Formalize the Hermite-mode population noise diagnostic for hardware characterization

Develop a rigorous hardware-characterization framework for the mode-population noise diagnostic based on transforming a state with the finite quantum Hermite transform, applying an idle noise channel, applying the inverse transform, and analyzing the resulting Hermite-mode populations.

Background

The paper proposes a transform–idle–inverse experiment in which the finite quantum Hermite transform maps a number-basis state to weighted position samples, a physical noise channel acts during an idle interval, and the inverse transform returns the state to the Hermite-number basis for measurement. The resulting mode populations can reveal how the noise redistributes oscillator-mode occupation.

The paper emphasizes that a practical characterization framework must distinguish noise during the idle interval from state-preparation, transform, and readout errors. It also notes that population measurements alone do not determine coherence transport: different noise generators can agree on all number-diagonal states while acting differently on superpositions. Consequently, a complete hardware diagnostic would need calibrated transform and measurement errors, a verified noise model, and phase-sensitive tests of coherence in addition to population measurements.

References

These supplementary proofs establish the analytic estimates, averaging-operator factorization, and assembly lemmas used in the main text. They also give the uniform-grid sampling contract, continuous-input error bounds, and introduce a mode-population noise diagnostic that we pose as an open problem for future hardware characterization.

— Efficient Non-Uniform Quantum Hermite Transform through Adaptive Sampling  (2609.20739 - Mayo et al., 17 Sep 2026) in Supplementary introduction; Section S-App:noise-diagnostic, especially subsection “Population and coherence tests”