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arXiv:2603.27532

Phase transitions in parametrized quantum circuits

Xiaoyang Wang, Han Xu, Lukas Broers, Tomonori Shirakawa, Seiji Yunoki

quant-phSubmitted Mar 29, 2026• Announced Mar 31, 2026• 0 citations
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Abstract

Phase transitions are among the most intriguing phenomena in physical systems, yet their behavior near criticality remain challenging to study using classical algorithms. Parameterized quantum circuits (PQCs) offer a promising approach to investigating such regimes on practical quantum computers. However, in order to use it to probe critical behavior, a PQC itself should be non-trivial and exhibit a phase transition and non-analyticity -- a property that has not yet been clearly identified. In this work, we identify a mechanism for generating non-analyticities intrinsically in PQCs. As a concrete realization, we construct a class of sequential PQCs whose observable expectation value is a non-analytic function of the circuit parameter in the infinite volume limit, showing that the prepared PQC states undergo a phase transition at the non-analytic points. The entanglement and the identified order parameter have distinct behaviors in different phases, revealing a phase diagram of the PQC state. We show that classical simulation of this PQC based on tensor networks and Pauli propagation gets less efficient in the vicinity of the phase transition point, indicating a physically motivated route towards practical quantum advantage using PQCs with phase transitions.

Comments
10 pages, 7 figures

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Phase transitions are among the most intriguing phenomena in physical systems, yet their behavior near criticality remain challenging to study using classical algorithms. Parameterized quantum circuits (PQCs) offer a promising approach to investigating such regimes on practical quantum computers. However, in order to use it to probe critical behavior, a PQC itself should be non-trivial and exhibit a phase transition and non-analyticity -- a property that has not yet been clearly identified. In this work, we identify a mechanism for generating non-analyticities intrinsically in PQCs. As a concrete realization, we construct a class of sequential PQCs whose observable expectation value is a non-analytic function of the circuit parameter in the infinite volume limit, showing that the prepared PQC states undergo a phase transition at the non-analytic points. The entanglement and the identified order parameter have distinct behaviors in different phases, revealing a phase diagram of the PQC state. We show that classical simulation of this PQC based on tensor networks and Pauli propagation gets less efficient in the vicinity of the phase transition point, indicating a physically motivated route towards practical quantum advantage using PQCs with phase transitions.
Efficient preparation of many-body ground states is key to harnessing the power of quantum computers in studying quantum many-body systems. In this work, we propose a simple method to design exact linear-depth parameterized quantum circuits which prepare a family of ground states across topological quantum phase transitions in 2D. We achieve this by constructing ground states represented by isometric tensor networks (isoTNS), which form a subclass of tensor network states that are efficiently preparable. By continuously tuning a parameter in the wavefunction, the many-body ground state undergoes quantum phase transitions, exhibiting distinct 2D quantum phases. We illustrate this by constructing an isoTNS path with bond dimension D = 2 interpolating between distinct symmetry-enriched topological (SET) phases. At the transition point, the wavefunction is related to a gapless point in the classical six-vertex model. Furthermore, the critical wavefunction supports a power-law correlation along one spatial direction while remaining long-range ordered in the other spatial direction. We provide an explicit parametrized local quantum circuit for the path and show that the 2D isoTNS can also be efficiently simulated by a holographic quantum algorithm requiring only an 1D array of qubits. Published by the American Physical Society 2024
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.
Understanding quantum phase transitions in physical systems is fundamental to characterize their behavior at low temperatures. Achieving this requires both accessing good approximations to the ground state and identifying order parameters to distinguish different phases. Addressing these challenges, our work introduces a hybrid algorithm that combines quantum optimization with classical machine learning. This approach leverages the capability of near-term quantum computers to prepare locally trapped states through finite optimization. Specifically, we apply LASSO for identifying conventional phase transitions and the Transformer model for topological transitions, utilizing these with a sliding window scan of Hamiltonian parameters to learn appropriate order parameters and locate critical points. We validated the method with numerical simulations and real-hardware experiments on Rigetti's Ankaa 9Q-1 quantum computer. This protocol provides a framework for investigating quantum phase transitions with shallow circuits, offering enhanced efficiency and, in some settings, higher precision-thus contributing to the broader effort to integrate near-term quantum computing and machine learning.
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Whether parameterized quantum circuits (PQCs) can be systematically constructed to be both trainable and expressive remains an open question. Highly expressive PQCs often exhibit barren plateaus, while several trainable alternatives admit efficient classical simulation. We address this question by deriving a finite-sample, dimension-independent concentration bound for estimating the variance of a PQC cost function, yielding explicit trainability guarantees. Across commonly used ansätze, we observe an anticorrelation between trainability and expressibility, consistent with theoretical insights. Building on this observation, we propose a property-based ansatz-search framework for identifying circuits that combine trainability and expressibility. We demonstrate its practical viability on a real quantum computer and apply it to variational quantum algorithms. We identify quantum neural network ansätze with improved effective dimension using over 6×6 \times6× fewer parameters, and for VQE on H2\mathrm{H}_2H2​ we achieve UCCSD-like accuracy at substantially reduced circuit complexity.
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.