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arXiv:2603.03071

From Reachability to Learnability: Geometric Design Principles for Quantum Neural Networks

Vishal S. Ngairangbam, Michael Spannowsky

quant-phcs.LGhep-exhep-phstat.MLSubmitted Mar 3, 2026• Announced Mar 4, 2026• 0 citations
v1 · Mar 3, 2026v2 · Mar 25, 2026
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Abstract

Classical deep networks are effective because depth enables adaptive geometric deformation of data representations. In quantum neural networks (QNNs), however, depth or state reachability alone does not guarantee this feature-learning capability. We study this question in the pure-state setting by viewing encoded data as an embedded manifold in $\mathbb{C}P^{2^n-1}$ and analysing infinitesimal unitary actions through Lie-algebra directions. We introduce Classical-to-Lie-algebra (CLA) maps and the criterion of almost Complete Local Selectivity (aCLS), which combines directional completeness with data-dependent local selectivity. Within this framework, we show that data-independent trainable unitaries are complete but non-selective, i.e. learnable rigid reorientations, whereas pure data encodings are selective but non-tunable, i.e. fixed deformations. Hence, geometric flexibility requires a non-trivial joint dependence on data and trainable weights. We further show that accessing high-dimensional deformations of many-qubit state manifolds requires parametrised entangling directions; fixed entanglers such as CNOT alone do not provide adaptive geometric control. Numerical examples validate that CLS-satisfying data re-uploading models outperform non-tunable schemes while requiring only a quarter of the gate operations. Thus, the resulting picture reframes QNN design from state reachability to controllable geometry of hidden quantum representations.

Comments
29 pages, 5 figures, 3 tables

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Classical deep networks are effective because depth enables adaptive geometric deformation of data representations. In quantum neural networks (QNNs), however, depth or state reachability alone does not guarantee this feature-learning capability. We study this question in the pure-state setting by viewing encoded data as an embedded manifold in CP2n−1\mathbb{C}P^{2^n-1}CP2n−1 and analysing infinitesimal unitary actions through Lie-algebra directions. We introduce Classical-to-Lie-algebra (CLA) maps and the criterion of almost Complete Local Selectivity (aCLS), which combines directional completeness with data-dependent local selectivity. Within this framework, we show that data-independent trainable unitaries are complete but non-selective, i.e. learnable rigid reorientations, whereas pure data encodings are selective but non-tunable, i.e. fixed deformations. Hence, geometric flexibility requires a non-trivial joint dependence on data and trainable weights. We further show that accessing high-dimensional deformations of many-qubit state manifolds requires parametrised entangling directions; fixed entanglers such as CNOT alone do not provide adaptive geometric control. Numerical examples validate that CLS-satisfying data re-uploading models outperform non-tunable schemes while requiring only a quarter of the gate operations. Thus, the resulting picture reframes QNN design from state reachability to controllable geometry of hidden quantum representations.
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Recent advancements in the discipline of quantum algorithms have displayed the importance of the geometry of quantum operators. Given this thrust, this paper develops a rigorous geometric framework to analyze how the Riemannian structure of data, under the manifold hypothesis, influences the subspace of quantum gates induced by quantum feature maps. While numerous encoding schemes have been proposed in quantum machine learning, little attention has been given to how data geometry is deformed when mapped into the Lie group of special unitary operators. Addressing this gap, we assume a point cloud forms a smooth Riemannian manifold and formally construct the induced Riemannian geometry of a broad class of Hamiltonian quantum feature maps, which encompasses the majority of derived schemes. Starting from first principles, we derive analytic and, consequently, computational formulas for fundamental geometric measurements, including curvature, volume forms, and harmonic maps, providing tools for systematic deformation analysis. Notably, the derivations of the formulas elucidates how changes along paths in the data manifold interplay to changes in the associated subspace of special unitary operators, thereby indicating a direct geometric effect of data on quantum circuits. This framework establishes the mathematical validity required for principled analysis beyond heuristic ansatz and enables future research into geometry-aware quantum algorithm design.
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Abstract We present a Lie-geometric framework for analyzing the trainability of quantum neural networks (QNNs) through the algebraic structure of their generator spaces. We prove that QNNs whose dynamics are restricted to polynomial-dimensional Lie subalgebras of su ( 2 n ) admit a polynomial lower bound on gradient variance, thereby avoiding barren plateaus by construction. In contrast, architectures exploring the full Lie algebra su ( 2 n ) exhibit exponentially vanishing gradient variance, O ( 2 − n ) , due to the exponential growth of the associated unitary orbit dimensions. Our results interpret barren plateaus as a geometric concentration phenomenon arising from high-dimensional Lie-group actions, rather than as an intrinsic limitation of parameterized quantum models. We further illustrate the mechanism using a finite-size collective-spin benchmark and a minimal two-qubit sanity check, both of which are consistent with the theoretical predictions.
Variational quantum algorithms (VQAs) are a leading approach to near-term quantum computation, but their utility is limited by barren plateaus and other pathologies in their loss landscapes. Existing landscape theories based on dynamical Lie algebras, Jordan-algebraic Wishart systems, approximate t-designs, and Haar-random circuits are foundational, but they often neglect the finite-depth geometry of realistic ansätze and are therefore ill-suited to the shallow-depth regime, where VQAs are poor approximators of 2-designs and trainability is most feasible. This work introduces Krylov algebras, algebraic structures induced by the Krylov span of a finite generator set acting on one or more seed vectors, as a framework for VQA landscape theory. We show that VQA reachable manifolds can be approximated in a numerically robust, geometrically faithful fashion by Krylov-Lie algebras and groups, and that these structures induce canonical invariant measures for computing expectation values and variances under general sampling measures. In particular, we derive weighted non-Haar variance formulas that recover the usual Lie-algebraic Haar formulas as a special case while isolating non-Haar effects into explicit correction terms. We also show that the common heuristic that sufficiently deep circuit ensembles must converge to Haar fails in general without additional hypotheses, identify concrete obstructions to naive Haar convergence, and recover convergence under natural necessary and sufficient ergodic conditions. Lastly, our formulas further imply that non-Haar contributions to landscape statistics may mitigate barren plateaus by reweighting the visible sectors of the loss landscape, suggesting that VQAs may be more trainable than recent literature has posited.