Learning Variational Quantum Circuit Parameters with Classical Artificial Intelligence for Quantum Phase Transition Detection

TL;DR

The paper tackles the challenge of detecting quantum phase transitions by learning from variational quantum circuit (VQE) parameters rather than quantum states. It introduces a classical AI framework that integrates an attention mechanism with a variational autoencoder to uncover nonlocal correlations among circuit parameters, enabling unsupervised phase recognition and a data-driven generalized order parameter. Through TFIM and Cluster-Ising model experiments, the authors show that latent representations reveal phase structure and support phase-diagram reconstruction, even when VQE optimizes to local minima. They further validate the latent space with a conditional diffusion model for generating quantum states conditioned on identified phases, highlighting robustness and practical applicability on NISQ hardware. Overall, the work provides a hardware-friendly pathway for QPT analysis that leverages learned parameter distributions to extract physical insights without full state tomography or observable measurements.

Abstract

Learning many-body quantum states and quantum phase transitions remains a major challenge in quantum many-body physics. Classical machine learning methods offer certain advantages in addressing these difficulties. In this work, we propose a novel framework that bypasses the need to measure physical observables by directly learning the parameters of parameterized quantum circuits. By integrating the attention mechanism from large language models (LLMs) with a variational autoencoder (VAE), we efficiently capture hidden correlations within the circuit parameters. These correlations allow us to extract information about quantum phase transitions in an unsupervised manner. Moreover, our VAE acts as a classical representation of parameterized quantum circuits and the corresponding many-body quantum states, enabling the efficient generation of quantum states associated with specific phases. We apply our framework to a variety of quantum systems and demonstrate its broad applicability, with particularly strong performance in identifying topological quantum phase transitions.

Figures (18)

• Figure 1: Schematic of VQE energy landscapes and endpoints for different parameters $\boldsymbol{x}$. A three-dimensional landscape is used to represent the energy landscape surface in high-dimensional parameter space. Different physical parameters $\boldsymbol{x}$ correspond to local minima under different energy landscapes. The final converged local minima are highlighted with red dots.
• Figure 2: Schematic diagram of the unsupervised quantum phase recognition framework. The arrow VQE indicates VQE optimization under the same initialization for all physical parameters $\boldsymbol{x}$. The arrow NN indicates the training of a classical neural network model on the parameters $\boldsymbol{\theta}^*(\boldsymbol{x})$ obtained from VQE optimization.
• Figure 3: Overview of the VAE framework. The encoder (left) processes input parameters through 1D-CNN, MHSA, and MLP modules to produce the parameters of a latent Gaussian distribution. The decoder (right) reconstructs parameters from latent samples using a mirrored deconvolutional architecture.
• Figure 4: VQE ansatz circuit diagram for solving TFIM ground states under different physical parameters. The parameterized circuit consists of alternating single-qubit $R_x, R_z$ gates and parity-interleaved $\text{CZ}$ gates.
• Figure 5: Distribution of 2001 TFIM samples with $N=12, p=14$ in the VAE latent space for $h \in [0, 2.0]$.