# Measurement-Budget Allocation in Quantum Learning with Finite-Shot Generalization Guarantees

arXiv: https://arxiv.org/abs/2608.24891 · HTML: https://mlsanity.com/paper/2608.24891/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 25 references that are in the map.
- First posted on arXiv 1 Jun 2026.

## Builds on (most cited first, 10 of 10)

- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Generalization in quantum machine learning from few training data](https://mlsanity.com/paper/2111.05292/) (2021)
- [Noise-Induced Barren Plateaus in Variational Quantum Algorithms](https://mlsanity.com/paper/2007.14384/) (2020)
- [Information-theoretic bounds on quantum advantage in machine learning](https://mlsanity.com/paper/2101.02464/) (2021)
- [Encoding-dependent generalization bounds for parametrized quantum circuits](https://mlsanity.com/paper/2106.03880/) (2021)
- [Understanding quantum machine learning also requires rethinking generalization](https://mlsanity.com/paper/2306.13461/) (2023)
- [On the statistical complexity of quantum circuits](https://mlsanity.com/paper/2101.06154/) (2021)
- [Reduction of finite sampling noise in quantum neural networks](https://mlsanity.com/paper/2306.01639/) (2023)
- [Generalization of Quantum Machine Learning Models Using Quantum Fisher Information Metric](https://mlsanity.com/paper/2303.13462/) (2023)

## Abstract (arXiv)

On near-term quantum hardware, estimating a Born probability requires repeated circuit executions. A quantum learning experiment with a fixed measurement budget B must therefore decide how many distinct training states n to use and how many shots S to allocate to each state. We study this tradeoff for binary quantum classifiers with fixed or independently selected measurement operators M, where the ideal score is Tr(Mρ). We prove a distribution-free generalization bound that separates the finite-sample and finite-shot contributions. The sample term scales as √d/n, while the shot term scales as √(log n)/S; under the constraint B=nS, these two terms move in opposite directions. Minimising a conservative closed-form surrogate of the bound gives the allocation rule nstar = 2√2dB/log(2B/δ) and Sstar = B/nstar. This surrogate has the same asymptotic scaling as the exact minimizer and yields a worst-case rate of B-1/4. The guarantee is intentionally conservative, since it applies to the full class of binary quantum measurements. We complement the theory with PennyLane simulations using 2-qubit and 4-qubit variational quantum circuits on nine synthetic binary classification benchmarks. In all tested configurations, the one-sided empirical generalization gap remains below the theoretical bound. The result provides a conservative statistical guideline for allocating measurement budgets in finite-shot evaluation and pre-experimental planning for near-term quantum learning systems, complementing hardware-level scheduling and circuit-design considerations. Extending the guarantee to fully adaptive shot-noisy training remains an open problem.