Purification and correction of quantum channels by commutation-derived quantum filters

Abstract: Reducing the effect of errors is essential for reliable quantum computation. Quantum error mitigation (QEM) and quantum error correction (QEC) are two frameworks that have been proposed to address this task, each with its respective challenges: sampling costs and inability to recover the state for QEM, and qubit overheads for QEC. In this work, we combine ideas from these two frameworks and introduce an information-theoretic machinery called a quantum filter that can purify or correct quantum channels. We provide an explicit construction of a filter that can correct arbitrary types of noise in an $n$-qubit Clifford circuit using $2n$ ancillary qubits based on a commutation-derived error detection circuit. We also show that this filtering scheme can partially purify noise in non-Clifford gates (e.g. T and CCZ gates). In contrast to QEC, this scheme works in an error-reduction sense because it does not require prior encoding of the input state into a QEC code and requires only a single instance of the target channel. Under the assumptions of clean ancillary qubits, this scheme overcomes the exponential sampling overhead in QEM because it can deterministically correct the error channel without discarding any result. We further propose an ancilla-efficient Pauli filter which can remove nearly all the low-weight Pauli components of the error channel in a Clifford circuit using only 2 ancillary qubits similar to flag error correction codes. We prove that for local depolarising noise, this filter can achieve a quadratic reduction in the {average} infidelity of the channel. The Pauli filter can also be used to convert an unbiased error channel into a completely biased error channel and thus is compatible with biased-noise QEC codes which have high code capacity. These examples demonstrate the utility of the quantum filter as an efficient error-reduction technique.