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

Abstract

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 $\sqrt{d/n}$, while the shot term scales as $\sqrt{(\log n)/S}$; 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 $\nstar = 2\sqrt{2dB/\log(2B/δ)}$ 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.

25 pages