Optimal-recovery ambiguity under truncated amplitude-damping models investigates a structural limitation of noise-adapted quantum error correction: a recovery optimized for a truncated noise model need not be uniquely determined with respect to its performance under the full physical channel.The work develops a geometric identifiability criterion for optimal quantum recoveries formulated as completely positive trace-preserving optimization problems in Choi form. It characterizes when the full-channel objective is constant across an optimal face of the truncated-design problem and when different equally optimal recoveries can instead produce different full-channel fidelities.The main application is the nine-qubit Shor code under canonical product amplitude-damping noise, with recovery models truncated by Kraus weight. For syndrome-structured recoveries, the study certifies nonzero full-channel fidelity intervals for truncation weights 1 through 5 and establishes their leading scaling with damping strength.For truncation weights 1 and 2, pairwise orthogonality of the retained encoded error ranges yields an exact syndrome-sufficiency result. The structured and unrestricted truncated-design optima coincide for all damping strengths between 0 and 1, showing that the observed recovery ambiguity is not caused by imposing the syndrome-block architecture in these cases.For truncation weight 3, the retained error ranges are no longer pairwise orthogonal. The exact unrestricted optimal-face diameter remains open, but dense unrestricted Choi witnesses at damping strength 0.05 reproduce 99.4202 percent of the corresponding structured full-channel separation while remaining close to the truncated-design optimum.The accompanying QER-FaceScan v2.7 reproducibility archive contains the manuscript source, numerical results, verification scripts, solver audits, structured-width certificates, unrestricted recovery witnesses, dual data, Petz and exact-support checks, and SHA-256 integrity manifests. Large Hermitian Choi matrices are stored using an exact lossless packed representation and are reconstructed by the supplied verification tools without reduced precision or lossy approximation.Related four-qubit calibration results and supporting code are available in the companion ALM-QEC archive: DOI 10.6084/m9.figshare.32080845.
Optimal-recovery ambiguity under truncated amplitude-damping models investigates a structural limitation of noise-adapted quantum error correction: a recovery optimized for a truncated noise model need not be uniquely determined with respect to its performance under the full physical channel.The work develops a geometric identifiability criterion for optimal quantum recoveries formulated as completely positive trace-preserving optimization problems in Choi form. It characterizes when the full-channel objective is constant across an optimal face of the truncated-design problem and when different equally optimal recoveries can instead produce different full-channel fidelities.The main application is the nine-qubit Shor code under canonical product amplitude-damping noise, with recovery models truncated by Kraus weight. For syndrome-structured recoveries, the study certifies nonzero full-channel fidelity intervals for truncation weights 1 through 5 and establishes their leading scaling with damping strength.For truncation weights 1 and 2, pairwise orthogonality of the retained encoded error ranges yields an exact syndrome-sufficiency result. The structured and unrestricted truncated-design optima coincide for all damping strengths between 0 and 1, showing that the observed recovery ambiguity is not caused by imposing the syndrome-block architecture in these cases.For truncation weight 3, the retained error ranges are no longer pairwise orthogonal. The exact unrestricted optimal-face diameter remains open, but dense unrestricted Choi witnesses at damping strength 0.05 reproduce 99.4202 percent of the corresponding structured full-channel separation while remaining close to the truncated-design optimum.The accompanying QER-FaceScan v2.7 reproducibility archive contains the manuscript source, numerical results, verification scripts, solver audits, structured-width certificates, unrestricted recovery witnesses, dual data, Petz and exact-support checks, and SHA-256 integrity manifests. Large Hermitian Choi matrices are stored using an exact lossless packed representation and are reconstructed by the supplied verification tools without reduced precision or lossy approximation.Related four-qubit calibration results and supporting code are available in the companion ALM-QEC archive: DOI 10.6084/m9.figshare.32080845.
Quantum error correction (QEC) is an essential concept for any quantum information processing device. Typically, QEC is designed with minimal assumptions about the noise process; this generic assumption exacts a high cost in efficiency and performance. In physical systems, errors are not likely to be arbitrary; rather we will have reasonable models for the structure of quantum decoherence. We may choose quantum error correcting codes and recovery operations that specifically target the most likely errors. We present a convex optimization method to determine the optimal (in terms of average entanglement fidelity) recovery operation for a given channel, encoding, and information source. This is solvable via a semidefinite program (SDP). We present computational algorithms to generate near-optimal recovery operations structured to begin with a projective syndrome measurement. These structured operations are more computationally scalable than the SDP required for computing the optimal; we can thus numerically analyze longer codes. Using Lagrange duality, we bound the performance of the structured recovery operations and show that they are nearly optimal in many relevant cases. We present two classes of channel-adapted quantum error correcting codes specifically designed for the amplitude damping channel. These have significantly higher rates with shorter block lengths than corresponding generic quantum error correcting codes. Both classes are stabilizer codes, and have good fidelity performance with stabilizer recovery operations. The encoding, syndrome measurement, and syndrome recovery operations can all be implemented with Clifford group operations.
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.
Quantum error correction (QEC) plays a critical role in preventing information loss in quantum systems and provides a framework for reliable quantum computation. Identifying quantum codes with nice code parameters for physically motivated noise models remains an interesting challenge. While past work has primarily focused on qubit codes, here we identify a [4,1] qudit error correcting code tailored to protect against amplitude-damping noise. We show that this four-qudit code satisfies the error correction conditions for all single-qudit and a few two-qudit damping errors up to the leading order in the damping parameter $\ensuremath{\gamma}$. We devise a protocol to extract syndromes that unambiguously identify this set of errors, leading to a noise-adapted recovery scheme that achieves a fidelity loss of $O({\ensuremath{\gamma}}^{2})$. For the d=2 case, our QEC scheme is identical to the known example of the four-qubit code and the associated syndrome-based recovery. We also assess the performance of this code using the Petz recovery map and note some interesting deviations from the qubit case. Finally, we generalize this construction to a family of [2M+2,M] qudit codes that can approximately correct all the single-qudit and a few two-qudit amplitude-damping errors.
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.
Quantum error correction (QEC) is essential for reliable quantum information processing. Targeting a particular error channel, both the encoding and the recovery channel can be optimized through a biconvex optimization to give a high-performance, noise-adapted QEC scheme. We solve the biconvex optimization by the technique of alternating semi-definite programming and identify a new four-qubit code for amplitude damping channel, one major noise in superconducting circuits and a good model for spontaneous emission and energy dissipation. We also construct analytical encoding and recovery channels that are close to the numerically optimized ones. We show that the new code notably outperforms the Leung-Nielsen-Chuang-Yamamoto four-qubit code in terms of the entanglement fidelity over an amplitude damping channel.
## Overview This paper studies when a reduction of a finite quantum error-history model preserves the local direction of quantum-code optimization, and when it reverses that direction. For finite coactive spectral branches, the paper first proves a second-order local truncation-safety certificate. A strict optimizer of a truncated model persists when the full-versus-truncated effective-jet discrepancy is smaller than the coercivity margin of the truncated objective. Opposite leading jets provide a certificate of local optimizer reversal. The paper then identifies an exact boundary between genuine temporal structure and removable path redundancy. For a finite projective unitary error group, any open-loop path operator has the form Wγ=eiθγUc(γ), where c(γ) is the final group product. Consequently, an arbitrary temporally correlated path law is isometrically equivalent, for complete complementary leakage and for every code, to the single-shot distribution of the final projective product: Lpath(V)=Lπ(V). For Pauli trajectories, this remains true even when ordered errors do not commute, because different orderings can differ only by projective phases after contraction to the final channel. Thus open-loop final-channel leakage cannot identify temporal organization beyond the final projective Pauli-product distribution. Intermediate measurements, controls, syndrome extraction, or adaptive recovery are required for genuinely intervention-resolved multi-time code geometry. ## Distribution-robust optimizer reversal For an arbitrary three-location bit-flip trajectory law, let π denote the induced final-parity distribution on F23, let c=(1,1,1), and define Λπ:=∑t∈F23πtπt⊕c. For the three-qubit repetition code E=span{∣000⟩,∣111⟩}, the complete complementary leakage is exactly Lπ(E)=Λπ. For every fixed full-support final-parity law, the intrinsic Fourier-syndrome graph coordinate w satisfies Lπ(V(w))=Λπ−2Λπ∣w∣2+o(∣w∣2). Hence the repetition code is a strict local maximum throughout this two-real-dimensional graph chart for every full-support final-parity distribution, independently of temporal correlations, stationarity, roundwise independence, identical error rates, or cyclic spatial symmetry. By contrast, a conditioned zero-and-one-fault truncation with positive coarse single-location weights has L≤1(E)=0 and strictly positive leakage at every nonzero sufficiently nearby graph code. The same reference code is therefore a strict local minimum for the truncated model and a strict local maximum for the complete process. The central conclusion is: > Exact aggregation of paths that implement the same projective physical operator preserves the entire code-optimization landscape, whereas deletion of higher-fault history fibers can reverse the local optimizer geometry. ## Scope The results concern random-unitary, open-loop finite-history models and complete complementary-channel leakage. The optimizer reversal is proved in a specified Fourier-syndrome graph chart around the three-qubit repetition code. The paper does not claim: - a full-Grassmannian classification for the three-qubit model;- a decoder or recovery-circuit construction;- a logical-error threshold;- a circuit-level hardware noise model;- collapse of an intervention-resolved process tensor;- equivalence in the presence of intermediate syndrome measurements, controls, or adaptive recovery. These intervention-resolved settings form the natural next boundary of the theory. ## Relation to the preceding research program This paper is a concluding bridge theorem paper connected to a six-paper exact finite-history QEC and spectral-minimax research line. ### 1. Compression-geometric foundation **Sharp Complementarity, Complete Attainable Region, and Metric Rigidity of Rank-Two Quantum Codes under Collective and Differential Dephasing** Version v0.5r2b DOI: 10.5281/zenodo.21523950 This paper determined the complete attainable compression-defect region for rank-two codes under collective and differential two-qubit dephasing, classified the equality geometry, and established sharp metric rigidity. ### 2. Balanced finite-history minimax **Environment-Record Leakage for Finite Quantum Error Histories: Balanced Global Minimax, Complete Minimizer Geometry, and Distributional Stability** Version v0.5r2 DOI: 10.5281/zenodo.21560397 This paper introduced the complete environment-record leakage objective for finite error histories, solved the balanced two-step minimax problem globally over Gr(2,4), classified all balanced minimizers, and established distributional stability. ### 3. Bias-induced geometric selection **Bias-Induced Non-Isoclinic Optimality in Finite Quantum Error Histories: Exact Separation, Thresholdless Bifurcation, and Near-Balanced All-Code Minimax** Version v0.3r2 DOI: 10.5281/zenodo.21598820 This paper proved exact non-isoclinic separation, an open non-isoclinic parameter region, thresholdless loss of isoclinic optimality under arbitrarily small bias, and the leading full-Grassmannian near-balanced minimax selector. ### 4. Fourth-order overlap selection **Flat-Interior Persistence and Singular Overlap Selection in Near-Balanced Quantum Error Histories: Fourth-Order Transition Curves and Full-Grassmannian Minimax Closure** Version v0.1r3 DOI: 10.5281/zenodo.21622029 This sequel resolved the higher-order overlap degeneracy, established flat-interior persistence, derived singular boundary-layer transition curves, and closed the near-balanced full-Grassmannian minimax expansion through fourth order. ### 5. Abstract reflected spectral-jet framework **Reflected Spectral-Jet Realization for Degenerate Minimax Problems: Hard-Normal Selection, Flat Tangential Persistence, and Coactive Boundary Transitions** Version v0.2r1 DOI: 10.5281/zenodo.21636022 This paper extracted a reusable abstract framework for hard-normal isolation, soft tangential persistence, reflected active branches, and coactive boundary transitions in degenerate spectral minimax problems. ### 6. Finite-orbit generalization and first QEC truncation reversal **Beyond Reflected Pairs: Finite-Orbit Spectral Jets, Quadratic Isolation, and Multi-Branch Coactivity in Degenerate Minimax Problems** Version v0.4r1 DOI: 10.5281/zenodo.21644364 This paper generalized the reflected-pair theory to finite transitive orbits, developed multi-branch coactivity and representation-theoretic jet obstructions, and exhibited an exact three-location QEC realization in which restoring omitted multi-fault histories reverses the local optimizer geometry. ### 7. Present bridge theorem The present paper determines which open-loop temporal path information is exactly removable and which history reductions remain unsafe. It extends the earlier single-process reversal from independent or cyclic models to arbitrary correlated projective-unitary paths and arbitrary full-support final-parity laws. The resulting research progression is compression geometry⟶balanced global minimax⟶bias-induced selection⟶fourth-order closure⟶reflected spectral jets⟶finite-orbit coactivity⟶projective path collapse and distribution-robust reversal. ## Reproducibility package The accompanying bundle contains: - the manuscript PDF;- the LaTeX source;- an exact symbolic verification script;- a machine-readable verification report;- a README describing the proof-artifact contract;- a SHA-256 manifest;- version and package-audit records. The verifier performs 72 exact symbolic checks, including projective Pauli phase identities, phase-covariant path compression, arbitrary-parity polar factors, translation orthogonality, Fourier-syndrome graph compressions, and coefficient bookkeeping for the distribution-robust local expansion. The verifier deliberately rejects optimized Python mode. PASS decisions do not rely on floating-point tolerances. Compactness, maximizing-direction localization, trace-norm differentiability, and uniform asymptotic assembly remain analytic parts of the manuscript proof rather than delegated computational claims.