Quantum Error Recovery (QER) uses knowledge of the error channel acting on a quantum system to find optimal recovery maps. The scheme restores the uncorrupted state with a fidelity exceeding that achieved by noise parameter independent quantum error correction. We use a generic coherent QER map implemented with a quantum circuit acting on the system together with ancillary qubits to recover quantum information stored in permutation invariant (PI) codes. PI codes admit tunable parameters to suit the noise model and benefit from simple recovery operation circuits with reduced addressability requirements, unlike stabilizer codes. We showcase the method by modeling QER in PI codes after collective and local symmetric correlated amplitude-damping (AD) noise, a non-Pauli noise process for which stabilizer codes often require additional overhead. We also propose a new PI code family called CAD codes with explicit examples on 4 and 9 qubits for global symmetric AD errors. We show that CAD9 (supported on 9 qubits) code beats many existing codes by more than one order of magnitude. For the CAD4 code, which perfectly corrects 1 global symmetric AD error, the compiled recovery circuit consists of 10 system and system-ancilla gates which can be realized from linear geometric phase gates. Our work provides a direct path from optimized recovery maps to experimentally implementable, low-overhead protocols.
Quantum Error Recovery (QER) uses knowledge of the error channel acting on a quantum system to find optimal recovery maps. The scheme restores the uncorrupted state with a fidelity exceeding that achieved by noise parameter independent quantum error correction. We use a generic coherent QER map implemented with a quantum circuit acting on the system together with ancillary qubits to recover quantum information stored in permutation invariant (PI) codes. PI codes admit tunable parameters to suit the noise model and benefit from simple recovery operation circuits with reduced addressability requirements, unlike stabilizer codes. We showcase the method by modeling QER in PI codes after collective and local symmetric correlated amplitude-damping (AD) noise, a non-Pauli noise process for which stabilizer codes often require additional overhead. We also propose a new PI code family called CAD codes with explicit examples on 4 and 9 qubits for global symmetric AD errors. We show that CAD9 (supported on 9 qubits) code beats many existing codes by more than one order of magnitude. For the CAD4 code, which perfectly corrects 1 global symmetric AD error, the compiled recovery circuit consists of 10 system and system-ancilla gates which can be realized from linear geometric phase gates. Our work provides a direct path from optimized recovery maps to experimentally implementable, low-overhead protocols.
Implementing quantum error correction (QEC) protocols is a challenging task in today's era of noisy intermediate-scale quantum devices. We present quantum circuits for a universal, noise-adapted recovery map, often referred to as the Petz map, which is known to achieve close-to-optimal fidelity for arbitrary codes and noise channels. While two of our circuit constructions draw upon algebraic techniques such as isometric extension and block encoding, the third approach breaks down the recovery map into a sequence of two-outcome POVMs. In each of the three cases we improve upon the resource requirements that currently exist in the literature. Apart from Petz recovery circuits, we also present circuits that can directly estimate the fidelity between the encoded state and the recovered state. As a concrete example of our circuit constructions, we implement Petz recovery circuits corresponding to a four-qubit code that is tailor-made to correct for amplitude-damping noise. The efficacy of our noise-adapted recovery circuits is then demonstrated through ideal and noisy simulations on the IBM QasmSimulator. Published by the American Physical Society 2024
Quantum error correcting (QEC) plays a crucial role in protecting quantum information against decoherence and enabling scalable, reliable quantum computing. One of the most realistic and ubiquitous sources of noise affecting quantum hardware today is generalized amplitude-damping (GAD) noise. Conventional, deterministic QEC codes struggle to correct for GAD noise because of their inherent structure, leading to fidelity losses that scale linearly with the damping strength. In this work, we introduce the framework of probabilistic approximate quantum error correction (PAQEC), that combines the flexibility of approximate QEC with the potential of post-selected recovery, enabling high-fidelity, resource-efficient error correction. We construct a five-qubit permutation-invariant code that, under probabilistic recovery, achieves a fidelity loss quadratic in the damping strength, thus outperforming existing QEC codes. Formulating PAQEC as an optimization problem, we present a numerical technique based on Charnes-Cooper and semidefinite programming to identify the optimal recovery map for any PAQEC code. Our results establish PAQEC as a powerful tool for developing resource-efficient, high-fidelity quantum codes tailored to realistic noise, with promising implications for near-term quantum devices and future fault-tolerant architectures.
We present for the first time a general theory of error correction for permutation invariant (PI) codes. Using representation theory of the symmetric group we construct efficient algorithms that can correct any correctible error on any PI code. These algorithms involve measurements of total angular momentum, quantum Schur transforms or logical state teleportations, and geometric phase gates. For erasure errors, or more generally deletion errors, on certain PI codes, we give a simpler quantum error correction algorithm.
Abstract We propose a novel optimization scheme designed to find optimally correctable subspace codes for a known quantum noise channel. To each candidate subspace code we first associate a universal recovery map, as if the code was perfectly correctable, and aim to maximize a performance functional that combines a modified channel fidelity with a tuneable regularization term that promotes simpler codes. With this choice optimization is performed only over the set of codes, and not over the set of recovery operators. The set of codes of fixed dimension is parametrized as a complex-valued Stiefel manifold: the resulting non-convex optimization problem is then solved by gradient-based local algorithms. When perfectly correctable codes cannot be found, a second optimization routine is run on the recovery Kraus map, also parametrized in a suitable Stiefel manifold via Stinespring representation. To test the approach, correctable codes are sought in different scenarios and compared to existing ones: three qubits subjected to bit-flip errors (single and correlated), four qubits undergoing local amplitude damping and five qubits subjected to local depolarizing channels. Approximate codes are found and tested for the previous examples as well pure non-Markovian dephasing noise acting on a 7 / 2 spin, induced by a 1 / 2 spin bath, and the noise of the first three qubits of IBM’s ibm_kyoto quantum computer. The fidelity results are competitive with existing iterative optimization algorithms, with respect to which we maintain a strong computational advantage, while obtaining simpler codes.
We formulate stabilizer-code error correction as a code- and recovery-dependent transformation of the noise channel. For a fixed recovery map, syndrome projection produces a block-diagonal logical channel whose diagonal Pauli weights follow from code-induced error classes, with no physical Pauli twirl; concatenation is then iterated redistribution, with the dominant logical error guiding the next code. The bit-flip and phase-flip three-qubit codes with standard recoveries supply a closed-form benchmark, and logically inequivalent recoveries appear as a design degree of freedom for shaping the channel presented to the next decoder. For amplitude damping, the three-qubit rules select C2(σ^y)C1(σ^x) and give fl=1−O(γ2), with an ideal-gate crossover at γc≈0.30 (f=1−1.72755γ2) and a favorable gate-count comparison with the five-qubit code under the same assumptions. Correlated noise requires joint class statistics, whereas coherent noise retains intra-class coherences that make the same transformation usable in reverse as a tomographic fingerprint; there the companion work proves local identifiability from a rank certificate evaluated inside the parameter domain, classifies the sole permutation-symmetry ambiguity as structural at any chain length, and closes it with one two-qubit marginal channel. The construction remains an analytic benchmark, not a universal prescription for an inner code beneath a surface code.
What must a quantum error-correction decoder know about temporally correlated noise, and what does it cost to learn? We study this question for fixed stabilizer memory experiments with known system-environment interactions. We formulate calibration in terms of the fidelity differences that select a Pauli recovery and relate missing information to the resulting fidelity loss. Our main example uses weak coherent X rotations in a class of CSS codes including Shor, Steane, and odd-distance rotated surface codes. All physical qubits share a sign either fixed throughout a run or independently redrawn each interval. The two models have identical syndrome statistics and the same spatiotemporal Pauli process (SPP), obtained by Pauli-twirling every noise interval. Yet at suitable storage times, they require different logical corrections. In the weak-noise limit, each model is almost perfectly correctable when known. Without model information, even the best final recovery has a worst-case fidelity loss approaching 1/2 relative to the noise-aware optimum, despite access to the full syndrome record. For this pair of models, at fixed code and fixed positive target loss below 1/2, we prove matching calibration bounds. With logical preparation and final readout, complete QEC cycles require a number of noise intervals scaling as the inverse rotation angle raised to the code's X distance. Allowing two uninterrupted intervals before one extraction reduces the optimal cost to inverse-square scaling. Physical Pauli twirling of every noise interval changes both recovery performance and its information requirements. Discarding the record of applied random Pauli operations makes the SPP sufficient for optimal recovery. Keeping this record can improve the attainable fidelity, but reaching that optimum may require additional noise information.