Identifying the Sign of Coherent Over-Rotations with Logarithmically Many Pauli Settings

Abstract

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 $n\ge 2$, 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 $5\le n\le 8$, 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 $|{\eta }_R|<\pi /2$, $0\le p_R<1/2$ 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.