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PLoS ONE

PsiAudit: An open-source toolkit for auditing symmetry-organised complexity in equivariant quantum neural networks

Hassan Ugail, Newton Howard

Published Jul 17, 2026• 1 citation
Publisher

Abstract

Parameterised quantum circuits are commonly assessed using measures such as expressibility, gradient behaviour, and entanglement. While useful, these measures do not indicate whether a circuit respects the symmetry it was designed to respect. This is especially important in equivariant quantum machine learning, where symmetry is central to the model. We introduce PsiAudit, an open-source Python toolkit for auditing symmetry-aware quantum neural network ansätze before training. Given an ansatz, a target symmetry, and a state trajectory, PsiAudit reports how the circuit occupies symmetry sectors, maintains coherence between them, fluctuates across the trajectory, and complies with the target symmetry. These outputs are combined into a configurable dashboard-style summary. PsiAudit supports phase, spin, and permutation symmetries, with the permutation audit implemented using Hamming-weight orbits. Tests on five ansatz families, across twenty random seeds and four to eight qubits, show, in the tested setting, that PsiAudit can identify inactive equivariant circuits, recover structure when multiple symmetry sectors are activated, and distinguish ansätze that appear similar under standard diagnostics. The toolkit also includes a unitary-level compliance check, reproducible examples, and a notebook for regenerating the reported results.

DOI
10.1371/journal.pone.0353739

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Parameterised quantum circuits are commonly assessed using measures such as expressibility, gradient behaviour, and entanglement. While useful, these measures do not indicate whether a circuit respects the symmetry it was designed to respect. This is especially important in equivariant quantum machine learning, where symmetry is central to the model. We introduce PsiAudit, an open-source Python toolkit for auditing symmetry-aware quantum neural network ansätze before training. Given an ansatz, a target symmetry, and a state trajectory, PsiAudit reports how the circuit occupies symmetry sectors, maintains coherence between them, fluctuates across the trajectory, and complies with the target symmetry. These outputs are combined into a configurable dashboard-style summary. PsiAudit supports phase, spin, and permutation symmetries, with the permutation audit implemented using Hamming-weight orbits. Tests on five ansatz families, across twenty random seeds and four to eight qubits, show, in the tested setting, that PsiAudit can identify inactive equivariant circuits, recover structure when multiple symmetry sectors are activated, and distinguish ansätze that appear similar under standard diagnostics. The toolkit also includes a unitary-level compliance check, reproducible examples, and a notebook for regenerating the reported results.
Useful quantum neural networks should not merely explore large Hilbert spaces but should organise their expressive capacity according to the symmetries of the learning problem. We introduce symmetry-organised complexity as an ansatz-level, representation-theoretic trajectory diagnostic for quantum neural networks. The diagnostic combines symmetry-sector organisation, cross-irreducible representation organised complexity, and symmetry metastability into a composite index, which is then multiplied by a compliance factor that penalises apparent complexity arising from symmetry violation. This compliance factor is defined at the level of the implemented trainable generators rather than as a representation-independent channel metric. The representation-theoretic basis of the construction is that, for an exactly equivariant network, the effective trainable operators lie in the commutant of the group action and are controlled by multiplicity dimensions rather than by the full Hilbert-space dimension. We show that joint sector collapse and state freezing force the index to vanish under an explicit multiplicity–purity condition and that networks with identical qubit and parameter counts can have different values of the index. Two analytically tractable four-qubit examples with excitation number and total spin symmetry illustrate how the diagnostic separates sector-collapsed, symmetry-organised, and symmetry-breaking behaviour. A controlled U(1)-compatible teacher–student classification task further shows that, in this validation setting, the ordering of the composite index across equivariant, hybrid, and non-equivariant ansatze agrees with the ordering of generalisation accuracy. The framework is most informative when the relevant symmetry of the learning problem is known.
Equivariant quantum neural networks are built to respect known symmetries in data or physical systems. In practice, however, a circuit is executed together with hardware noise, and this noise can break the intended symmetry even when the ideal noiseless circuit is exactly symmetric. In this paper, we introduce a channel-level diagnostic for detecting this kind of symmetry failure before training is performed. The diagnostic compares a noisy quantum channel with the target symmetry action and reports the mismatch as a bounded compliance score. The method does not require labels, loss functions, benchmark tasks, or a completed training loop. We validate it on reproducible examples with phase and rotation symmetries under depolarising noise, amplitude damping, coherent over-rotation, and mixed noise. The results show that the diagnostic distinguishes symmetry-preserving from symmetry-breaking noise, and that this distinction depends on the chosen representation. In our tests, amplitude damping preserves the phase symmetry but breaks the rotation symmetry. A comparison with ansatz-level commutation checks, process-distance measures, randomised benchmarking, and task-level metrics shows that the proposed score measures a distinct representation-specific channel property. This property is not captured by those alternatives. Furthermore, a controlled downstream symmetry-sensitive classification task and a larger-system runtime study reaching six qubits show that the score tracks symmetry-specific output inconsistency in the operating regime where the classifier remains informative, while also making the dense-scaling limits of the reference implementation explicit. The proposed method is a pre-training audit intended to complement downstream task-performance studies and existing channel diagnostics. A reference implementation with code and accompanying data is provided.
Abstract Equivariant quantum neural networks are designed to preserve symmetries in quantum machine-learning data, but this property is usually verified only before hardware compilation. We show that a transpiler can preserve the expected unitary symmetry relation while compiling symmetry-related inputs into circuits with different two-qubit gate counts and physical-qubit layouts, so ideal-circuit equivariance does not guarantee symmetry at the level of compiled circuit structure. We formalise this gap with a three-level compliance hierarchy and a taxonomy separating topology-input ansätze, where the symmetry changes the gate topology the compiler sees, from parameter-input ansätze, where it changes only numerical gate parameters and provably leaves controlled- Z counts unchanged under topology-based transpilation. Auditing four published ansatz families, we observe the effect only in the topology-input case. For the Skolik permutation-equivariant weighted-graph ansatz compiled to the IBM Heron Fez simulator target at twelve qubits, sparse graph inputs produce a mean relative controlled- Z asymmetry of 7.3% across fifty random vertex permutations, with a worst case of 22.5%. The effect persists across all four Qiskit optimisation levels and is not removed by matched-layout compilation, implicating routing-stage sensitivity to the ordered sparse-edge gate sequence. A density sweep shows that the asymmetry is comparable across intermediate graph densities and vanishes for complete graphs, where the ansatz degenerates into the parameter-input class, and stochastic baselines attribute the realised differences to heuristic routing sensitivity rather than to differing optimal compiled costs. Finally, we propose compile-once-and-relabel, a deployment pattern that removes controlled- Z count asymmetry by construction and reduces compilation cost for orbit-based workflows.
Abstract Variational quantum algorithms are the leading candidate for advantage on near-term quantum hardware. When training a parametrized quantum circuit in this setting to solve a specific problem, the choice of ansatz is one of the most important factors that determines the trainability and performance of the algorithm. In quantum machine learning (QML), however, the literature on ansatzes that are motivated by the training data structure is scarce. In this work, we introduce an ansatz for learning tasks on weighted graphs that respects an important graph symmetry, namely equivariance under node permutations. We evaluate the performance of this ansatz on a complex learning task, namely neural combinatorial optimization, where a machine learning model is used to learn a heuristic for a combinatorial optimization problem. We analytically and numerically study the performance of our model, and our results strengthen the notion that symmetry-preserving ansatzes are a key to success in QML.
Quantum physics-informed neural networks (QPINNs) have recently emerged as a promising framework for the solution of partial differential equations (PDEs), with several studies reporting improved convergence and accuracy relative to classical physics-informed neural networks (PINNs) at reduced training cost. Motivated by these advances, we introduce geometric quantum physics-informed neural networks (GQPINNs), a symmetry-aware extension of QPINNs in which the geometric structure of the underlying PDE is incorporated directly into the quantum-circuit ansatz. Building on the framework of geometric quantum machine learning, we construct parametrized circuits that encode finite-group and compact Lie-group symmetries as inductive biases through problem-specific equivariant generator sets . Using a twirling-based construction, we derive symmetry-preserving gates that ensure that the model predictions respect the symmetries of the governing equation whenever the boundary and initial data are symmetry compatible. We benchmark GQPINNs against standard QPINNs and symmetry-adapted classical PINN baselines under matched training protocols across a representative set of linear and nonlinear PDEs. Across these benchmarks, GQPINNs achieve improved solution accuracy, as quantified by lower mean absolute error, while requiring substantially fewer trainable parameters. These results identify symmetry-aware quantum-circuit design as an effective route toward improved efficiency and generalization in quantum PDE solvers and provide a systematic framework for incorporating geometric inductive biases into quantum-enhanced scientific machine learning.
Quantum neural network architectures that have little to no inductive biases are known to face trainability and generalization issues. Inspired by a similar problem, recent breakthroughs in machine learning address this challenge by creating models encoding the symmetries of the learning task. This is materialized through the usage of equivariant neural networks the action of which commutes with that of the symmetry. In this work, we import these ideas to the quantum realm by presenting a comprehensive theoretical framework to design equivariant quantum neural networks (EQNNs) for essentially any relevant symmetry group. We develop multiple methods to construct equivariant layers for EQNNs and analyze their advantages and drawbacks. Our methods can find unitary or general equivariant quantum channels efficiently even when the symmetry group is exponentially large or continuous. As a special implementation, we show how standard quantum convolutional neural networks (QCNNs) can be generalized to group-equivariant QCNNs where both the convolution and pooling layers are equivariant to the symmetry group. We then numerically demonstrate the effectiveness of a S U ( 2 ) -equivariant QCNN over symmetry-agnostic QCNN on a classification task of phases of matter in the bond-alternating Heisenberg model. Our framework can be readily applied to virtually all areas of quantum machine learning. Lastly, we discuss about how symmetry-informed models such as EQNNs provide hopes to alleviate central challenges such as barren plateaus, poor local minima, and sample complexity. Published by the American Physical Society 2024