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.

Quantum Physics (quant-ph); Machine Learning (cs.LG)

arXiv:2611.01314 [quant-ph]

(or arXiv:2611.01314v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2611.01314 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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