Partial Error Correction with Error Mitigation

Fault-tolerant quantum computing remains constrained by the resources of current quantum devices. However, quantum error mitigation techniques combined with partial quantum error correction (QEC) could be employed to protect near-term quantum systems. Using a modified Pauli twirling scheme, we demonstrate how propagating quantum error can be effectively shaped to favor specific QEC codes. Using cycle benchmarking, we show that our twirling technique increases the effectiveness of partial error correction and error detection schemes, while simultaneously reducing overhead. We propose a protocol that integrates probabilistic error cancellation (PEC) with partial QEC to reduce logical errors in noisy quantum circuits. We optimize the PEC procedure by using a classical algorithm to estimate the total noise $\Lambda_{\rm tot}$, then apply PEC to cancel $\Lambda_{\rm tot}$ at the end of circuit, rather than after each layer, reducing the depth-dependent scaling of the sampling overhead from an exponential to polynomial scale. This reduction effectively exchanges classical resources for exponential quantum resources. Finally, we implement our protocol on a non-Clifford VQE circuit which estimates the ground state energy of $\rm H_2$ using a $[[4,2,2]]$ quantum error detection code.