Calibrating a quantum processor means estimating gate-error parameters from data, and a hard-to-estimate parameter is usually assumed to leave a weak signature more repetitions will resolve. Coherent over-rotations break that premise. For commuting single- and two-qubit transverse over-rotations with known support on a computational-basis input, the passive histogram is exactly invariant under a sign group acting on the coherent angles, of order $2^{n+1}$ for the complete family with $n\ge2$, so no estimator resolves the signs uniformly at any sample size. A calibration correction applied with the wrong sign doubles the rotation error it should remove. At zero angle the obstruction turns continuous, with a Fisher information singular along every coherent direction and an infinite Cramér-Rao bound. At generic angles the continuous defect clears for $5\le n\le8$, the range we compute, while the sign degeneracy persists. Covering the support removes both, and a design resolves the signed angles uniformly exactly when it covers. On $|η_R|<π/2$, $0\le p_R<1/2$ coverage is constructive, returning each angle in closed form from a ratio of two measured expectations without knowing the rates. A twirl-free code of $\lceil\log_2(n+1)\rceil$ added product-Pauli settings covers, and for the complete family none uses fewer. Conditioning then sets the Cramér-Rao cost, and we give its floor explicitly. Exact computation matches the theory, the passive fit stays at chance on the signs at every budget while the closed-form inverse recovers them, and angle magnitudes on two IBM Heron processors follow the conditioning ordering.
Calibrating a quantum processor means estimating gate-error parameters from data, and a hard-to-estimate parameter is usually assumed to leave a weak signature more repetitions will resolve. Coherent over-rotations break that premise. For commuting single- and two-qubit transverse over-rotations with known support on a computational-basis input, the passive histogram is exactly invariant under a sign group acting on the coherent angles, of order 2n+1 for the complete family with n≥2, so no estimator resolves the signs uniformly at any sample size. A calibration correction applied with the wrong sign doubles the rotation error it should remove. At zero angle the obstruction turns continuous, with a Fisher information singular along every coherent direction and an infinite Cramér-Rao bound. At generic angles the continuous defect clears for 5≤n≤8, the range we compute, while the sign degeneracy persists. Covering the support removes both, and a design resolves the signed angles uniformly exactly when it covers. On ∣ηR∣<π/2, 0≤pR<1/2 coverage is constructive, returning each angle in closed form from a ratio of two measured expectations without knowing the rates. A twirl-free code of ⌈log2(n+1)⌉ added product-Pauli settings covers, and for the complete family none uses fewer. Conditioning then sets the Cramér-Rao cost, and we give its floor explicitly. Exact computation matches the theory, the passive fit stays at chance on the signs at every budget while the closed-form inverse recovers them, and angle magnitudes on two IBM Heron processors follow the conditioning ordering.
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.
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.
Here's what this paper means for actual humans: WHAT THIS IS ABOUT. When you build or use a quantum device, everything rests on knowing what its parts are: which degrees of freedom are "qubit 1," which are "qubit 2." The companion paper ("Forged Frames") showed that simple mathematical criteria cannot certify this; convincing fakes exist. This paper answers the question that result leaves standing: what is the MINIMUM experimental information that genuinely pins down a system's subsystem structure, and when can that information silently fail? WHAT WE FOUND, IN PLAIN TERMS. Four things. (1) EXACT LAWS INSTEAD OF MEASUREMENTS. Any wrong choice of parts must visibly distort the family of unentangled states, and we derive exactly how much, at every system size: closed-form constants (verified to seven decimal places), an explicit error bound, and a surprise. The two natural ways to measure "wrongness" (state distortion versus operator scrambling) turn out to be literally the same measurement, up to a fixed number, whenever the two parts have equal size. Wrongness cannot hide. (2) THE TWO-OPERATION THRESHOLD. You can pin down what your subsystems are by implementing just TWO generic operations and checking their statistics, while a THOUSAND mutually commuting operations tell you nothing at all. Diversity of control, not volume of control, is what fixes what your qubits are. The residual uncertainty is proportional to your measurement tolerance, with a constant we derive exactly and show how to compute in advance, so an experimenter can screen candidate operation pairs and pick a strong one (random pairs are typically several times worse, occasionally catastrophically so). (3) THE ALIASING LIMIT: CERTIFICATION CAN PASS WHILE TELLING YOU NOTHING. If you sample your certification data at special "resonant" times (set by the operations' energy scales), a maximally wrong assignment of subsystems passes the test perfectly. What your qubits are depends on when you looked. We characterize these blind spots exactly (for qubits they are universal and computable in advance; the danger windows shrink with the square root of your tolerance), and the fix costs almost nothing: sample at two different times, at least one away from the resonant set. With that, certification is restored; we prove it exactly and measure it. (4) REAL-LAB BUDGETS. We work out what certification costs in actual measurement shots (the error falls off unusually fast, inversely with the total number of shots rather than with its square root), how hardware noise floors the procedure, and how a simple reference measurement removes that floor to first order. A single seven-row table in the paper is effectively a lab protocol: it shows, condition by condition, which kinds of experimental access identify the subsystem structure (two generic operations: yes; one operation, commuting operations, resonant sampling: no). WHY IT MATTERS. For quantum hardware: this is a concrete recipe for certifying that the qubits you think you have are the qubits you physically have, with explicit tolerances, a sampling-time warning that is a real failure mode on calibrated hardware, and shot budgets. For foundations: the contested question "does physics determine its own subsystems?" gets a quantitative operational answer. Physics determines an operational subsystem CLASS (not one preferred coordinate representation of it), and that class is fixed by two operations sampled at two times, and by nothing less that we tested. WHAT IT OPENS UP. The seven-row identifiability table is a blueprint for a tabletop experiment on any small quantum platform; running it on hardware would move this from simulation to laboratory fact. On the theory side the paper leaves sharply posed open problems: the exact global constants when the two parts differ in size, the number-theoretic structure of long-time sampling (the failure times form a small-divisor hierarchy), and the extension from two parts to many. METHODS NOTE, BECAUSE IT MATTERS FOR TRUST. Every quantitative claim was pre-registered in the companion scripts before its run. The full ledger, including predictions that failed and one published-in-note constant that our own later theory overturned, is an appendix of the paper. All code, data, and working notes are in the deposit, seeded for exact reproduction. WHO AND HOW. A human-AI collaboration: research direction and standards of evidence by Ryan Cardwell (Crystalline Labs); design, code, derivations, and drafting by Claude (Anthropic), including autonomous sessions under the program's pre-registration rules, with adversarial reviews by other AI models incorporated and credited.
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.
The Qang () framework [1] expresses single-qubit rotations through two measurement-level quantities, the polar bias and the outcome entropy . We report results obtained with the open-source reference implementation qang. First, the two metrics are linked by an exact, branch-free identity, , which holds for any single-qubit state. Consequently, any new multi-qubit information must come from the joint outcome distribution. We name the resulting Z-basis total correlation and give its bounds and its scope: it is not an entanglement measure. Second, we identify exact blind spots: graph-state entanglement and readout-only dephasing are invisible to every Z-diagonal statistic, and qg-space updates cannot reach optima outside because of the range. On H unsigned qg updates recover none of the correlation energy. Keeping the sign of as a branch label removes this limit and reaches the FCI energy, but the best signed update reduces to -space descent, so qg coordinates bring no optimization gain. Third, we study a pole-damped gradient step whose damping schedule is . On a 300-landscape benchmark, H, and a 4-qubit LiH Hamiltonian with a mixed ansatz, it trades 3–20 more iterations at well-tuned learning rates for convergence where plain gradient descent diverges. Fourth, as a benchmark, tracks heavy output probability () and needs no calibration, unlike linear XEB. However, it is not monotonic under amplitude damping: under device-calibrated noise with added idle relaxation it falls below its own noiseless value. Pairing with the register-averaged removes the ambiguity in 23 of 24 circuits tested. A summary section lists fifty-two follow-up studies (QML encoding, control quantization, noise-type detection, circuit knitting, chemistry, error mitigation, thermal states, error correction, Hubbard dynamics, constrained QAOA, qubit characterization, syndrome tracking, coherence, few-shot estimation, IonQ noise models, shot-noise propagation, neural quantum states, coherent gate errors, symmetry checks in rotated bases, Heisenberg-chain dynamics, classical shadows, noisy Grover search, GHZ metrology, circuit-level surface codes, a lattice gauge theory, and the choice between filter and ZNE by shot budget), each with its classical or standard control; for the qg quantities, direct measurement is 3–41 cheaper than classical shadows, and the per-qubit reading loses to the outcome histogram when the answer is a bitstring; the symmetry-witness results are developed in a companion paper. Every number is reproduced by a script and pinned by a regression test in the public repository.