Hidden selection rules and amplitude ceilings in quantum Fourier models
Abstract
Quantum models with Pauli data encodings compute trigonometric polynomials whatever channels, ancillas and outcome-averaged measurements follow, so their expressivity depends on the Fourier coefficients a circuit can reach. We identify two kinds of structure that settle this before training. An antiunitary time-reversal rule and a unitary reflection rule remove frequencies. For gate-level circuits, deciding whether such a rule exists reduces to a linear system over GF(2), solvable in polynomial time, and the time-reversal rule holds in 17.6% of 6,120 configurations of standard templates. Single-upload encodings cap the top coefficient at 2−m for every channel, and we compute the tight ceiling for every frequency and a depth-independent re-uploading bound. Frozen before training, these bounds were never violated on a non-linearity benchmark, and they predicted trained performance on real data. On IBM processors the rule held on a 65-qubit chain, and we switched rule-breaking relaxation noise on and off by design.
Related
- Hidden amplitude and dynamical information in phase-separation spectra
- Coarse-Graining Hidden Representations: Unsupervised Neuron Selection via Mapping Entropy
- Convergence Analysis of the Quantum Blahut-Arimoto Algorithm and the Contraction Coefficient of Qubit Amplitude-Damping Channels
- Non-Orthogonal Amplitude Amplification for Hybrid CV-DV Quantum Processors
- Measurement-enabled online quantum processing with amplitude encoding
BibTeX
@misc{sulimov2026hidden,
title = {{Hidden selection rules and amplitude ceilings in quantum Fourier models}},
author = {Pavel Sulimov and Claude Lehmann},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01314},
eprint = {2611.01314},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01314},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hidden selection rules and amplitude ceilings in quantum Fourier models
AU - Pavel Sulimov
AU - Claude Lehmann
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01314
UR - https://arxiv.org/abs/2611.01314
AB - Quantum models with Pauli data encodings compute trigonometric polynomials whatever channels, ancillas and outcome-averaged measurements follow, so their expressivity depends on the Fourier coefficients a circuit can reach. We identify two kinds of structure that settle this before training. An antiunitary time-reversal rule and a unitary reflection rule remove frequencies. For gate-level circuits, deciding whether such a rule exists reduces to a linear system over GF(2), solvable in polynomial time, and the time-reversal rule holds in 17.6% of 6,120 configurations of standard templates. Single-upload encodings cap the top coefficient at 2^−m for every channel, and we compute the tight ceiling for every frequency and a depth-independent re-uploading bound. Frozen before training, these bounds were never violated on a non-linearity benchmark, and they predicted trained performance on real data. On IBM processors the rule held on a 65-qubit chain, and we switched rule-breaking relaxation noise on and off by design.
ER -