Abstract We present a general-purpose quantum error correction primitive based on state purification via the SWAP test, which we refer to as purification quantum error correction (PQEC). This method operates on N noisy copies, requires minimally O(Mlog 2 N) data qubits to process the M-qubit inputs. In a similar way to standard QEC, the purification steps may be interleaved within a quantum algorithm to suppress the logical error rate. No postselection is performed and no knowledge of the state is required. We analyze its performance under a variety of error channels and find that PQEC is highly effective at boosting fidelity and reducing logical error rates, particularly for the depolarizing channel. Error thresholds for the local depolarizing channel are found to be 75% for any register size. For local dephasing, the error threshold is reduced to 50% but may be boosted using twirling.
Abstract We present a general-purpose quantum error correction primitive based on state purification via the SWAP test, which we refer to as purification quantum error correction (PQEC). This method operates on N noisy copies, requires minimally O(Mlog 2 N) data qubits to process the M-qubit inputs. In a similar way to standard QEC, the purification steps may be interleaved within a quantum algorithm to suppress the logical error rate. No postselection is performed and no knowledge of the state is required. We analyze its performance under a variety of error channels and find that PQEC is highly effective at boosting fidelity and reducing logical error rates, particularly for the depolarizing channel. Error thresholds for the local depolarizing channel are found to be 75% for any register size. For local dephasing, the error threshold is reduced to 50% but may be boosted using twirling.
Quantum error correction physically removes errors from a quantum state, while quantum error mitigation improves observable estimates by processing noisy measurement data. We introduce \emph{purification quantum error suppression} (PQES), a hybrid approach that uses multiple noisy copies of an unknown state to combine these two ideas. The protocol uses SWAP tests to physically reduce errors by purification, while the full outcome record is used to combine all branches without postselection. In this way, PQES avoids the fixed-success-outcome requirement of standard SWAP-test purification while still accessing the power-purified state ρN. The SWAP identities allow purification steps to be interleaved with unitary circuit blocks, so errors can be suppressed during a computation rather than only at the final measurement. We provide both a parallel binary-tree implementation and a more compact register-recycled implementation using O(Mℓ) coherent data qubits for an M-qubit register and N=2ℓ input copies. We analyze the resulting error thresholds under representative noise models. For local depolarizing noise on the product-state family studied here, the threshold is pth=3/4 for any register size, while local dephasing of ∣+⟩⊗M has a threshold of pth=1/2. Local Clifford twirling can be used to convert dephasing into a depolarization channel and restore the higher threshold.
Quantum computers promise transformative speedups, but environmental noise destroys their fragile states. Conventional quantum error correction (QEC) encodes information redundantly across physical qubits, yet fails above a threshold of about 1% and incurs polynomial qubit overhead. A recent theorem from the resource theory of coherence shows that catalytic covariant operations amplify coherence at an unbounded rate, but this result has never been cast as an operational protocol. The challenge is to turn an asymptotic theorem into a recovery scheme that works at any noise strength with realistic resources. Here we show that catalytic coherence amplification can be cast as an error-correction primitive, Catalytic Quantum Error Correction (CQEC), which recovers a known target state from noisy copies without any error magnitude threshold whenever the target's coherent modes are preserved. In an effective model of the recovery map, fidelity exceeds 0.99 across 200 noise configurations spanning d = 4-64; the catalyst cost drops from the constructive bound n* ~ d^4 e^(2 gamma) to 32 copies at matched fidelity (10^4- to 10^9-fold) via a pipeline of dynamical decoupling, Clifford twirling, and recursive swap-test purification. A first explicitly CPTP, exactly covariant joint-channel implementation validates genuine recovery under dephasing at small dimension (0.54 -> 0.77) while showing that shallow circuits consume the catalyst, quantifying the model-vs-channel gap. These results turn an abstract resource-theoretic statement into a concrete protocol candidate complementary to stabilizer- and purification-based QEC; an open-source package reproducing the benchmarks accompanies this work (arXiv:2603.25774, https://github.com/deeptell-inc/cqec).
Quantum error mitigation is a key approach for extracting target state properties on state-of-the-art noisy machines and early fault-tolerant devices. Using the ideas from flag fault tolerance and virtual state purification, we develop the virtual-channel-purification (VCP) protocol, which consumes similar qubit and gate resources as virtual state purification but offers stronger error suppression with increased system size and more noisy operation copies. The application of VCP does not require specific knowledge about the target quantum state, the target problem and the gate noise model in the target circuit, and can still offer rigorous performance guarantees for practical noise regimes as long as the noise is incoherent. Further connections are made between VCP and quantum error correction to produce the virtual error-correction (VEC) protocol, one of the first protocols that combine quantum error correction (QEC) and quantum error mitigation beyond directly applying error-mitigation protocols on top of logical qubits. Assuming perfect syndrome extraction, VEC can virtually remove all correctable noise in the channel while paying only the same sampling cost as low-order purification. It can achieve QEC-level protection on an unencoded register when transmitting it through a noisy channel, removing the associated encoding qubit overhead. Another variant of VEC can mimic the error-suppression power of the surface code by inputting only a bit-flip and a phase-flip code. Our protocol can also be adapted to key tasks in quantum networks like channel capacity activation and entanglement distribution.
The extreme sensitivity of qubits to environmental noise constitutes the central bottleneck impeding the practical deployment of near-term quantum computing. Quantum error correction suppresses physical errors by encoding logical qubits into redundant physical qubits, yet its corrective gain deteriorates sharply when physical error rates approach the fault-tolerant threshold. This paper proposes a purification-assisted quantum error correction framework that systematically embeds a purification preprocessing module between the encoding layer and the physical layer, founded on the permutation symmetry of multiple noisy state copies. From an information-theoretic perspective, this framework treats the purification process as an entropy reduction mechanism that filters noise entropy from quantum states, concentrating quantum information into lower-entropy subspaces and achieving entropy suppression before downstream error correction processing. By projecting the joint state of identical copies onto the symmetric subspace, the module achieves noise entropy filtering, thereby concentrating quantum information and exponentially compressing the effective physical error rate before it enters the error-correcting code, establishing a compound exponential logical error rate suppression mechanism under ideal assumptions. Analytical derivations under depolarizing noise yield closed-form expressions for the purification fidelity and the enhanced equivalent fault-tolerant threshold, demonstrating that purification with three copies elevates the surface code threshold from approximately 1.1% to approximately 2.0%. Building on this framework, an Iterative Purification-assisted Error Correction (IPEC) algorithm is designed, which dynamically adjusts the purification depth via real-time syndrome feedback to achieve an adaptive balance between fidelity gain and resource consumption. Monte Carlo simulations under both independent depolarizing noise and circuit-level noise models validate the theoretical predictions: the IPEC algorithm reduces the logical error rate by approximately 46-fold at code distance 7 with a physical error rate of 1.0%, achieves an approximately 8-fold improvement on quantum LDPC codes, and maintains robust performance in non-stationary noise environments through its adaptive mechanism. From an entropy-theoretic perspective, the proposed framework constitutes an entropy management mechanism for quantum computation, in which the purification process compresses state entropy at the input stage and error correction maintains the low-entropy condition during computation, together forming a closed-loop entropy flow control system.
Quantum state purification is a process in which decoherence is partially reversed by using multiple copies of the input states that have been subject to the same decoherence effect. This thesis focuses on purifying the decoherence caused by the depolarizing channel. In the first half of the thesis, the purification problem is formally introduced and one efficient purification procedure featuring the swap-test is presented and analyzed. The rest of the thesis formulates the optimal purification problem as an optimization problem and applies it to qubit and qutrit purification.
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.