Efficient Noise and Fidelity Estimation of Fault-Tolerant Quantum Circuits

Benchmarking physical devices and verifying logical algorithms are important tasks for scalable fault-tolerant quantum computing. While many protocols exist for benchmarking devices prior to algorithm execution, we investigate whether in situ benchmarking can be performed for both physical and logical errors in fault-tolerant circuits. Using the spacetime code formalism, we map general fault-tolerant Clifford circuits to subsystem codes and propose a scheme for learning Pauli noise—up to logical equivalence—directly from syndrome data collected during active error correction. Furthermore, under standard fault-tolerance conditions for circuit-level Pauli noise model, we prove that the logical error rates of Clifford circuits with magic state injection can be learned from syndrome data, and that output fidelities can be efficiently and accurately estimated using proxy metrics. We also introduce a method to estimate physical noise and logical fidelity in circuits built from X, CNOT, and diagonal gates in the third level of the Clifford hierarchy. We validate our methods using experimental syndrome data from both Clifford and logical IQP circuits, demonstrating their practical applicability and accuracy. Our approach requires only a polynomial number of samples, even when the logical error rate is exponentially suppressed by the code distance. This provides a way of characterizing physical components of the device to help gate tune-up, improve decoding accuracy, and verify logical circuits without running extra circuits, but relying solely on efficient classical processing of existing syndrome data.