error mitigation techniques, such as Probabilistic Error Cancellation (PEC), Zero Noise Extrapolation (ZNE) and Symmetry Verification (SV) which have been developed for mitigating errors in single expected value measurements and generalise them for mitigating errors in classical shadows. We find that PEC is the most natural candidate and thus develop a thorough theoretical framework for PEC shadows with the following rigorous theoretical guarantees: PEC shadows are an unbiased estimator for the ideal quantum state $\rho_{id}$; the sample complexity for simultaneously predicting many linear properties of $\rho_{id}$ is identical to that of the conventional shadows approach up to a multiplicative factor which is the sample overhead due to error mitigation. Due to efficient post-processing of shadows, this overhead does not depend directly on the number of qubits but rather grows exponentially with the number of noisy gates. The broad set of tools introduced in this work may be instrumental in exploiting near-term and early fault-tolerant quantum computers: We demonstrate in detailed numerical simulations a range of practical applications of quantum computers that will significantly benefit from our techniques.