Calibrating a quantum processor means estimating error parameters, and estimation theory usually assumes a parameter hard to estimate is faint: its signal is weak but present, so more repetitions or a richer model will recover it. This assumption fails for a leading hardware fault. A coherent over-rotation is a small systematic gate miscalibration. Measured through the cheapest data a device returns--one fixed-basis histogram--it is not faint but absent: to first order it leaves the distribution unchanged, indistinguishable from a compensating stochastic error, exactly as two numbers cannot be separated from their sum. For commuting single- and two-qubit transverse over-rotations, with known support on the canonical input, the histogram's Fisher information is singular along the fault's direction at zero angle, its Cramer-Rao bound is infinite, and no finite-variance, locally unbiased estimator recovers it. At a generic nonzero angle the degeneracy partly lifts; beyond four qubits it clears entirely, leaving conditioning, not absence, as the obstruction. The cure is a richer measurement, not a richer model: a fixed, logarithmically small set of extra settings makes every such fault visible. Visibility alone is not enough. The sampling cost is set by conditioning, not coverage, through a floor whose complete-family closed form is exponentially small in the qubit count. We prove the impossibility and cure, confirm both in exact simulation, show conditioning predicts recovery error across hundreds of designs, and observe a 3-5x bias gap on IBM Heron hardware as a consistency check. Non-commuting faults and unknown support remain open.
Calibrating a quantum processor means estimating error parameters, and estimation theory usually assumes a parameter hard to estimate is faint: its signal is weak but present, so more repetitions or a richer model will recover it. This assumption fails for a leading hardware fault. A coherent over-rotation is a small systematic gate miscalibration. Measured through the cheapest data a device returns--one fixed-basis histogram--it is not faint but absent: to first order it leaves the distribution unchanged, indistinguishable from a compensating stochastic error, exactly as two numbers cannot be separated from their sum. For commuting single- and two-qubit transverse over-rotations, with known support on the canonical input, the histogram's Fisher information is singular along the fault's direction at zero angle, its Cramer-Rao bound is infinite, and no finite-variance, locally unbiased estimator recovers it. At a generic nonzero angle the degeneracy partly lifts; beyond four qubits it clears entirely, leaving conditioning, not absence, as the obstruction. The cure is a richer measurement, not a richer model: a fixed, logarithmically small set of extra settings makes every such fault visible. Visibility alone is not enough. The sampling cost is set by conditioning, not coverage, through a floor whose complete-family closed form is exponentially small in the qubit count. We prove the impossibility and cure, confirm both in exact simulation, show conditioning predicts recovery error across hundreds of designs, and observe a 3-5x bias gap on IBM Heron hardware as a consistency check. Non-commuting faults and unknown support remain open.
Coherent noise calls for a reusable physical model, but Pauli twirling erases it and full process tomography scales exponentially. We turn measurement-free quantum error correction into a probe: bit-flip and phase-flip codes polarize few-qubit coherent noise into single-logical-qubit channels, and the resulting fingerprint identifies an 11-parameter three-qubit family. An exact rational-arithmetic rank certificate, evaluated at a rational point in the interior of the parameter domain, proves local injectivity on a dense open subset of that domain. The map is nevertheless not globally injective: a compensating rotation combined with a site exchange yields exact collisions, which we classify and then remove by adding one two-qubit marginal channel. The site exchange is the only nontrivial permutation symmetry of the n-qubit chain generator at any n, so the classification is structural rather than a three-qubit accident. A Cramér--Rao analysis, checked at the level of individual measurement settings, reaches 0.01~rad per parameter at a representative point, and decoder priors rebuilt from such an estimate retain most of their correlation gain. Propagating the generator into a distance-three surface code gives a 14.8-fold encoding gain when the generator is known exactly, and 1.21-fold of the correlation gain survives when the prior is rebuilt from an estimate at the stated precision, closing the loop from characterization to decoding.
One should not assume that reconstructing an electronic observable with a smaller set of measurement settings will cut the number of repetitions in proportion. Each setting is simply a basis change and qubit readout, but the savings are not guaranteed to be commensurate. Take two electrons of opposite spin in two spatial orbitals: the orbital degree of freedom of each electron acts as a logical qubit and rotations are independent in the spin sectors. From the observable's one- and two-qubit coefficients we derive an exact floor for the setting count and a decomposition to meet it in no more than three settings. With fewer settings, the minimum error in the two-qubit correlation coefficients is set by the singular values left out. A case in point is an exchange with equal nonzero couplings on the three Pauli axes. This demands three orbital settings yet only one joint measurement. We offer a Clifford circuit for four physical qubits to effect the necessary mapping of Pauli products under conjugation. In the case of a one-parameter family of logical singlet-triplet mixtures, the orbital-to-Clifford shot cost ratio is three at maximum mixing and trends toward 3/2 near the logical singlet. Where spin sectors may be mixed, isotropic exchange needs one orbital setting. For spin-conserving Hamiltonians with N spatial orbitals encoded via Jordan-Wigner, the X-rank, the binary rank of the X parts of its strings, of any commuting Pauli string family is no more than 2(N−1). Searches with the Robust Adaptive Nature-inspired Global Explorer (RANGE) reach this limit for CH4, with the rank of the Hamiltonian support vouching for the maximality of the search values. And for all nonidentity Pauli strings on n qubits, semidefinite witnesses from the Lovász theta number give a lower bound on the Clifford setting count that agrees with the integer cover and counting bounds at 2n+1.
Abstract Quantum and hybrid quantum-classical workflows repeatedly need to predict how ideal circuit distributions appear as finite-shot noisy hardware histograms, but refitting high-dimensional noise simulators at each calibration point can be costly. We cast this task as finite-shot distribution learning for structured quantum outputs: given ideal distributions and noisy counts, learn a low-dimensional Markov surrogate and evaluate held-out distributional risk. Existing fast closed-form kernels lack identifiability guarantees, while richer physics-motivated simulators fit approximately 20 BO-tuned noise parameters without parameter-identifiability guarantees. We instantiate this middle ground with SA-IBF, a two-parameter closed-form kernel mixing independent bit-flip noise with a Hamming-sector-preserving component. For the support-face regimes induced by symmetry-preserving circuits, we show a support-face bridge and a rank-conditioned local guarantee at fitted operating points; open-interior genericity establishes non-degeneracy of the full-simplex model. We also show a small-q Fisher upper bound in which the Fisher information for alpha conditional on q scales on the order of N times q squared for N shots and per-qubit flip rate q, certified for 2 to 8 qubits, yielding Cramer-Rao divergence as q approaches zero; analytic-Jacobian diagnostics verify the fitted-point rank premise at all seven representative operating points and all 230 panel fits. On an 890-execution IBM Heron hardware dataset spanning four backends, SA-IBF gives a diagnosable accuracy-cost Pareto point: in matched fitted-physics comparisons, it meets the sector-aware 21-parameter Ji-BO Hellinger accuracy budget on three of four molecular qubit settings while reducing refit cost by 231-447 times under matched implementations. The remaining setting flags a uniform-flip-rate boundary where the tested Ji-BO comparator's heterogeneous parameters lower risk. Supporting calibration evidence shows lower held-out Hellinger distance than plain bit-flip noise on all seven primary configurations and throughout a 400-execution cross-backend molecular panel. These results make SA-IBF a lightweight, identifiability-guided learned distribution-level surrogate for repeated quantum noise simulation in symmetry-structured quantum and hybrid quantum-classical workloads, including symmetry-preserving quantum-machine-learning settings.
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.
Recovery from a coherent rotation with an unobserved angle depends on both the encoding and what calibration reveals about the noise. We determine their combined limit for a known finite-dimensional generator and one final recovery. Two polynomial projections organize the answer. A physical Krylov chain separates correctable error amplitudes from their first state-dependent distortion; projection onto lower-angle powers identifies the part that a coherent decoder can cancel. Under explicit rank and moment conditions, their combination gives the exact leading optimal entanglement infidelity at weak noise and a recovery attaining it for any finite logical dimension. The optimal worst-input error lies between one and d times that infidelity at leading order, for logical dimension d. When the angle distribution is unknown, we optimize one decoder for every distribution compatible with the calibration. When spectral and calibration frequencies are integer multiples of a common frequency, moment positivity turns this into an exact finite optimization. Small measured means can still allow rare finite rotations. If the sole calibration datum is βr=E[sin2r(θ/2)], with θ the wrapped rotation phase at that common frequency, the optimal worst-prior infidelity is O(βr) exactly when generator powers through degree r−1 are correctable. When the first uncorrectable power has degree m<r, the optimal loss scales as βrm/r, with a sharp coefficient fixed by a universal moment problem. In a loss-free four-photon memory, population measurements at two calibration durations retain the fully informed quartic error coefficient for a fixed symmetric narrow-noise family with positive variance and finite fourth moment; a single duration permits loss quadratic in the noise width.
The textbook CHSH measurement configuration {a, a', b, b'} = {0, pi/2, pi/4, -pi/4} with Form I evaluates to S = 0 on all four Bell states in the spin-1/2 convention used by gate-based quantum processors. This "failure mode", documented as a state-formula-angle mismatch in a companion pedagogical note (doi:10.5281/zenodo.20146219), is shown here to have a clean algebraic origin: the corresponding Bell operator reduces to sqrt(2) * (sigma_z (x) sigma_x + sigma_x (x) sigma_z), whose spectrum is {-2*sqrt(2), 0, 0, +2*sqrt(2)}, with the kernel spanned by {|Phi+>, |Psi->} and the non-kernel pair {|Phi->, |Psi+>} mapped to each other. Main results. The note establishes three properties of the failure-mode configuration: (1) The CHSH value S is preserved at exactly zero under arbitrary symmetric local Pauli channels (depolarizing, dephasing, bit-flip) and under local Rx and Rz rotations on either qubit, for every Bell state. Verified numerically with 1000 random Pauli channels per state and across the full range of standard noise channels. (2) Local Ry(theta) rotations break S = 0 only for the pair {|Phi->, |Psi+>}, with closed form S = 2*sqrt(2)*|sin(theta)|, reaching the Tsirelson bound at theta = pi/2. (3) The asymmetry has a direct algebraic origin in the correlation tensor: the kernel pair has equal XX and ZZ moments (t_1 = t_3), while the broken pair has opposite signs (t_1 = -t_3). This connects to and gives algebraic meaning to the Bell-state taxonomy of the companion paper. Diagnostic interpretation. The same CHSH measurement that fails in textbook form becomes a no-extra-resource diagnostic for coherent Ry-type errors restricted to a specific Bell-state family. If S > 0 in this configuration, then (i) the state is in the {|Phi->, |Psi+>} family, and (ii) a coherent Ry error of magnitude theta_err = arcsin(S/(2*sqrt(2))) is present. Incoherent Pauli noise alone cannot produce this signal. Calibration of novelty. The note explicitly searches the self-testing literature (Supic-Bowles 2020 review) and the CHSH-under-noise literature (Kofman 2012, Kofman-Korotkov 2008) and reports that this specific asymmetric breaking pattern was not located in those communities. Priority is explicitly disclaimed with respect to gate-set-tomography and randomized-benchmarking literatures, which were not exhaustively surveyed. The result is presented as either a novel observation or, if previously catalogued elsewhere, an independent algebraic rederivation with closed-form expression in terms of the correlation tensor. Reproducibility. The repository contains the LaTeX source, the bibliography, both figures, and seven Python scripts that reproduce every numerical claim from scratch using only NumPy and Matplotlib. Each script is self-contained and runs in seconds. Companion paper. This work extends the pedagogical note "State-dependent CHSH inequality on gate-based quantum processors" (Perez-Eugenio 2026, doi:10.5281/zenodo.20146219), which documents the original state-formula-angle compatibility issue and the corrected configurations. The present note can be read independently.