# Certification cost of quantum models: measurement correlation, not parameter count

arXiv: https://arxiv.org/abs/2609.14424 · HTML: https://mlsanity.com/paper/2609.14424/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 13 references that are in the map.
- First posted on arXiv 13 Sep 2026.

## Builds on (most cited first, 9 of 9)

- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Stealing Machine Learning Models via Prediction APIs](https://mlsanity.com/paper/1609.02943/) (2016)
- [The power of quantum neural networks](https://mlsanity.com/paper/2011.00027/) (2020)
- [Power of data in quantum machine learning](https://mlsanity.com/paper/2011.01938/) (2020)
- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Exponential concentration in quantum kernel methods](https://mlsanity.com/paper/2208.11060/) (2022)
- [Better than classical? The subtle art of benchmarking quantum machine learning models](https://mlsanity.com/paper/2403.07059/) (2024)
- [Understanding quantum machine learning also requires rethinking generalization](https://mlsanity.com/paper/2306.13461/) (2023)
- [When cheap gradients fail: the measurement cost of attacking quantum classifiers](https://mlsanity.com/paper/2607.11095/) (2026)

## Abstract (arXiv)

Reporting the Fisher geometry of a trained variational quantum model is routine; quoting the shot budget that would establish it is not. Certifying an empirical Fisher matrix to relative Frobenius error varepsilon under coordinate-wise parameter shift costs Theta(B p2 V/(varepsilon2 G)) circuit executions, where V is the measured readout variance and G the measured squared gradient norm, with uniform allocation optimal in that class. One constant reproduces the cost of two circuit families whose exponents differ by a full power of p. The exponent is an identity in how nV and nG scale with the register, holding family by family to 0.001 across 624 matrix-product-state cells once the finite-p prefactor is removed. The cubic cost is therefore a finite-size window, set by whether the readout light cone grows with the register. A product family to 256 qubits gives 1.966 (95% CI 1.934--1.997); a brickwork entangler falls from 2.853 below ten qubits to 1.715 beyond sixty-four; a blocked entangler gives 1.984 at a fixed cone width against 3.034 at a proportional one. Fixed device connectivity fixes the cone, so a cubic budget from a small simulation overestimates a large machine, on top of hardware multipliers 2.07× (1.41--3.02), 2.38× and 1.91× on ibmmarrakesh, ibmfez and ibmkingston. Cost-optimal readout weights cut the measured shot budget by 2.67× (1.33--4.00) on hardware, flat from four to twelve qubits. A discrepancy model fitted on cheap circuits transfers its mean inside the calibration grid and, at six larger sizes named before the data, does not: nominal 90% intervals cover 36%, and split conformal is the only rung that stays near nominal.