Haar-random pure states confine O(n^k 2^{-n}) of quantum Fisher information to weight-k computational-basis readouts

The arXiv preprint derives exact Beta laws by comparing the quantum Fisher information F_Q against the full bitstring Fisher F_full and the largest variance-normalized response I_A available to any diagonal readout space of centered dimension r. When the joint state-tangent frame is Haar-random, the two successive ratios are independent Beta random variables; their expectations immediately imply that the joint span of all computational-basis Pauli strings up to fixed weight k retains only O(n^k 2^{-n}) of the total information. Exact-statevector simulations across five non-conserving circuit families approach this hierarchy with increasing depth, while a number-conserving ensemble deviates even after support and rank corrections, showing isotropy is required in addition to rank.

This result sharpens known barren-plateau and information-scrambling analyses by quantifying how little of the tangent space survives a fixed computational-basis measurement. Earlier numerical studies of variational quantum eigensolvers reported similar sensitivity loss but lacked the closed-form Beta statistics that now explain why increasing circuit depth alone cannot restore readout efficiency. The departure of the conserving family further indicates that symmetry-protected subspaces can preserve higher effective rank, a nuance missed by generic Haar averaging arguments.

Future experiments will need adaptive or non-diagonal measurements to approach the full F_Q; otherwise, scaling to hundreds of qubits will leave most parameter information inaccessible under standard bitstring statistics. The chief limitation is the strict isotropy assumption; relaxing it to structured ensembles or noisy circuits would strengthen applicability to near-term hardware.

What comes next is targeted tests of the Beta-law predictions on superconducting or trapped-ion processors with up to 20 qubits, measuring the empirical distribution of F_full / F_Q ratios as depth increases.