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

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.