具有有限采样泛化保证的量子学习中的测量预算分配
Measurement-Budget Allocation in Quantum Learning with Finite-Shot Generalization Guarantees
- University of Stavanger（斯塔万格大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对近期量子硬件固定测量预算下量子学习的训练态数与单态采样数权衡问题，提出带有限采样泛化保证的预算分配规则，经仿真验证间隙低于理论界，为量子学习实验规划提供统计指导。
AI中文摘要：
在近期量子硬件上，估计一个玻恩概率需要重复执行电路。因此，在测量预算固定为$B$的量子学习实验中，必须确定使用多少个不同的训练态$n$，以及为每个态分配多少次采样$S$。我们针对测量算子$M$固定或独立选取的二分类量子分类器研究了这一权衡关系，这类分类器的理想得分为$\tr(M\rho)$。我们证明了一个不依赖分布的泛化界，该界将有限样本贡献与有限采样贡献分离开来。样本项的尺度为$\nobreak{\frac{d}{n}}$的平方根，而采样项的尺度为$\nobreak{\frac{\nobreak{\nobreak{\text{log}}\nobreak{n}}}{S}}$的平方根；在$B=nS$的约束下，这两项呈反向变化。对该界的一个保守闭式代理函数进行最小化，可得到分配规则$n^\text{star} = 2\boldsymbol{\frac{2dB}{\nobreak{\text{log}}\nobreak{(2B/\nobreak{\nobreak{\nobreak{\text{δ}}}\nobreak)}}}$的平方根，且$S^\text{star} = B/n^\text{star}$。该代理函数与精确最小化函数具有相同的渐近尺度，且能达到$B^{-1/4}$的最坏情况速率。这一保证是刻意保守的，因为它适用于全部二分类量子测量类别。我们通过PennyLane仿真对理论进行了补充，仿真在9个合成二分类基准上使用2量子比特和4量子比特变分量子电路。在所有测试配置中，单侧经验泛化间隙均低于理论界。该结果为近期量子学习系统的有限采样评估和实验前规划中的测量预算分配提供了保守的统计指导，是对硬件级调度和电路设计考量的补充。将该保证扩展到完全自适应的采样噪声训练仍是一个开放问题。
英文摘要：
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.
