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arXiv:2512.24019

One-Shot Structured Pruning of Quantum Neural Networks via $q$-Group Engineering and Quantum Geometric Metrics

Haijian Shao, Wei Liu, Xing Deng, Yingtao Jiang

quant-phcs.CVSubmitted Dec 30, 2025• Announced Dec 31, 2025• 0 citations
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Abstract

Quantum neural networks (QNNs) suffer from severe gate-level redundancy, which hinders their deployment on noisy intermediate-scale quantum (NISQ) devices. In this work, we propose q-iPrune, a one-shot structured pruning framework grounded in the algebraic structure of $q$-deformed groups and task-conditioned quantum geometry. Unlike prior heuristic or gradient-based pruning methods, q-iPrune formulates redundancy directly at the gate level. Each gate is compared within an algebraically consistent subgroup using a task-conditioned $q$-overlap distance, which measures functional similarity through state overlaps on a task-relevant ensemble. A gate is removed only when its replacement by a subgroup representative provably induces a bounded deviation on all task observables. We establish three rigorous theoretical guarantees. First, we prove completeness of redundancy pruning: no gate that violates the prescribed similarity threshold is removed. Second, we show that the pruned circuit is functionally equivalent up to an explicit, task-conditioned error bound, with a closed-form dependence on the redundancy tolerance and the number of replaced gates. Third, we prove that the pruning procedure is computationally feasible, requiring only polynomial-time comparisons and avoiding exponential enumeration over the Hilbert space. To adapt pruning decisions to hardware imperfections, we introduce a noise-calibrated deformation parameter $λ$ that modulates the $q$-geometry and redundancy tolerance. Experiments on standard quantum machine learning benchmarks demonstrate that q-iPrune achieves substantial gate reduction while maintaining bounded task performance degradation, consistent with our theoretical guarantees.

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10 pages, 2 figures

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One-Shot Structured Pruning of Quantum Neural Networks via qqq-Group Engineering and Quantum Geometric Metrics
Quantum neural networks (QNNs) suffer from severe gate-level redundancy, which hinders their deployment on noisy intermediate-scale quantum (NISQ) devices. In this work, we propose q-iPrune, a one-shot structured pruning framework grounded in the algebraic structure of qqq-deformed groups and task-conditioned quantum geometry.
Unlike prior heuristic or gradient-based pruning methods, q-iPrune formulates redundancy directly at the gate level. Each gate is compared within an algebraically consistent subgroup using a task-conditioned qqq-overlap distance, which measures functional similarity through state overlaps on a task-relevant ensemble. A gate is removed only when its replacement by a subgroup representative provably induces a bounded deviation on all task observables.
We establish three rigorous theoretical guarantees. First, we prove completeness of redundancy pruning: no gate that violates the prescribed similarity threshold is removed. Second, we show that the pruned circuit is functionally equivalent up to an explicit, task-conditioned error bound, with a closed-form dependence on the redundancy tolerance and the number of replaced gates. Third, we prove that the pruning procedure is computationally feasible, requiring only polynomial-time comparisons and avoiding exponential enumeration over the Hilbert space.
To adapt pruning decisions to hardware imperfections, we introduce a noise-calibrated deformation parameter λλλ that modulates the qqq-geometry and redundancy tolerance. Experiments on standard quantum machine learning benchmarks demonstrate that q-iPrune achieves substantial gate reduction while maintaining bounded task performance degradation, consistent with our theoretical guarantees.
Quantum neural networks (QNNs) and parameterized quantum circuits (PQCs) are key building blocks for near-term quantum machine learning. However, their scalability is constrained by excessive parameters, barren plateaus, and hardware limitations. We propose LiePrune, the first mathematically grounded one-shot structured pruning framework for QNNs that leverages Lie group structure and quantum geometric information. Each gate is jointly represented in a Lie group--Lie algebra dual space and a quantum geometric feature space, enabling principled redundancy detection and aggressive compression. Experiments on quantum classification (MNIST, FashionMNIST), quantum generative modeling (Bars-and-Stripes), and quantum chemistry (LiH VQE) show that LiePrune achieves over 10×10\times10× compression with negligible or even improved task performance, while providing provable guarantees on redundancy detection, functional approximation, and computational complexity.
The NISQ era places stringent constraints on quantum computation, where noise and decoherence fundamentally limit performance. In classical deep learning, model robustness and resilience to perturbations are well studied: deep neural networks (DNNs) maintain high performance despite pruning, noise injection, and structural perturbations due to inherent redundancy in their representations. A central challenge in quantum machine learning is to transfer this notion of robustness to quantum neural networks (QNNs) under realistic NISQ noise. While classical deep learning exhibits robustness through structural redundancy, analogous principles for QNNs remain underdeveloped. We propose JGRA: a framework for assessing robustness in noise-aware QNNs via Jacobian geometry, capturing model sensitivity to parameter perturbations induced by noise. Our method includes entropy-matched noise calibration, noise-aware training, and noise-conditioned Jacobian extraction, yielding geometric descriptors that link clean-regime structure to noisy inference behaviour. We also empirically demonstrate that these descriptors encode predictive information about robustness under unseen noise.
We present a routing-aware pruning strategy for quantum circuits that selectively removes parametric two-qubit gates whose computational contribution is outweighed by the fidelity cost of their execution, and assess it across both Noisy Intermediate-Scale Quantum (NISQ) and fault-tolerant quantum computing (FTQC) architectures. Each two-qubit gate of a circuit is evaluated by comparing an architecture-independent lower bound on the fidelity cost of omitting the gate against a hardware-specific model for the routing execution cost in each setting. We additionally provide an analytical characterization of the pruning boundary as a function of rotation angle, qubit distance, and hardware noise. Simulations on benchmark circuits demonstrate gate count reductions of up to 48.6% and fidelity improvements of up to 47.7% in the NISQ regime, while a feasibility analysis on FT hardware indicates that pruning is expected to remain relevant across a range of code distances and physical error rates. These results establish routing-aware pruning as a practical and unified compilation strategy across the quantum computing stack.
The convergence of quantum computing and machine learning offers significant potential for processing complex data distributions. However, the realization of this potential in the Noisy Intermediate-Scale Quantum (NISQ) era is constrained by physical hardware limitations, particularly restricted qubit counts and system noise. This thesis addresses the challenge of the ”Representation Bottleneck” by establishing a framework for resource-efficient quantum representation learning. The research demonstrates that effective learning on near-term devices can be achieved by optimizing circuit architectures and objective functions to align with hardware constraints. First, to address resource limitations in high-dimensional visual recognition, we propose the Multi-Gate Quantum Convolutional Neural Network (MG-QNN). By employing a Multi-Gate strategy, this architecture optimizes the utilization of quan tum resources within the circuit design. Empirical evaluations on face recognition benchmarks indicate that this approach maintains competitive recognition accuracy while reducing the qubit requirement compared to standard quantum convolutional methods. Second, extending the framework to non-Euclidean data, we introduce the Single-Qubit Graph Neural Network (sQGNN). Utilizing Discrete-Time Quantum Walks, this method encodes graph topologies and relational patterns onto a single qubit. This approach circumvents the need for multi-qubit entangling gates, thereby reducing the computational overhead and noise sensitivity typically associated with processing graph-structured data. Third, in the domain of generative modeling, we present the Single-Qubit Quan tum Convolutional Variational Autoencoder (QCVAE). Addressing the difficulty of generative tasks on constrained devices, this model integrates the Information Bottleneck principle with efficient data re-uploading techniques. The study demonstrates that a single qubit possesses sufficient expressivity to model and reconstruct image distributions effectively when guided by appropriate information theoretic constraints. Finally, to resolve the fundamental limitations of classical loss functions in quantum learning, we establish the Hilbert-Schmidt Information Bottleneck (HSIB). We identify that standard Euclidean metrics often fail to respect the geometric structure of the Hilbert space. The HSIB framework employs the Hilbert Schmidt distance as a tractable geometric measure, providing a learning objective that maximizes information retention while adhering to the intrinsic geometry of quantum states. Collectively, these contributions illustrate that practical quantum machine learning advantage in the NISQ era does not rely solely on scaling hardware, but on the strategic design of resource-efficient architectures and geometrically valid metrics.
Deploying quantum neural networks (QNNs) on sparse hardware requires preserving task accuracy while reducing compiler-induced two-qubit gates, SWAPs, and depth. We present ConQUESTA (\underline{C}onstrained \underline{Qu}antum \underline{E}ntangler \underline{S}election with \underline{T}opology \underline{A}wareness), a post training pruning framework that combines task saliency with compiler cost, screens candidate entanglers, and searches complete pruning masks under an explicit accuracy constraint. Unlike one-shot ranking, each mask is transpiled as a whole across multiple coupling topologies. Across 4-, 6-, and 8-qubit QNNs and eight model seeds at 55% pruning, ConQUESTA achieves mean resource reductions of 61.6% on Breast Cancer, 72.0% on Digits01, and 82.2% on Iris, exceeding the strongest matched simple baseline by 3.5, 1.7, and 8.9 percentage points, respectively. Fixed-mask recovery raises feasibility from 16/24 to 22/24 runs. In a fixed-k=4 scaling study on a scaling probe of 4 to 12-qubits, the raw mask space grew from 495 to 12.1M while the evaluated masks increase only from 189 to 305. Resource-weight changes alter depth/routing priorities with at most a 0.44-point mean validation shift.
Quantum Machine Learning (QML) is fundamentally limited by two challenges: barren plateaus (exponentially vanishing gradients) and the fragility of parameterized quantum circuits under noise. Despite extensive empirical studies, a unified theoretical framework remains lacking. We introduce LieTrunc-QNN, an algebraic-geometric framework that characterizes trainability via Lie-generated dynamics. Parameterized quantum circuits are modeled as Lie subalgebras of u(2^n), whose action induces a Riemannian manifold of reachable quantum states. Expressivity is reinterpreted as intrinsic manifold dimension and geometry. We establish a geometric capacity-plateau principle: increasing effective dimension leads to exponential gradient suppression due to concentration of measure. By restricting to structured Lie subalgebras (LieTrunc), the manifold is contracted, preventing concentration and preserving non-degenerate gradients. We prove two main results: (1) a trainability lower bound for LieTrunc-QNN, and (2) that the Fubini-Study metric rank is bounded by the algebraic span of generators, showing expressivity is governed by structure rather than parameter count. Compact Lie subalgebras also provide inherent robustness to perturbations. Importantly, we establish a polynomial trainability regime where gradient variance decays polynomially instead of exponentially. Experiments (n=2-6) validate the theory: LieTrunc-QNN maintains stable gradients and high effective dimension, while random truncation leads to metric rank collapse. At n=6, full metric rank is preserved (rank=16). Results support a scaling law between gradient variance and effective dimension. This work provides a unified geometric framework for QNN design, linking Lie algebra, manifold geometry, and optimization.