Sample Complexity for Generative Learning of Quantum Measurement Data

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 $E(s,N)=B(s)+V(s,N)$, 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.

How should model capacity grow with the number of quantum-measurement records, and when should generative-learning error decay as a power of sample size rather than logarithmically? The paper studies expected population KL error through $\mathcal E(s,N)=B(s)+V(s,N)$, combining representation bias with learner-dependent estimation cost. Assuming $V\simeq[NC(s)]^{-1}$ near the optimum, exponential or algebraic capacity dependence gives four scaling templates. For unrestricted radius-$R$ local autoregressive tables, the bias equals the summed conditional mutual information discarded by context truncation. For independently estimated conditionals with Krichevsky–Trofimov smoothing, an exact finite-sample risk expression averages Bernoulli KL errors over binomial context and outcome counts. At fixed radius with positive, nondeterministic cells, it yields $V=d_R/(2N)+O(N^{-2})$, where $d_R=(n-R+1)2^R-1$; a single bulk site has $2^R$ parameters. Computational-basis transverse-field Ising ground-state distributions provide numerical tests using free-fermion probabilities and independent training realizations. Approximately exponential gapped bias produces effective optimal-error laws $N^{-0.56}$ and $N^{-0.68}$ at $g=0.5$ and $1.5$. Critical bias consistent with $b(R)\sim0.0589/R$ predicts $f^\star\sim0.0589\ln2/\ln N$, with substantial logarithmic corrections. Exact finite-sample calculations reproduce measured errors and nearly all optimal radii. These are population benchmarks requiring target probabilities, not data-only capacity-selection rules or high-probability guarantees. The risk formula is histogram-specific; gapped exponents describe finite ranges, and critical asymptotics remain consistency tests involving tail continuation and a fixed-radius expansion whose validity near the growing optimum is required.