We introduce a technique to estimate error-mitigated expectation values on noisy quantum computers. Our technique performs shadow tomography on a logical state to produce a memory-efficient classical reconstruction of the noisy density matrix. Using efficient classical post-processing, one can mitigate errors by projecting a general nonlinear function of the noisy density matrix into the codespace. The subspace expansion and virtual distillation can be viewed as special cases of the new framekwork. We show our method is favorable in the quantum and classical resources overhead. Relative to subspace expansion which requires $O\left(2^{N} \right)$ samples to estimate a logical Pauli observable with $[[N, k]]$ error correction code, our technique requires only $O\left(4^{k} \right)$ samples. Relative to virtual distillation, our technique can compute powers of the density matrix without additional copies of quantum states or quantum memory. We present numerical evidence using logical states encoded with up to sixty physical qubits and show fast convergence to error-free expectation values with only $10^5$ samples under 1% depolarizing noise.
Time-Reversal Selection Rules for Quantum Error Correction
Quantum Initiative, Florida State University（佛罗里达州立大学量子倡议）；Department of Mathematics, Florida State University（佛罗里达州立大学数学系）；Department of Mathematics, University of California, San Diego（加州大学圣地亚哥分校数学系）
Physics-Inspired Extrapolation for efficient error mitigation and hardware certification
LG Electronics Toronto AI Lab（LG电子多伦多人工智能实验室）；Instituto de Física Fundamental IFF-CSIC（基础物理研究所（西班牙科学理事会下属））；Instituto Tecnológico de Galicia（加利西亚理工学院）；LG Electronics, AI Lab, CTO Div（LG电子，人工智能实验室，首席技术官部门）
Quantum subspace verification for error correction codes
Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院量子信息中心）；Pritzker School of Molecular Engineering, The University of Chicago（芝加哥大学普里茨克分子工程学院）；Department of Computer Science, University of Hong Kong（香港大学计算机科学系）；Key Laboratory for Information Science of Electromagnetic Waves (Ministry of Education), Fudan University（复旦大学电磁波信息科学教育部重点实验室）
