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arXiv:2610.38028

Sample Complexity for Generative Learning of Quantum Measurement Data

Shuhan Zhang, Yi-Zhuang You

quant-phcond-mat.stat-mechSubmitted Oct 27, 2026• Announced Oct 29, 2026• 0 citations
v1 · Oct 27, 2026v2 · Oct 30, 2026
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Abstract

Generative learning of a quantum measurement distribution from finite data requires choosing a model capacity that balances representation accuracy against estimation cost. We develop a bias-variance framework that decomposes the expected population Kullback-Leibler error at model capacity $s$ and sample size $N$ as $\mathcal E(s,N)=B(s)+V(s,N)$, with representation bias $B$ and a variance term $V$ capturing the mean estimation error. In the well-sampled regime, $V(s,N)\simeq[NC(s)]^{-1}$, and exponential or algebraic laws for $B$ and $C$ yield four scaling templates for the optimal error, from inverse powers of $N$ to inverse powers of $\ln N$. For local autoregressive models, the context radius sets the model capacity, and $B$ is the summed conditional mutual information lost by truncating the context. For a smoothed histogram learner, we derive an exact finite-sample estimation cost that accounts for rarely observed contexts and recovers the parameter-count law in the well-sampled regime. For transverse-field Ising Born distributions, the independently evaluated bias and estimation cost predict both the optimal context radius and the optimal error. In the well-sampled regime, the estimation cost grows exponentially with context radius. Its competition with approximately exponential bias decay in the gapped regimes predicts power-law error decay in $N$, while algebraic bias decay at criticality predicts asymptotic inverse-power decay in $\ln N$. Bulk-site results are consistent with the corresponding finite-sample predictions. These results connect the structure of quantum measurement distributions to the model capacity justified by finite data.

Comments
14 pages, 8 figures

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