Efficient benchmarking of logical noise

Characterizing the performance of noisy quantum circuits is crucial for the calibration of quantum devices, but it can be costly in terms of experimental efforts.  In the context of quantum error correction (QEC), a central component of device calibration is to learn the logical noise induced by the physical noise process. In the context of local Pauli noise models, Wagner et al. [PRL 2023] provides an initial framework where the logical Pauli noise can be estimated from syndrome measurement data, which is usually collected during error correction anyway. In this work, we provide the necessary and sufficient conditions for learnability of logical Pauli noise from syndrome data, from a purely code-theoretical perspective. We also extend the framework to corporate circuit-level noise where we explicitly characterize the learnable degrees of freedom with circuit Pauli faults, even if it is impossible to learn every fault probability of the entire circuit. Furthermore, by utilizing the mathematical tool of compressed sensing, we provide an efficient protocol for estimating the logical Pauli channel with provable sample complexity and computational efficiency. Finally, we present the estimation framework in learning logical noise in an end-to-end protocol and demonstrate its performance in several examples of syndrome extraction circuits.