Readout-Rank Laws for Isotropic Quantum Tangents

• URL: http://arxiv.org/abs/2608.07628v1
• Date: Fri, 07 Aug 2026 11:09:51 GMT
• Title: Readout-Rank Laws for Isotropic Quantum Tangents
• Abstract summary: Deep parameterized quantum circuits may remain sensitive to a parameter change while the observables retained by a learning model barely respond.<n>We show that even the joint span of all computational-basis Pauli strings through any fixed weight $k$ retain only $O(nk2-n)$ of the full-record information.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Deep parameterized quantum circuits may remain sensitive to a parameter change while the observables retained by a learning model barely respond. We study this separation for a fixed computational-basis measurement. For a pure-state tangent, we compare the quantum Fisher information $F_Q$, the Fisher information $F_{\rm full}$ in the complete bitstring distribution, and the largest variance-normalized response $\mathcal I_{\mathcal A}$ available to a diagonal readout space $\mathcal A$. If the joint state--tangent frame is Haar random, we prove that the two successive information fractions are independent Beta variables whose means are $1/2$ and $r/(2^n-1)$, where $r$ is the centered dimension of the readout. Consequently, even the joint span of all computational-basis Pauli strings through any fixed weight $k$ retain only $O(n^k2^{-n})$ of the full-record information. Exact-statevector experiments across six circuit families show increasing finite-size agreement with this hierarchy in five nonconserving ensembles as the circuit depth grows. A number-conserving family departs strongly from the isotropic prediction even after correcting the support and readout rank, showing that rank alone is insufficient without tangent isotropy.

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