# Readout-Rank Laws for Isotropic Quantum Tangents

arXiv: https://arxiv.org/abs/2608.07628 · HTML: https://mlsanity.com/paper/2608.07628/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 35 references that are in the map.
- First posted on arXiv 7 Aug 2026.

## Builds on (most cited first, 8 of 8)

- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Noise-Induced Barren Plateaus in Variational Quantum Algorithms](https://mlsanity.com/paper/2007.14384/) (2020)
- [Connecting ansatz expressibility to gradient magnitudes and barren plateaus](https://mlsanity.com/paper/2101.02138/) (2021)
- [Quantum Natural Gradient](https://mlsanity.com/paper/1909.02108/) (2019)
- [Entanglement Induced Barren Plateaus](https://mlsanity.com/paper/2010.15968/) (2020)
- [Stochastic gradient descent for hybrid quantum-classical optimization](https://mlsanity.com/paper/1910.01155/) (2019)

## Abstract (arXiv)

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 FQ, the Fisher information Frm full in the complete bitstring distribution, and the largest variance-normalized response mathcal Imathcal 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/(2n-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(nk2-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.