Z-basis metrics for parameterized quantum circuits: exact identities, structural blind spots, pole-damped optimization, and a relaxation-aware entropy benchmark
Vicente Humberto Monteverde · Zenodo (CERN European Organization for Nuclear Research) · 2026
The Qang (qg) framework [1] expresses single-qubit rotations through two measurement-level quantities, the polar bias qgZ = ⟨σz⟩ and the outcome entropy qgS = H(p0). We report results obtained with the open-source reference implementation qang. First, the two metrics are linked by an exact, branch-free identity, qgS = H (1 + qgZ)/2 , 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 [0, π] because of the arccos range. On H2 they recover none of the correlation energy. Third, we study a pole-damped gradient step whose damping schedule is |dqgZ/dθ|. On a 300-landscape benchmark, H2, and a 4-qubit LiH Hamiltonian with a mixed Ry/Rx ansatz, it trades 3-20× more iterations at well-tuned learning rates for convergence where plain gradient descent diverges. Fourth, as a benchmark, qgS tracks heavy output probability (r = −0.93) and needs no calibration, unlike linear XEB. However, it is ÙOÍ monotonic under amplitude damping: under device-calibrated noise with added idle relaxation it falls below its own noiseless value. Pairing qgS with the register-averaged qgZ removes the ambiguity in 23 of 24 circuits tested. Every number is reproduced by a script and pinned by a regression test in the public repository.