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arXiv:2604.11857

Blind Catalytic Quantum Error Correction: Target-State Estimation and Fidelity Recovery Without A Priori Knowledge

Hikaru Wakaura

quant-phSubmitted Apr 13, 2026• Announced Apr 14, 2026• 0 citations• Revised Oct 7, 2026 (v4)
v1 · Apr 13, 2026v2 · May 7, 2026v3 · May 8, 2026v4 · Oct 7, 2026
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Abstract

Catalytic quantum error correction (CQEC) amplifies residual coherence with a reusable catalyst, giving threshold-free recovery whenever the target coherent modes survive in the noisy state; its original protocol, however, requires complete knowledge of the ideal target, which fails for variational and iterative algorithms whose output is unknown to the correction module. Here we show that this requirement can be removed by estimating the target from the noisy output alone, in a two-stage protocol we call blind CQEC. At the density-matrix level studied here the recovery map is the mode-inclusion-restricted projection of the estimate, so recovery fidelity obeys F_rec >= 1 - 2||rho_est - rho_target||_1 - 2 Delta_mode: the design of blind CQEC reduces to a classical estimation problem under a recovery ceiling set by mode survival. We benchmark five estimation strategies across three noise channels, four quantum algorithms (d = 4-64), Haar-random states, and mixed targets, assuming oracle access to the noisy density matrix. With exactly calibrated noise, channel inversion coincides with the oracle and both are capped by the mode-survival ceiling (F_rec = 0.77 at d = 64 with the 10^-10 mode threshold); noise-model-free coherence maximization matches the ceiling to within 1.5% at d <= 16; the choice between them is set by calibration accuracy, with the dephasing rate exponentially dominant. Unlike error-mitigation methods, blind CQEC returns the state itself rather than corrected expectation values. A noisy-VQE demonstration for H2 yields a 3.4x energy-error reduction. These results chart the estimator design space for blind, threshold-free recovery and identify the two open problems, an operational measurement model and a circuit-level catalytic map, that remain before deployment.

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Hikaru Wakaura, Taiki Tanimae

We characterize where, in the noise parameter space of the uniform dephasing--depolarizing channel $\mathcal N_γ^δ=\mathcal E_δ\!\circ\!\mathcal D_γ$, a deterministic, \emph{nonlinear, target-informed} denoising heuristic yields its largest fidelity gain. The procedure pulls the off-diagonal magnitudes of the noisy state toward those of a known target, with efficiency set by a SWAP-test-purified catalyst; it is not a quantum channel, and target access is an explicit classical resource, so the results describe a benchmark procedure, not blind error correction. Our main tool is the covariant purification map $\mathcal P_\mathrm{cov}(ρ)=(ρ+ρ^2)/(1+\mathrm{Tr}\,ρ^2)$, an exact closed form for one SWAP-test purification round (a rederivation of symmetrization purification: Barenco \emph{et al.}, Cirac--Ekert--Macchiavello) that reduces the catalyst to a scalar eigenvalue iteration. With it we derive the $d\to\infty$ fidelity-gain peak location on Haar-random pure states (Theorem~3): $γ_{\rm peak}(d)\toγ^\star(r,δ)$, with $γ^\star(2,0.1)=0.4725$ and limiting magnitude $0.2262$. Bootstrap-quantified sweeps to $d=256$ are consistent with both limits. The peak is resource- and protocol-dependent: a catalyst-only reference moves it from $\approx0.50$ to $\approx0.34$, and one purification round instead of two to $\approx0.39$. Bell and uniform $d=4$ states admit unique-peak theorems for the $r=0$, $δ=0$ protocol member. All results are reproducible from the open-source \texttt{organic-qc-bench} package with seed~42.

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Accurate optimal quantum error correction thresholds from coherent information

Luis Colmenárez, Ze-Min Huang, Sebastian Diehl +1 more

Quantum error correcting (QEC) codes protect quantum information from decoherence, as long as error rates fall below critical error thresholds. In general, obtaining thresholds implies simulating the QEC procedure using, in general, sub-optimal decoding strategies. In a few cases and for sufficiently simple noise models, optimal decoding of QEC codes can be framed as a phase transition in disordered classical spin models. In both situations, accurate estimation of thresholds demands intensive computational resources. Here we use the coherent information of the mixed state of noisy QEC codes to accurately estimate the associated optimal QEC thresholds already from small-distance codes at moderate computational cost. We show the effectiveness and versatility of our method by applying it first to the topological surface and color code under bit-flip and depolarizing noise. We then extend the coherent information based methodology to phenomenological and quantum circuit level noise settings. For all examples considered we obtain highly accurate estimates of optimal error thresholds from small, low-distance instances of the codes, in close accordance with threshold values reported in the literature. Our findings establish the coherent information as a reliable competitive practical tool for the calculation of optimal thresholds of state-of-the-art QEC codes under realistic noise models.

arXiv:2312.06664•Jan 1, 2023•0 citations
Catalytic quantum error correction (CQEC) amplifies residual coherence with a reusable catalyst, giving threshold-free recovery whenever the target coherent modes survive in the noisy state; its original protocol, however, requires complete knowledge of the ideal target, which fails for variational and iterative algorithms whose output is unknown to the correction module. Here we show that this requirement can be removed by estimating the target from the noisy output alone, in a two-stage protocol we call blind CQEC. At the density-matrix level studied here the recovery map is the mode-inclusion-restricted projection of the estimate, so recovery fidelity obeys F_rec >= 1 - 2||rho_est - rho_target||_1 - 2 Delta_mode: the design of blind CQEC reduces to a classical estimation problem under a recovery ceiling set by mode survival. We benchmark five estimation strategies across three noise channels, four quantum algorithms (d = 4-64), Haar-random states, and mixed targets, assuming oracle access to the noisy density matrix. With exactly calibrated noise, channel inversion coincides with the oracle and both are capped by the mode-survival ceiling (F_rec = 0.77 at d = 64 with the 10^-10 mode threshold); noise-model-free coherence maximization matches the ceiling to within 1.5% at d <= 16; the choice between them is set by calibration accuracy, with the dephasing rate exponentially dominant. Unlike error-mitigation methods, blind CQEC returns the state itself rather than corrected expectation values. A noisy-VQE demonstration for H2 yields a 3.4x energy-error reduction. These results chart the estimator design space for blind, threshold-free recovery and identify the two open problems, an operational measurement model and a circuit-level catalytic map, that remain before deployment.
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 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.
Most previous efforts on quantum error correction focused on either extending classical error-correction schemes to the quantum regime by performing a perfect correction on a subset of errors or seeking a recovery operation to maximize the fidelity between an input state and its corresponding output state of a noisy channel. There are few results concerning quantum error precompensation. Here we design an error-precompensated input state for an arbitrary quantum noisy channel and a given target output state. By following a procedure, the required input state, if it exists, can be analytically obtained in single-partite systems. Furthermore, we also present semidefinite programs to numerically obtain the error-precompensated input states with maximal fidelities between the target state and the output state. The numerical results coincide with the analytical results.
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
The γcγ_cγc​-Peak: Covariant Recovery on Four Organic Qubit Platforms
We characterize where, in the noise parameter space of the uniform dephasing--depolarizing channel Nγδ=Eδ ⁣∘ ⁣Dγ\mathcal N_γ^δ=\mathcal E_δ\!\circ\!\mathcal D_γNγδ​=Eδ​∘Dγ​, a deterministic, \emph{nonlinear, target-informed} denoising heuristic yields its largest fidelity gain. The procedure pulls the off-diagonal magnitudes of the noisy state toward those of a known target, with efficiency set by a SWAP-test-purified catalyst; it is not a quantum channel, and target access is an explicit classical resource, so the results describe a benchmark procedure, not blind error correction. Our main tool is the covariant purification map Pcov(ρ)=(ρ+ρ2)/(1+Tr ρ2)\mathcal P_\mathrm{cov}(ρ)=(ρ+ρ^2)/(1+\mathrm{Tr}\,ρ^2)Pcov​(ρ)=(ρ+ρ2)/(1+Trρ2), an exact closed form for one SWAP-test purification round (a rederivation of symmetrization purification: Barenco \emph{et al.}, Cirac--Ekert--Macchiavello) that reduces the catalyst to a scalar eigenvalue iteration. With it we derive the d→∞d\to\inftyd→∞ fidelity-gain peak location on Haar-random pure states (Theorem~3): γpeak(d)→γ⋆(r,δ)γ_{\rm peak}(d)\toγ^\star(r,δ)γpeak​(d)→γ⋆(r,δ), with γ⋆(2,0.1)=0.4725γ^\star(2,0.1)=0.4725γ⋆(2,0.1)=0.4725 and limiting magnitude 0.22620.22620.2262. Bootstrap-quantified sweeps to d=256d=256d=256 are consistent with both limits. The peak is resource- and protocol-dependent: a catalyst-only reference moves it from ≈0.50\approx0.50≈0.50 to ≈0.34\approx0.34≈0.34, and one purification round instead of two to ≈0.39\approx0.39≈0.39. Bell and uniform d=4d=4d=4 states admit unique-peak theorems for the r=0r=0r=0, δ=0δ=0δ=0 protocol member. All results are reproducible from the open-source \texttt{organic-qc-bench} package with seed~42.
Quantum error correcting (QEC) codes protect quantum information from decoherence, as long as error rates fall below critical error thresholds. In general, obtaining thresholds implies simulating the QEC procedure using, in general, sub-optimal decoding strategies. In a few cases and for sufficiently simple noise models, optimal decoding of QEC codes can be framed as a phase transition in disordered classical spin models. In both situations, accurate estimation of thresholds demands intensive computational resources. Here we use the coherent information of the mixed state of noisy QEC codes to accurately estimate the associated optimal QEC thresholds already from small-distance codes at moderate computational cost. We show the effectiveness and versatility of our method by applying it first to the topological surface and color code under bit-flip and depolarizing noise. We then extend the coherent information based methodology to phenomenological and quantum circuit level noise settings. For all examples considered we obtain highly accurate estimates of optimal error thresholds from small, low-distance instances of the codes, in close accordance with threshold values reported in the literature. Our findings establish the coherent information as a reliable competitive practical tool for the calculation of optimal thresholds of state-of-the-art QEC codes under realistic noise models.