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arXiv:2606.12211

Quantum Occam Learning: Sample-Supported Expressibility for Circuit-Based Quantum Learning

Jeongho Bang, Kyoungho Cho, Jeongwoo Jae

quant-phcs.LGSubmitted Jun 10, 2026• Announced Jun 11, 2026• 0 citations
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Abstract

A central principle in quantum machine learning is that an ansatz should be expressive enough to represent the quantum data of interest. Yet, the expressibility is statistically meaningful only insofar as it can be learned from finitely many copies of an unknown quantum state. In this work, we develop an information-theoretic Occam theory for quantum data generated by finite-size quantum circuits. For the class $S_{n,G}$ of $n$-qubit pure states preparable with at most $G$ two-qubit gates, a metric-entropy argument gives the realizable sample law $\widetildeΘ(G/ε^2)$ in the circuit-limited regime. For an arbitrary source $\hatρ$, we introduce the best $G$-gate approximation error $d_G(\hatρ)$ and the approximate circuit complexity $C_η(\hatρ)$. We prove an agnostic quantum Occam theorem: with $M$ copies, one can learn up to the best $G$-gate approximation error plus a statistical penalty $\widetilde{O}(\sqrt{G/M})$. We then remove the need to know $G$ in advance through an adaptive model-selection theorem whose oracle inequality selects the circuit complexity justified by the data. Matching lower bounds yield a sample-supported expressibility law: at trace-distance accuracy $ε$, $M$ samples can support only $G_{\rm supported} \simeq Mε^2$ gates, up to logarithmic factors and tomography saturation at $2^n$. Thus, the circuit complexity becomes an adaptive statistical resource rather than a static promise. Our framework turns bounded circuit complexity into a model-selection principle for quantum machine learning.

Comments
22 pages (main text + appendix), 2 figures

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A central principle in quantum machine learning is that an ansatz should be expressive enough to represent the quantum data of interest. Yet, the expressibility is statistically meaningful only insofar as it can be learned from finitely many copies of an unknown quantum state. In this work, we develop an information-theoretic Occam theory for quantum data generated by finite-size quantum circuits. For the class Sn,GS_{n,G}Sn,G​ of nnn-qubit pure states preparable with at most GGG two-qubit gates, a metric-entropy argument gives the realizable sample law Θ~(G/ε2)\widetildeΘ(G/ε^2)Θ(G/ε2) in the circuit-limited regime. For an arbitrary source ρ^\hatρρ^​, we introduce the best GGG-gate approximation error dG(ρ^)d_G(\hatρ)dG​(ρ^​) and the approximate circuit complexity Cη(ρ^)C_η(\hatρ)Cη​(ρ^​). We prove an agnostic quantum Occam theorem: with MMM copies, one can learn up to the best GGG-gate approximation error plus a statistical penalty O~(G/M)\widetilde{O}(\sqrt{G/M})O(G/M​). We then remove the need to know GGG in advance through an adaptive model-selection theorem whose oracle inequality selects the circuit complexity justified by the data. Matching lower bounds yield a sample-supported expressibility law: at trace-distance accuracy εεε, MMM samples can support only Gsupported≃Mε2G_{\rm supported} \simeq Mε^2Gsupported​≃Mε2 gates, up to logarithmic factors and tomography saturation at 2n2^n2n. Thus, the circuit complexity becomes an adaptive statistical resource rather than a static promise. Our framework turns bounded circuit complexity into a model-selection principle for quantum machine learning.
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The vast and complicated many-qubit state space forbids us to comprehensively capture the dynamics of modern quantum computers via classical simulations or quantum tomography. Recent progress in quantum learning theory prompts a crucial question: can linear properties of a many-qubit circuit with d tunable RZ gates and G − d Clifford gates be efficiently learned from measurement data generated by varying classical inputs? In this work, we prove that the sample complexity scaling linearly in d is required to achieve a small prediction error, while the corresponding computational complexity may scale exponentially in d. To address this challenge, we propose a kernel-based method leveraging classical shadows and truncated trigonometric expansions, enabling a controllable trade-off between prediction accuracy and computational overhead. Our results advance two crucial realms in quantum computation: the exploration of quantum algorithms with practical utilities and learning-based quantum system certification. We conduct numerical simulations to validate our proposals across diverse scenarios, encompassing quantum information processing protocols, Hamiltonian simulation, and variational quantum algorithms up to 60 qubits. It is currently unclear whether machine learning model could efficiently learn the dynamics of many-qubit bounded-gate quantum circuits. This work reveals the learnability of predicting linear properties of large- scale quantum circuits and proposes a learning algorithm that enables resource-efficient solutions for many practical quantum applications.
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Abstract Quantum data loading plays a central role in quantum algorithms and quantum information processing. Many quantum algorithms hinge on the ability to prepare arbitrary superposition states as a subroutine, with claims of exponential speedups often predicated on access to an efficient data-loading oracle. In practice, constructing a circuit to prepare a generic n-qubit quantum state typically demands computational efforts scaling as O ( 2 n ) , posing a significant challenge for quantum algorithms to outperform their classical counterparts. To address this critical issue, various hybrid quantum–classical approaches have been proposed. However, many of these solutions favor simplistic circuit architectures, which are susceptible to substantial optimization challenges. In this study, we harness quantum circuits as Born machines to generate probability distributions. Drawing inspiration from methods used to investigate electronic structures in quantum chemistry and condensed matter physics, we propose a framework called Adaptive Circuit Learning of Born Machine, which dynamically expands the ansatz circuit. Our algorithm is designed to selectively integrate two-qubit entangled gates that best capture the intricate entanglement present within the target state. Empirical experiments underscore the efficacy of our approach in encoding real-world data through amplitude embedding, demonstrating not only compliance with but also enhancement over the performance benchmarks set by prior research.
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