Variational Monte Carlo (VMC) with row-update Projected Entangled-Pair States (PEPS) and its applications in quantum spin glasses

Abstract

Solving the quantum many-body ground state problem remains a central challenge in computational physics. In this context, the Variational Monte Carlo (VMC) framework based on Projected Entangled Pair States (PEPS) has witnessed rapid development, establishing itself as a vital approach for investigating strongly correlated two-dimensional systems. However, standard PEPS-VMC algorithms predominantly rely on sequential local updates. This conventional approach often suffers from slow convergence and critical slowing down, particularly in the vicinity of phase transitions or within frustrated landscapes. To address these limitations, we propose an efficient autoregressive row-wise sampling algorithm for PEPS that enables direct, rejection-free sampling via single-layer contractions. By utilizing autoregressive single-layer row updates to generate collective, non-local configuration proposals, our method significantly reduces temporal correlations compared to local Metropolis moves. We benchmark the algorithm on the two-dimensional transverse-field Ising model and the quantum spin glass. Our results demonstrate that the row-wise scheme effectively mitigates critical slowing down near the Ising critical point. Furthermore, in the rugged landscape of the quantum spin glass, it yields improved optimization stability and lower ground-state energies. These findings indicate that single-layer autoregressive row updates provide a flexible and robust improvement to local PEPS-VMC sampling and may serve as a basis for more advanced sampling schemes.

Figures (4)

• Figure 1: Schematic illustration of the single-layer autoregressive row-update algorithm. (a) The projected entangled pair state (PEPS) ansatz representing the quantum state $|\Psi\rangle$. (b) Construction of the effective environment for the target row $y$. The tensor network is partitioned, and the sub-networks above and below row $y$ are contracted into boundary matrix product states (MPS) with bond dimension $\chi$. (c1)-(c3) Sequential autoregressive sampling along row $y$. The conditional probability for spin $s_{y,i}$ is derived by contracting the local tensor with the environment comprising the boundary MPS and the previously sampled spins $\mathbf{s}_{y,<i}$. This procedure yields the marginal partition function (rightmost panel), enabling direct sampling without Metropolis rejection.
• Figure 2: Markov chain equilibration dynamics initialized from a random high-energy state. The panels display the relaxation of the variational energy toward the pre-optimized ground state for (a) the 2D TFIM at the critical point $\Gamma_c \approx 3.044J$ and (b) the 2D QSG at $\Gamma=1.0J$. Main panels: Equilibration step $\tau$ as a function of system size $L$. Insets: Energy trajectories for the largest system sizes ($L=20$ for TFIM, $L=10$ for QSG).
• Figure 3: Convergence analysis of variational optimization across independent realizations. The evolution of the mean variational energy and its standard deviation (shaded regions) is shown for (a) $L=10$ (100 independent seeds) and (b) $L=16$ (20 independent seeds) QSG lattices. Both the pure Autoregressive Row-Update and the Hybrid methods demonstrate superior convergence properties, achieving significantly lower asymptotic energies with minimal variance compared to the Metropolis benchmark, which exhibits strong initialization dependence.
• Figure 4: Distribution of variational energies obtained after 5000 optimization steps for the QSG model: (a) $L=10$ and (b) $L=16$ systems. Histograms show the final energies from independent optimization runs, with the vertical dashed line indicating the mean value. Compared to the Metropolis sampler, the autoregressive row-update and hybrid methods produce narrower energy distributions with lower mean and minimum energies.