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.