Hybrid quantum-classical neural network for sample-efficient recognition of topological phases

Abstract: With increasing maturity of quantum computers, standard methods for characterizing global properties of their output quantum states via direct measurements and classical post-processing are becoming increasingly impractical due to large measurement costs. Although quantum neural networks could directly process quantum states to identify underlying characteristics with reduced measurement efforts, they often require deep quantum circuits that cannot be implemented on existing devices. To overcome these challenges, we introduce a hybrid quantum-classical neural network that consists of a shallow parameterized quantum circuit, measurements, and a classical neural network. The parameterized quantum circuit performs a nonlocal transformation of the measurement basis, which is jointly trained with the classical neural network to maximize the statistical distance between data obtained by measuring different quantum states. Using supervised learning, we demonstrate that the hybrid neural network distinguishes the topological phase of the surface code from a symmetry-enriched topological phase and random product states. Moreover, this hybrid neural network reduces both inference and training sample complexities of recognizing the topological phase by approximately one order of magnitude compared to a classical neural network trained on randomized Pauli measurements. As this hybrid neural network features a shallow quantum circuit that can be readily implemented on existing quantum computers, it enables the efficient characterization of complex quantum states.

• The paper introduces a hybrid quantum-classical neural network that significantly reduces sample complexity for recognizing topological phases.
• It leverages shallow parameterized quantum circuits paired with a classical neural network to enhance the distinguishability of quantum states.
• The framework achieves inference errors below 0.01% with far fewer measurements compared to traditional classical methods.

Hybrid Quantum-Classical Neural Network for Topological Phase Recognition

Introduction

The paper "Hybrid quantum-classical neural network for sample-efficient recognition of topological phases" (2606.28199) addresses the challenge of characterizing quantum many-body states with nonlocal order, in particular the identification of topological phases, using near-term quantum devices. Existing classical approaches based on direct measurement and post-processing demonstrate exponential scaling with respect to sample complexity, especially for global observables in nontrivial quantum phases, making them intractable as system size increases. While quantum neural networks (QNNs) are theoretically able to process quantum states to reduce sample demands, their deep circuit requirements typically exceed present quantum hardware capabilities. This work presents a hybrid quantum-classical architecture that leverages shallow parameterized quantum circuits integrated with classical neural networks, achieving an order-of-magnitude reduction in training and inference sample complexity over purely classical methods.

Architecture and Training Paradigm

The hybrid model consists of three stages: quantum state preparation, a shallow parameterized quantum circuit, projective measurements, and a classical feed-forward neural network acting on the measurement data. The quantum circuit is specifically engineered to maximize statistical distinguishability (measured via Jensen-Shannon divergence) between classes of quantum states, tailored towards phase recognition tasks. The classical network, with a non-trivial representational capacity, is trained to serve as a near-optimal Bayesian classifier given the distribution processed by the quantum device.

Figure 1: A schematic of the hybrid neural network: lattice-encoded qubit topology, parameterized quantum circuit layers, and measurement-driven classical post-processing.

The training utilizes a two-loop optimization: an inner loop for supervised classical network fitting (with cross-entropy loss) on the projective outcomes, and an outer loop for variational quantum circuit parameter update by minimizing the residual classification cost. Circuit optimization is accomplished via evolution strategies, compatible with gradient-free quantum hardware.

Statistical Working Principles

The classical neural subnetwork, provided sufficient expressivity, converges to an optimal estimator for the posterior class probability given the observed bitstring, achieving classification error tightly bounded by the statistical overlap of measurement distributions from distinct quantum phases. The quantum circuit acts to variationally transform the input state such that the measurement-induced probability distributions (for different phases) become maximally distinguishable, thus lowering the intrinsic Bayes error.

Figure 2: Probability distributions for bitstrings: overlap (gray) is pronounced with random quantum circuit parameters but minimized after variational training, amplifying classification margin.

The combined system therefore realizes a synergistic quantum-classical learning flow—quantum preprocessing tailors the informational content of measurement data, and the classical network converts optimized distributions into sharp phase discrimination.

Results: Surface Code and Symmetry-Enriched Topological Phase Learning

The core benchmarks involve learning to distinguish between:

1. Topological ground states of the surface code (with $\mathbb{Z}_2$ topological order) and a featureless ensemble (1-design) of random product states, which is globally maximally mixed over local measurements.
2. Surface code states vs. symmetry-enriched topological (SET) states, derived from a parameterized family of tensor network ground states [Liu et al.].

The quantum circuits are constructed either as parameterized inverses of known preparation unitaries (c.f., toric/surface code) or as generalizations thereof, with depths scaling as $\mathcal{O}(\sqrt{N})$ for up to $N=25$ qubits.

Numerically, the trained hybrid model achieves inference error rates below $0.01\%$ with as few as $5$–$13$ measurement shots per test state for both phase boundaries and unseen $g$ parameters. In comparison, classical neural networks (with randomized Pauli-6 POVM measurement inputs) require over ten times as many measurements to achieve comparable performance.

Figure 3: Multi-shot inference errors drop below $0.01\%$ after only a handful of measurement shots with the hybrid model.

When distinguishing SET phases (nonlocal, both lacking local order parameters), the hybrid model again demonstrates superior statistical efficiency. The quantum circuit learns basis rotations that enhance the discriminative features necessary for the classical network to resolve non-local topological data.

Figure 4: Classification output as a function of the interpolating parameter $g$: test and training data smoothly interpolate phase boundaries with an unambiguous decision margin.

Sample Complexity Benchmark and Scalability

The hybrid architecture systematically outperforms state-of-the-art classical post-processing over randomized measurements regarding sample efficiency. For system sizes up to $N=25$, the hybrid framework attains near-perfect classification ($\mathcal{O}(\sqrt{N})$0 error) with a training shot budget $\mathcal{O}(\sqrt{N})$1, whereas classical approaches require $\mathcal{O}(\sqrt{N})$2 or higher.

Figure 5: Multi-shot error rates as a function of training sample budget for various system sizes and inference shot numbers; the hybrid model maintains consistently lower sample complexity across scales.

Figure 6: Sample complexity comparison for SET phase models; hybrid neural network achieves target accuracy at an order of magnitude lower cost than the classical benchmark.

This reduction is robust across system size (tested for $\mathcal{O}(\sqrt{N})$3) and model expressivity parameters. The advantage is realized in both training and inference phases, attributable primarily to the quantum circuit’s role in concentrating nonlocal information into measurement-accessible features.

Implications and Future Directions

This work establishes a scalable, implementable protocol for topological phase detection on NISQ devices. Practically, it enables significantly more efficient certification and investigation of complex quantum matter, directly using quantum hardware, by minimizing measurement overhead and classical post-processing. The methodology is generalizable to other global property measurements where classical shadows or randomized measurement schemes are exponentially inefficient.

From a theoretical standpoint, the architecture’s trainability—entirely data-driven circuit learning without explicit analytic circuit design—suggests promise for broader hybrid quantum-classical ML workflows operating on data of authentic quantum complexity. Further optimization of quantum circuit expressivity vs. depth trade-offs, e.g., hardware-efficient ansätze, and integration with nonlinear classical post-processing remain open for future exploration. Extension to nonlinear quantum observables (requiring higher-order classical shadows or t-designs), robustness under noise and decoherence, and experimental validations are critical next steps.

Conclusion

The hybrid quantum-classical neural network introduced in this work presents a concrete advancement in quantum-assisted phase recognition, bridging current quantum device capabilities with demanding condensed matter learning tasks. By coupling shallow parameterized quantum circuits with expressive classical neural networks, the method surmounts the measurement bottleneck inherent in classical statistical schemes. The reduction of both inference and training sample complexity by an order of magnitude enables efficient characterization of topological quantum phases, positioning hybrid architectures as essential components of NISQ-era quantum information processing.