[Paper Review] Finding Quantum Critical Points with Neural-Network Quantum States

Remmy Zen, Long D. H. My+6 more|arXiv (Cornell University)|Feb 7, 2020

Quantum many-body systems73 references4 citations

TL;DR

This paper proposes a novel method to locate quantum critical points in the quantum Ising model using analytically constructed restricted Boltzmann machine (RBM) neural-network quantum states, transfer learning, and unsupervised learning. The approach achieves higher efficiency and accuracy than cold-start training and traditional methods, with critical point estimates converging toward known values in one-, two-, and three-dimensional systems, especially in 2D where the result (0.302) closely matches the quantum Monte Carlo benchmark (0.32847).

ABSTRACT

Finding the precise location of quantum critical points is of particular importance to characterise quantum many-body systems at zero temperature. However, quantum many-body systems are notoriously hard to study because the dimension of their Hilbert space increases exponentially with their size. Recently, machine learning tools known as neural-network quantum states have been shown to effectively and efficiently simulate quantum many-body systems. We present an approach to finding the quantum critical points of the quantum Ising model using neural-network quantum states, analytically constructed innate restricted Boltzmann machines, transfer learning and unsupervised learning. We validate the approach and evaluate its efficiency and effectiveness in comparison with other traditional approaches.

Motivation & Objective

• To develop an efficient and accurate method for locating quantum critical points in quantum many-body systems, particularly in the quantum Ising model.
• To overcome the exponential scaling of Hilbert space dimension by leveraging machine learning techniques such as neural-network quantum states.
• To improve upon cold-start training by introducing innate knowledge into RBM-based quantum states through analytical construction.
• To validate the method’s effectiveness and efficiency against traditional approaches and established benchmarks in one-, two-, and three-dimensional systems.

Proposed method

• Construct innate restricted Boltzmann machine (RBM) neural-network quantum states by analytically encoding known physical properties of the system in each phase, embedding prior knowledge directly into the architecture.
• Apply transfer learning across different system parameters to refine critical point estimates, initializing from pre-trained states rather than random weights.
• Use unsupervised learning to optimize the RBM parameters by minimizing the energy variance and identifying phase transitions via order parameter inflection points.
• Employ finite-size scaling to extrapolate critical points to the thermodynamic limit by fitting the inflection points of order parameters (e.g., ferromagnetic and antiferromagnetic magnetization) as a function of system size.
• Compare results with tensor network methods and quantum Monte Carlo benchmarks to evaluate accuracy and convergence speed.
• Use the ferromagnetic and antiferromagnetic magnetization order parameters, as well as correlation functions, to detect phase transitions and assess the method’s robustness across dimensions.

Experimental results

Research questions

• RQ1 Can analytically constructed RBM neural-network quantum states improve the accuracy and efficiency of quantum critical point detection compared to randomly initialized models?
• RQ2 How effective is transfer learning in reducing training time and improving convergence to the true critical point in quantum Ising models across different dimensions?
• RQ3 To what extent do the critical point estimates from the proposed method converge toward established benchmarks (e.g., quantum Monte Carlo values) in one-, two-, and three-dimensional systems?
• RQ4 How do different order parameters (magnetization, correlation functions) affect the identification of phase transitions using this approach?
• RQ5 Can the method be generalized to other quantum many-body systems beyond the Ising model?

Key findings

• The proposed method with innate RBM-NQS and transfer learning achieves higher accuracy and faster convergence than cold-start training, particularly in estimating the critical point in two-dimensional systems.
• In two dimensions, the method estimates the critical point at $ J/|h| = 0.302 $, which is very close to the quantum Monte Carlo benchmark of $ 0.32847 $, indicating strong convergence.
• In one-dimensional systems, the method correctly identifies the critical point at $ J/|h| = 1 $, consistent with exact results.
• In three-dimensional systems, the estimate of $ J/|h| = 0.256 $ is sizeably different from the quantum Monte Carlo value of $ 0.1887 $, suggesting that larger system sizes are needed for accurate estimation.
• The antiferromagnetic magnetization order parameter yields a critical point estimate of $ -0.266 $, which is closer to the benchmark $ -0.32847 $ than the ferromagnetic case in 2D, though still not fully converged.
• For one-dimensional systems, the tensor network method outperforms RBM-NQS-ITT in correlation-based order parameter estimation, indicating potential limitations in correlation capture for certain observables.