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arXiv:2605.00026

The $γ_c$-Peak: Covariant Recovery on Four Organic Qubit Platforms

Hikaru Wakaura, Taiki Tanimae

q-bio.NCquant-phSubmitted Apr 22, 2026• Announced Apr 23, 2026• 0 citations• Revised May 13, 2026 (v2)
v1 · Apr 22, 2026v2 · May 13, 2026v3 · May 15, 2026v4 · Aug 14, 2026
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Abstract

The Petz recovery map (1986) provably reverses a noisy quantum channel on a reference state, but its algorithmic relevance to real, dissipation-dominated platforms has remained unclear. Using the open-source \texttt{organic-qc-bench} simulation package, we benchmark a Petz-style covariant-purification quantum error correction (CQEC) protocol across four engineered organic qubit platforms operated \emph{without any magnetic field}: a flavin-nitroxide radical-pair reservoir (P1); perchlorotriphenylmethyl radicals in a covalent organic framework (P2); the SVILC qubit [Wakaura2017] on $κ$-(BEDT-TTF)$_2$Cu[N(CN)$_2$]Br (P3, conditional on SVILC confirmation); and a Su-Schrieffer-Heeger soliton on \emph{trans}-polyacetylene (P4). Across five quantum algorithms (QKAN, qDRIFT, control-free QPE, Shor-Regev, Bernstein-Vazirani) and two ML tasks, CQEC gains are significant ($p\!<\!10^{-5}$; Wilcoxon, Bonferroni $α\!=\!0.05/44$) for all sixteen path$\times$algorithm pairs. The central finding is the \emph{$γ_c$-peak}: the fidelity gain $ΔF$ is maximised \emph{at} the entanglement-breaking threshold $γ_c$, with $ΔF_{\rm max}\!=\!+0.303$ at $d\!=\!64$ and a linear $\log_2 d$ scaling over $d=2$-$64$ -- algorithmically confirming the prediction [Wakaura2026LQBH] that Petz recovery preserves coherence beyond this threshold. Bernstein-Vazirani also yields a $7.6$-$31\times$ provable quantum advantage at $n\!=\!3$-$5$, diarylethene-photoswitch CZ fidelities reach $F_{CZ}\!\ge\!0.987$ for P2-P4, and projected manufacturing costs are 10-40$\times$ lower with 10-200$\times$ less operating power than superconducting platforms. The $γ_c$-peak establishes Petz-style recovery as a practically relevant primitive at the dissipation-coherence boundary and identifies PTM-COF (P2) as the highest-priority experimental target.

Related papers

Catalytic Quantum Error Correction: Theory, Efficient Catalyst Preparation, and Numerical Benchmarks

Hikaru Wakaura, Taiki Tanimae

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).

quant-pharXiv:2603.25774•Mar 27, 2026•0 citations

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

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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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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 $ρ^N$. 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 $O(M\ell)$ coherent data qubits for an $M$-qubit register and $N=2^\ell$ 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 $p_{\mathrm{th}}=3/4$ for any register size, while local dephasing of $|+\rangle^{\otimes M}$ has a threshold of $p_{\mathrm{th}}=1/2$. Local Clifford twirling can be used to convert dephasing into a depolarization channel and restore the higher threshold.

quant-pharXiv:2603.11568•Mar 13, 2026•0 citations

Scaling-optimal purification of noisy qubit unitary channels

Ryotaro Niwa, Satoshi Yoshida, Koki Ono +5 more

We consider the problem of purifying noisy qubit unitary channels. Given the ability to apply an unknown qubit unitary channel followed by depolarizing noise, we aim to construct a superchannel that purifies the noisy unitary back to the original unknown unitary. We first provide numerical evidence that sequential strategies can strictly outperform parallel strategies when the number of channel uses is finite, highlighting the fundamental distinction from state purification. We then provide a concrete $\mathrm{U}(2)$-covariant parallel protocol based on a novel entanglement-assisted quantum error-correcting code that suppresses the first-order noise strength as $O(1/n)$ with $n$ channel uses and show this scaling is asymptotically optimal in the low-noise regime, even when sequential strategies are allowed.

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Yu Guo, Yuqi Chen, Jinming Guo +3 more

Abstract The quantum switch is known to remove noise probabilistically, but whether this constitutes an advantage over causally ordered processing has remained contentious: postselection, weak classical benchmarks, and task-dependent comparisons have all drawn criticism. Here we give an exact analysis of the question for a qubit under Pauli noise. For any two qubit Pauli channels, measuring the switch’s control qubit implements a commutation syndrome : it sorts each noise realization by whether the two Kraus errors commuted, syndrome information that an unencoded qubit does not otherwise possess, and the optimal measure-and-feed-forward protocol has a closed form. The analytical results have two levels of generality. For the exactly solvable pair, an isotropic depolarizing channel and a flip channel along an arbitrary axis, we prove a complete ceiling theorem: every definite-order protocol using each channel once, with arbitrary memoryless intermediate operations and receiver-side post-processing, deterministic or heralded, has entanglement fidelity at most ma x P c P , and the switch exceeds this ceiling deterministically , by p q / 3 over most of parameter space. For arbitrary qubit Pauli pairs, the ceiling is proven against arbitrary receiver-side post-processing, and the gain is strictly positive if and only if the two syndrome branches call for different Pauli corrections, confining the exceptions to error-dominated noise. Side systems are covered whenever they are non-signaling about the message. Beyond these theorems the evidence is numerical: strict gain in 99.8 % of 5000 random Pauli pairs, and no case, within a 200 -pair sample, in which an optimized midpoint echo overturns the gain. The state-resolved gain obeys an exact geometric law, Δ F = 2 p q / 3 | r ^ ⋅ n ^ | 2 : alignment is the condition under which the degenerate two-error syndrome becomes perfectly correctable. The separation survives control dephasing down to a threshold visibility that falls with noise strength, and at strong noise it outperforms coherent control of the two orderings even with twice the channel uses.

Physica Scripta•Sep 7, 2026•0 citations

Real Classical Shadows with Noise

Atharva Hingane, Dax Enshan Koh

The real classical shadows protocol of West et al. replaces the unitary (Clifford) ensemble of the Huang--Kueng--Preskill scheme by the orthogonal (real Clifford) ensemble, and for symmetric observables achieves strictly smaller estimator variances: a factor approaching two for global evolution and an exponential factor $(3/2)^k$ for $k$-local real Pauli observables. Real hardware, however, never implements the ideal evolution. Building on the noisy classical shadows framework of Koh and Grewal, we give a complete theory of the real classical shadows protocol in the presence of a known completely positive trace-preserving noise channel acting after the orthogonal evolution. We derive the noisy global and local orthogonal shadow channels from first principles using the Weingarten calculus of the orthogonal group, prove that each is a depolarizing channel acting on the symmetric (respectively locally symmetric) component of its input, and derive from it the exact single-shot variance in closed form, together with the associated shadow seminorm, two-sided bounds on it, and the resulting sample-complexity guarantees. We prove that the noiseless sample-complexity advantages survive intact under noise. Because the variances are exact rather than bounded, the ratio is controlled by a single dimensionless parameter, which gives a closed-form criterion for when the factor of two is attainable: both the second-moment and the variance ratio reach it exactly when the observable's norm profile grows, and the noise enters that limit only through a factor lying within $2/(d+2)$ of two, so the advantage is uniform in the noise. For rank-one targets it is provably unattainable, saturating strictly below two. The local real-Pauli advantage remains $(3/2)^k$. We treat complex measurement bases through a reality parameter and a transposed-noise scalar, recovering unitary shadows in the appropriate limit.

quant-pharXiv:2608.18935•Aug 20, 2026•0 citations
The γcγ_cγc​-Peak: Covariant Recovery on Four Organic Qubit Platforms
The Petz recovery map (1986) provably reverses a noisy quantum channel on a reference state, but its algorithmic relevance to real, dissipation-dominated platforms has remained unclear. Using the open-source \texttt{organic-qc-bench} simulation package, we benchmark a Petz-style covariant-purification quantum error correction (CQEC) protocol across four engineered organic qubit platforms operated \emph{without any magnetic field}: a flavin-nitroxide radical-pair reservoir (P1); perchlorotriphenylmethyl radicals in a covalent organic framework (P2); the SVILC qubit [Wakaura2017] on κκκ-(BEDT-TTF)2_22​Cu[N(CN)2_22​]Br (P3, conditional on SVILC confirmation); and a Su-Schrieffer-Heeger soliton on \emph{trans}-polyacetylene (P4).
Across five quantum algorithms (QKAN, qDRIFT, control-free QPE, Shor-Regev, Bernstein-Vazirani) and two ML tasks, CQEC gains are significant (p ⁣< ⁣10−5p\!<\!10^{-5}p<10−5; Wilcoxon, Bonferroni α ⁣= ⁣0.05/44α\!=\!0.05/44α=0.05/44) for all sixteen path×\times×algorithm pairs. The central finding is the \emph{γcγ_cγc​-peak}: the fidelity gain ΔFΔFΔF is maximised \emph{at} the entanglement-breaking threshold γcγ_cγc​, with ΔFmax ⁣= ⁣+0.303ΔF_{\rm max}\!=\!+0.303ΔFmax​=+0.303 at d ⁣= ⁣64d\!=\!64d=64 and a linear log⁡2d\log_2 dlog2​d scaling over d=2d=2d=2-646464 -- algorithmically confirming the prediction [Wakaura2026LQBH] that Petz recovery preserves coherence beyond this threshold. Bernstein-Vazirani also yields a 7.67.67.6-31×31\times31× provable quantum advantage at n ⁣= ⁣3n\!=\!3n=3-555, diarylethene-photoswitch CZ fidelities reach FCZ ⁣≥ ⁣0.987F_{CZ}\!\ge\!0.987FCZ​≥0.987 for P2-P4, and projected manufacturing costs are 10-40×\times× lower with 10-200×\times× less operating power than superconducting platforms. The γcγ_cγc​-peak establishes Petz-style recovery as a practically relevant primitive at the dissipation-coherence boundary and identifies PTM-COF (P2) as the highest-priority experimental target.
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).
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 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 ρNρ^NρN. 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)O(M\ell)O(Mℓ) coherent data qubits for an MMM-qubit register and N=2ℓN=2^\ellN=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=3/4p_{\mathrm{th}}=3/4pth​=3/4 for any register size, while local dephasing of ∣+⟩⊗M|+\rangle^{\otimes M}∣+⟩⊗M has a threshold of pth=1/2p_{\mathrm{th}}=1/2pth​=1/2. Local Clifford twirling can be used to convert dephasing into a depolarization channel and restore the higher threshold.
We consider the problem of purifying noisy qubit unitary channels. Given the ability to apply an unknown qubit unitary channel followed by depolarizing noise, we aim to construct a superchannel that purifies the noisy unitary back to the original unknown unitary. We first provide numerical evidence that sequential strategies can strictly outperform parallel strategies when the number of channel uses is finite, highlighting the fundamental distinction from state purification. We then provide a concrete U(2)\mathrm{U}(2)U(2)-covariant parallel protocol based on a novel entanglement-assisted quantum error-correcting code that suppresses the first-order noise strength as O(1/n)O(1/n)O(1/n) with nnn channel uses and show this scaling is asymptotically optimal in the low-noise regime, even when sequential strategies are allowed.
Abstract The quantum switch is known to remove noise probabilistically, but whether this constitutes an advantage over causally ordered processing has remained contentious: postselection, weak classical benchmarks, and task-dependent comparisons have all drawn criticism. Here we give an exact analysis of the question for a qubit under Pauli noise. For any two qubit Pauli channels, measuring the switch’s control qubit implements a commutation syndrome : it sorts each noise realization by whether the two Kraus errors commuted, syndrome information that an unencoded qubit does not otherwise possess, and the optimal measure-and-feed-forward protocol has a closed form. The analytical results have two levels of generality. For the exactly solvable pair, an isotropic depolarizing channel and a flip channel along an arbitrary axis, we prove a complete ceiling theorem: every definite-order protocol using each channel once, with arbitrary memoryless intermediate operations and receiver-side post-processing, deterministic or heralded, has entanglement fidelity at most ma x P c P , and the switch exceeds this ceiling deterministically , by p q / 3 over most of parameter space. For arbitrary qubit Pauli pairs, the ceiling is proven against arbitrary receiver-side post-processing, and the gain is strictly positive if and only if the two syndrome branches call for different Pauli corrections, confining the exceptions to error-dominated noise. Side systems are covered whenever they are non-signaling about the message. Beyond these theorems the evidence is numerical: strict gain in 99.8 % of 5000 random Pauli pairs, and no case, within a 200 -pair sample, in which an optimized midpoint echo overturns the gain. The state-resolved gain obeys an exact geometric law, Δ F = 2 p q / 3 | r ^ ⋅ n ^ | 2 : alignment is the condition under which the degenerate two-error syndrome becomes perfectly correctable. The separation survives control dephasing down to a threshold visibility that falls with noise strength, and at strong noise it outperforms coherent control of the two orderings even with twice the channel uses.
The real classical shadows protocol of West et al. replaces the unitary (Clifford) ensemble of the Huang--Kueng--Preskill scheme by the orthogonal (real Clifford) ensemble, and for symmetric observables achieves strictly smaller estimator variances: a factor approaching two for global evolution and an exponential factor (3/2)k(3/2)^k(3/2)k for kkk-local real Pauli observables. Real hardware, however, never implements the ideal evolution. Building on the noisy classical shadows framework of Koh and Grewal, we give a complete theory of the real classical shadows protocol in the presence of a known completely positive trace-preserving noise channel acting after the orthogonal evolution. We derive the noisy global and local orthogonal shadow channels from first principles using the Weingarten calculus of the orthogonal group, prove that each is a depolarizing channel acting on the symmetric (respectively locally symmetric) component of its input, and derive from it the exact single-shot variance in closed form, together with the associated shadow seminorm, two-sided bounds on it, and the resulting sample-complexity guarantees. We prove that the noiseless sample-complexity advantages survive intact under noise. Because the variances are exact rather than bounded, the ratio is controlled by a single dimensionless parameter, which gives a closed-form criterion for when the factor of two is attainable: both the second-moment and the variance ratio reach it exactly when the observable's norm profile grows, and the noise enters that limit only through a factor lying within 2/(d+2)2/(d+2)2/(d+2) of two, so the advantage is uniform in the noise. For rank-one targets it is provably unattainable, saturating strictly below two. The local real-Pauli advantage remains (3/2)k(3/2)^k(3/2)k. We treat complex measurement bases through a reality parameter and a transposed-noise scalar, recovering unitary shadows in the appropriate limit.