Symmetric Channel Verification for Purifying Noisy Quantum Channels

Symmetry inherent in quantum states has been widely used to reduce the effect of noise in quantum error correction and a quantum error-mitigation technique known as symmetry verification. However, these symmetry-based techniques exploit symmetry in quantum rather than quantum , limiting their application to cases where the entire circuit shares the same symmetry. In this work, we propose symmetric channel verification (SCV), a channel purification protocol that leverages the symmetry inherent in quantum channels. By introducing different phases to each symmetric subspace and employing a quantum phase estimation-like circuit, SCV can detect and correct symmetry-breaking noise in quantum channels. We further propose a hardware-efficient implementation of SCV at the virtual level, which requires only a single-qubit ancilla and is robust against the noise in the ancilla qubit. Our protocol is applied to various Hamiltonian simulation circuits and phase estimation circuits, resulting in a significant reduction of errors. Furthermore, in setups where only Clifford unitaries can be used for noise purification, which is relevant in the early fault-tolerant regime, we show that SCV under Pauli symmetry represents the optimal purification method.