Quantum Phase Recognition via Quantum Attention Mechanism

TL;DR

The paper tackles quantum phase recognition (QPR) in many‑body systems by introducing a hybrid quantum–classical attention mechanism that constructs an attention matrix from swap‑test measurements to capture intrinsic correlations. A parameterized quantum circuit maps ground states and a classical FFN performs phase classification, trained jointly in a variational framework. The approach achieves high accuracy on the cluster‑Ising benchmark with limited labeled data (as few as ~20 training pairs) and reveals phase‑specific attention patterns and an emergent effective correlation length $\xi$, offering interpretable diagnostics of $SPT$, AFM, and paramagnetic phases. This data‑efficient, correlation‑focused method provides a scalable pathway for QPR in complex quantum matter and can be extended to other models and higher dimensions, with caveats about hardware noise and scalability.

Abstract

Quantum phase transitions in many-body systems are fundamentally characterized by complex correlation structures, which pose computational challenges for conventional methods in large systems. To address this, we propose a hybrid quantum-classical attention model. This model uses an attention mechanism, realized through swap tests and a parameterized quantum circuit, to extract correlations within quantum states and perform ground-state classification. Benchmarked on the cluster-Ising model with system sizes of 9 and 15 qubits, the model achieves high classification accuracy with less than 100 training data and demonstrates robustness against variations in the training set. Further analysis reveals that the model successfully captures phase-sensitive features and characteristic physical length scales, offering a scalable and data-efficient approach for quantum phase recognition in complex many-body systems.

Figures (7)

• Figure 1: (a) Over framework of attention model. The input state $\ket{\psi}$ is fed into a trainable parametric quantum circuits (PQC) followed by $n(n-1)/2$ swap tests, yielding the attention matrix. The attention values are then fed into a classical feed-forward neural network (FFN) to produce the final output. (b) The single-layer ansatz on $n$ qubits consist of $R_Y$ and $CR_X$. For $l$ layers of circuit on $n$ qubits, the number of parameterised gates (and parameters) is $4nl$. (c) The phase diagram shows the ground-state expectation values of the string order parameter $\langle S \rangle$ in a cluster-Ising model, as a function of the parameters $h_1$ and $h_2$ for a system of size $N=9$. The white dashed lines mark the boundaries between the symmetry-protected topological (SPT) phase and the paramagnetic and antiferromagnetic phases, respectively.
• Figure 2: Classification accuracy of QCNN and attention-based models with 9 and 15 qubits as a function of the training set size. Each data point is obtained by averaging over 10 independent runs with randomly selected training datasets. Error bars indicate the 95% confidence interval.
• Figure 3: Heatmaps of attention matrices revealing distinct correlation patterns in different quantum phases. The axes correspond to qubit indices, and the color intensity represents the correlation strength. SPT phase: uniformly large matrix elements, indicating nonlocal entanglement. Paramagnetic phase: partially decoupled pattern, with one qubit effectively decoupled from the rest, reflecting dominant local polarization. Antiferromagnetic phase: staggered structure with alternating strong and weak correlations, characteristic of Néel-type order.
• Figure 4: Contrast value $C = (q_\text{near}-q_\text{long})/(q_\text{near}+q_\text{long})$ as a function of the control parameter $h_2$ at fixed $h_1 \approx 0.39$ for (a) 9-qubit and (b) 15-qubit systems. From left to right, the system exhibits the AFM, SPT, and paramagnetic phases. Sharp sign changes of $C$ clearly delineate the phase boundaries, providing an order-parameter-like signature of the quantum phase transitions.
• Figure 5: Effective correlation length $\xi$ from the attention matrix versus $h_2$ at fixed $h_1 \approx 0.39$ for (a) 9-qubit and (b) 15-qubit systems. From left to right, the system exhibits the AFM, SPT, and paramagnetic phases. In the AFM phase, $\xi$ is one order smaller than that in the SPT phase, indicating that while long-range correlations exist, they are less dominant than the non-local topological order in the SPT phase; in the SPT phase, $\xi$ is strongly enhanced and reaches its peak, reflecting nonlocal topological order; in the paramagnetic phase, $\xi$ is in the order of $10$, consistent with localized correlations.