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arXiv:2608.24891

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

Ferhat Özgür Çatak

Published Jun 1, 2026• 0 citations
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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 $Tr(M\rho)$. 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 $n^{*} = 2\sqrt{2dB/\log(2B/\delta)}$ and $S^{*} = B/nstar$. 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.

DOI
10.48550/arxiv.2608.24891

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On near-term quantum hardware, estimating a Born probability requires repeated circuit executions. A quantum learning experiment with a fixed measurement budget BBB must therefore decide how many distinct training states nnn to use and how many shots SSS to allocate to each state. We study this tradeoff for binary quantum classifiers with fixed or independently selected measurement operators MMM, where the ideal score is Tr(Mρ)Tr(M\rho)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\sqrt{d/n}d/n​, while the shot term scales as (log⁡n)/S\sqrt{(\log n)/S}(logn)/S​; under the constraint B=nSB=nSB=nS, these two terms move in opposite directions. Minimising a conservative closed-form surrogate of the bound gives the allocation rule n∗=22dB/log⁡(2B/δ)n^{*} = 2\sqrt{2dB/\log(2B/\delta)}n∗=22dB/log(2B/δ)​ and S∗=B/nstarS^{*} = B/nstarS∗=B/nstar. This surrogate has the same asymptotic scaling as the exact minimizer and yields a worst-case rate of B−1/4B^{-1/4}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.
Quantum kernel estimation on near-term hardware is shot-budgeted: every entry of the kernel Gram matrix is a Bernoulli expectation that must be sampled with a finite number of circuit executions. Recent work on quantum kernel classification has shown that allocating shots non-uniformly across kernel entries, weighted by their downstream task sensitivity, can reduce the shot budget required to reach a target accuracy. We extend this idea to Gaussian process (GP) regression, a setting whose downstream quantities (full-spectrum posterior variance, log-determinant, marginal likelihood) couple to kernel error more tightly than the sign-only outputs of classification. We derive three closed-form pair-level sensitivities predictive coupling ∣αiαj∣|α_iα_j|∣αi​αj​∣, leave-one-out residual, and marginal-likelihood gradient and plug them into a Neyman-style minimum-variance allocation rule. To prevent catastrophic over-concentration when the warm-up sensitivity estimate is itself noisy, we add a high uniform coverage floor justified by a Frobenius lower bound on the missing-entry perturbation. On four UCI benchmarks and two synthetic RBF + Bernoulli controlled studies, the resulting allocator delivers 101010--21%21\%21% test-RMSE improvement over uniform allocation across the moderate-budget regime. The gain transfers (i) to genuine ZZ and Pauli-Z quantum kernels on quantum-natural data (−13-13−13--15%15\%15% at low budget, p<0.05p<0.05p<0.05 paired) and (ii) to four downstream tasks (Bayesian quadrature, heteroscedastic regression, hyperparameter learning, multi-output Cokriging). On UCI features embedded into a ZZ kernel the gain disappears, consistent with the exponential-concentration regime where shot allocation has nothing to exploit.
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Training parameterized quantum circuits requires balancing optimization progress with limited evaluation budgets and the statistical uncertainty of near-term quantum hardware. We present a modular benchmarking framework that separates quantum search-direction estimation from classical parameter-update rules, allowing their interactions and sensitivities to be examined independently. We evaluate a suite of gradient-based, stochastic, and derivative-free optimizers across combinatorial optimization using the Quantum Approximate Optimization Algorithm (QAOA), supervised quantum machine learning using Iris classification and a quantum convolutional neural network (QCNN) for binary MNIST classification, and quantum chemistry using the variational quantum eigensolver (VQE) for molecular hydrogen. By separating objective evaluations from sample-circuit costs under finite-shot simulation and physical execution on a 156-qubit processor, we compare optimizer behavior across workloads with different parameter counts and measurement requirements. Four selected simulator and hardware case studies report terminal and best observed objectives separately, alongside post-training classifier accuracies. The comparisons include selected MaxCut and hydrogen trajectories and illustrate workload-dependent objective changes and evaluation costs, while the simulator benchmark compares both peak and mean performance across three seeds. The hardware runs do not establish an optimizer ranking or isolate a causal effect of device noise.
Extracting quantum information from a quantum state is a fundamental task of quantum computation, often requiring the estimation of many non-commuting observables under a finite measurement budget. For both near-term and early fault-tolerant settings, the measurement protocol must balance statistical efficiency against implementation resources such as circuit depth, connectivity, and entangling-gate count. Many existing strategies focus on two extremes: hardware-friendly product measurements with high sampling cost, and fully commuting measurements with deep circuits. Here we recast resource-constrained measurement design as a generative learning problem. We introduce FlowMeas, which uses a generative flow network to directly sample finite ensembles of shallow Clifford measurement circuits subject to a prescribed shot budget and hardware constraints. At zero entangling depth, FlowMeas learns qubit-wise commuting measurement schedules and already matches or improves leading product-measurement methods on nearly all molecular benchmarks. Allowing one or two entangling gate layers yields further reductions in energy estimation error of up to 27%27\%27% relative to the strongest state-independent product-measurement baseline. The learned policy can also be reused across related Hamiltonians, substantially accelerating retraining along a molecular potential-energy surface. We further obtain results for molecular Hamiltonians with up to 20 qubits and apply the framework to a compactly encoded 54-qubit interacting fermionic model, extending the demonstrated scale beyond prior molecular benchmarks. These results establish generative learning as a flexible and unified framework for quantum measurement design under practical resource constraints.
Adversarial perturbations threaten machine learning classifiers, including variational quantum classifiers. We show that finite quantum measurement statistics (shot noise) act as a built-in defense against gradient-based test-time attacks whose cost scales unfavorably for the attacker. Because every gradient component must be inferred from repeated circuit executions under any unbiased gradient-estimation rule, white-box extraction consumes a dimension-dependent measurement budget that measurement grouping cannot remove in expressive circuits. Under stated assumptions, single-step attacks need at least quadratically many shots in the input dimension ddd, growing as d5/2d^{5/2}d5/2 under norm-concentration scaling, with a sufficient-budget analysis for iterative attacks via stochastic gradient Langevin dynamics. Simulations up to 784 input dimensions validate the law: the realized total budget is the d5/2d^{5/2}d5/2 geometric floor for plateau-mitigated models and grows as d3.00d^{3.00}d3.00 for the tested deep circuits, whose gradient norms decay with dimension absent barren-plateau mitigation; folding the measured gradient norm back in recovers the parameter-free d3/2d^{3/2}d3/2 shot-noise geometry. Against a matched classical baseline whose attack overhead is dimension-independent (the cheap-gradient principle of automatic differentiation), the quantum gradient cost ratio grows empirically as d3.00d^{3.00}d3.00, so the attacker's relative cost diverges as the model scales. Experiments on a 156-qubit IBM processor (ibm_boston, 4-qubit circuits, d=12d=12d=12) reproduce the effect: at matched budgets the device attack tracks the ideal within a few percent, with the high-shot gradient faithful to the exact one. The defense operates precisely when the forward map is classically hard to simulate: only then is a white-box attacker denied the simulate-and-backpropagate shortcut and must pay the measurement cost we quantify.