Symmetry Discovery in Quantum Learning: Observable-Level and Task-Level Inference from Finite Measurements

Zeyu Chen

quant-phSep 11, 2026

Symmetry Discovery in Quantum Learning: Observable-Level and Task-Level Inference from Finite Measurements

Authors: Zeyu Chen

Abstract

Symmetry reduces the capacity of a quantum learning model, but the imposed group must match both the measured information and the label transformation. We establish a finite-measurement theory for inferring this group from candidate transformations. The central structural result identifies observable-invisible transformations with the stabilizer of a projected state whenever the probe span is invariant. It turns recovered generators into a valid subgroup and identifies the continuous invisible space with its Lie algebra. For finite dictionaries, an unbiased shadow statistic distinguishes zero from positive squared expectation discrepancies with an inverse-gap measurement rate, improving the inverse-square-gap rate of uniform discrepancy estimation. A commuting qubit lower bound proves the gap dependence optimal at fixed snapshot scale, and simultaneous intervals support data-dependent tolerances. Task validation then tests either the joint distribution through a characteristic kernel or its encoded mean through a classical--quantum discrepancy. An exact group-average identity relates the latter to joint-state asymmetry and specifies its conversion to binary task breaking mass. Projection bias quantifies the cost of excessive symmetry, while an $\ell_1$ readout bound quantifies the capacity gained by relaxing it. At an invariant pure-state backbone, retained and nontrivial breaking sectors are Fisher-orthogonal. Ising-chain calculations connect finite-shot recovery, label-dependent symmetry, and physical sector drift. These results determine which symmetry the measurements support and provide the statistical and geometric basis for a subsequent release decision.

Figures & tables

Panel of Figure 1 — Figure 1: Illustrative cyclic-symmetry recovery curve for $m=12$ and the hidden subgroup $K=\langle r^{4}\rangle$ . Each point is the empirical success probability over 2000 multinomial simulations from $p^{(\epsilon)}$ , using the threshold rule in Proposition 3.11 .

Panel of Figure 2 — Figure 2: Exact-diagonalization baseline for the transverse-field Ising chain. The dots show $F^{\mathrm{phys}}_{MM}=4\operatorname{Var}(\widehat{M})$ for $L\in\{8,10,12,14\}$ , and the dashed lines are power-law visual fits. The fitted exponents are approximately $1.048$ for $h=1.5$ , $1.759$ for $h=1.0$ , and $1.993$ for $h=0.5$ , matching the expected disordered, critical, and ordered trends.

Panel of Figure 3 — Figure 3: Observable-level dictionary recovery from Pauli classical shadows on the $L=8$ chain at $h=1$ , over $500$ independent repetitions per point. The estimated invisible set is declared by thresholding at $\gamma_{\mathcal{O}}/2$ ; a repetition counts as a success only if the recovered set equals the population invisible set exactly. The population margin is supplied to both threshold rules. The unbiased $U$ -statistic performs better on this tested grid; the plug-in square has positive finite-sample bias. These curves compare the estimators at the stated margins and do not establish a different asymptotic recovery rate for the plug-in rule.

Symmetry Discovery

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Learning Operators for Symmetry Discovery: From Lie Groups to Local Symmetry Geometry

Pavan Karjol, Scindhiya Laxmi, Prathosh AP

Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore, India · Department of Mathematics and Computing, IIT (ISM) Dhanbad, India

Symmetry Discovery Operator Learning Lie Group Methods

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Continuous symmetry discovery is typically formulated as a separate estimation problem for each function or dataset. We instead investigate whether the map from a function to its symmetry structure can be learned across a family of functions from finite observations. Using the infinitesimal differential characterization of continuous symmetries, we formulate this map as an inverse symmetry operator, analogous to an inverse PDE problem. For global Lie-group symmetries, we establish a representation-theoretic population characterization: the symmetry Lie algebra is the nullspace of a positive-semidefinite operator constructed from representation coefficients and the infinitesimal group action. We also show that the inverse symmetry operator is equivariant under coordinate changes through the adjoint action. This characterization separates the geometric symmetry map from finite-sample estimation, motivating a hybrid approach that amortizes statistically difficult components while retaining analytic structure. On SO(5) benchmarks, amortized equivariant estimation of harmonic coefficients improves symmetry recovery from sparse observations, whereas ordinary least squares suffices for SL(2,R). We extend this viewpoint to local symmetries, where symmetry directions form smoothly varying tangent distributions rather than a fixed Lie algebra. Under the Lie-bracket consistency condition, Frobenius' theorem guarantees local integral submanifolds along which observables remain unchanged. We recover this geometry using a hybrid estimator that learns observation weights while preserving analytic differential and geometric recovery. Across global, non-compact, local, and physically motivated synthetic benchmarks, our results support learning the statistically difficult finite-sample components while preserving the known geometric structure of symmetry recovery.

Abstract

Symmetry Discovery

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