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

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.

Physics-Aware Learnability: From Set-Theoretic Independence to Operational Constraints

Beyond binary classification, learnability can become a logically fragile notion: in EMX, even the class of all finite subsets of $[0,1]$ is learnable in some models of ZFC and not in others. We argue the paradox is operational. The standard definitions quantify over arbitrary set-theoretic learners that implicitly assume non-operational resources (infinite precision, unphysical data access, and non-representable outputs). We introduce physics-aware learnability (PL), which defines the learnability relative to an explicit access model -- a family of admissible physical protocols. Finite-precision coarse-graining reduces continuum EMX to a countable problem, via an exact pushforward/pullback reduction that preserves the EMX objective, making the independence example provably learnable with explicit $(ε,δ)$ sample complexity. For quantum data, admissible learners are exactly POVMs on $d$ copies, turning sample size into copy complexity and yielding Helstrom(-type) lower bounds. For finite no-signaling and quantum models, PL feasibility becomes linear or semidefinite and is therefore decidable.