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

TL;DR

This work introduces quantum filters, an information-theoretic tool that purifies or corrects quantum channels by exploiting commutation structure, thereby addressing error suppression with fewer resources than traditional quantum error correction or post-processing-only methods. The authors develop the commutation-derived filter, show how to detect or deterministically correct errors with a single query to the noisy circuit using Clifford operations, and extend the framework to a general superchannel formulation. They demonstrate complete purification of Clifford channels with 2n clean ancillas and partial purification for non-Clifford gates, along with an ancilla-efficient Pauli filter that removes low-weight Pauli errors using only two ancillas, achieving a quadratic reduction in infidelity under local depolarizing noise. Numerical results confirm regimes where quantum filters outperform other AQEM strategies, and theoretical analyses provide scaling laws and resource considerations; collectively, the approach offers a scalable, tunable alternative bridging QEM and QEC, with practical relevance for near-term and early fault-tolerant devices.

Abstract

Reducing errors is essential for reliable quantum computation. Quantum error mitigation (QEM) and quantum error correction (QEC) are two leading approaches for this task, each with challenges: QEM suffers from high sampling costs and cannot recover states, while QEC incurs large qubit and gate overheads. We combine ideas from both and introduce an information-theoretic device called a quantum filter that can purify or correct quantum channels. We present an explicit construction capable of correcting arbitrary noise in an n-qubit Clifford circuit using 2n ancillary qubits through a commutation-derived error-detection circuit. This scheme can also partially purify noise in non-Clifford gates such as T and CCZ. Unlike QEC, it achieves deterministic error reduction without encoding the input state. Under the assumption of clean ancillas, it overcomes the exponential sampling overhead in QEM using a single query to the channel. We also propose an ancilla-efficient Pauli filter that removes nearly all low-weight erroneous Pauli components in noisy Clifford circuits using only two ancillas. For local depolarizing noise, it achieves a quadratic reduction in average infidelity. Beyond existing QEM methods, our approach enables systematic error correction as the infidelity can be exponentially reduced with each added ancilla. Through numerical simulations under ancilla noise, we identify regimes where quantum filters outperform other techniques, demonstrating their effectiveness as a scalable error-reduction tool for quantum information processing.

Figures (19)

• Figure 1: Successive filtration scheme for purifying a single-qubit error channel.
• Figure 2: An example of a Quantum Filter for channel correction.
• Figure 3: Quantum filter outlining the filter supermap. The output is given by \ref{['eqn:vanilla_filter_output']}. If the filtering qubits are discarded, this will result in a non-unitary quantum filter. We denote the corresponding superchannel by $\widetilde{\boldsymbol{\mathcal{F}}}$.
• Figure 4: A schematic of a quantum filter specified by the prepare unitary V and the controlled unitaries $\textsc{Select}_P$ and $\textsc{Select}_Q$.
• Figure 5: Schematic of a Circuit Compiled in the Clifford+$T$ gate-set. $\boldsymbol{C}_i$ and $\boldsymbol{T}_i$ are circuit layers composed purely of Clifford and T gates respectively. $\mathcal{D}_p$ is a local depolarising channel with error-rate $p$.

Theorems & Definitions (4)

• Proposition 1
• proof
• Theorem 1: Quadratic reduction in the infidelity with ancilla-efficient Pauli filters
• Theorem 2: Filtration of low-weight Pauli errors