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)≃[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.
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BibTeX
@misc{zhang2026sample,
title = {{Sample Complexity for Generative Learning of Quantum Measurement Data}},
author = {Shuhan Zhang and Yi-Zhuang You},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38028},
eprint = {2610.38028},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38028},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sample Complexity for Generative Learning of Quantum Measurement Data
AU - Shuhan Zhang
AU - Yi-Zhuang You
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.38028
UR - https://arxiv.org/abs/2610.38028
AB - 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)≃[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.
ER -