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arXiv:2109.07356 · Physical review. B./Physical review. B

Direct sampling of projected entangled-pair states

Tom Vieijra, Jutho Haegeman, Frank Verstraete, Laurens Vanderstraeten

Published Dec 21, 2021• 37 citations
arXivHTMLPDF

Abstract

Variational Monte Carlo studies employing projected entangled-pair states (PEPS) have recently shown that they can provide answers to long-standing questions such as the nature of the phases in the two-dimensional ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model. The sampling in these Monte Carlo algorithms is typically performed with Markov chain Monte Carlo algorithms employing local update rules, which often suffer from long autocorrelation times and interdependent samples. We propose a sampling algorithm that generates independent samples from a PEPS, bypassing all problems related to finite autocorrelation times. This algorithm is a generalization of an existing direct sampling algorithm for unitary tensor networks. We introduce an auxiliary probability distribution from which independent samples can be drawn, and combine it with importance sampling in order to evaluate expectation values accurately. We benchmark our algorithm on the classical Ising model and on variational optimization of two-dimensional quantum spin models.

DOI
10.1103/physrevb.104.235141

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Variational Monte Carlo studies employing projected entangled-pair states (PEPS) have recently shown that they can provide answers to long-standing questions such as the nature of the phases in the two-dimensional J1\ensuremath-J2{J}_{1}\text{\ensuremath{-}}{J}_{2}J1​\ensuremath-J2​ model. The sampling in these Monte Carlo algorithms is typically performed with Markov chain Monte Carlo algorithms employing local update rules, which often suffer from long autocorrelation times and interdependent samples. We propose a sampling algorithm that generates independent samples from a PEPS, bypassing all problems related to finite autocorrelation times. This algorithm is a generalization of an existing direct sampling algorithm for unitary tensor networks. We introduce an auxiliary probability distribution from which independent samples can be drawn, and combine it with importance sampling in order to evaluate expectation values accurately. We benchmark our algorithm on the classical Ising model and on variational optimization of two-dimensional quantum spin models.
Based on the scheme of variational Monte Carlo sampling, we develop an accurate and efficient two-dimensional tensor-network algorithm to simulate quantum lattice models. We find that Monte Carlo sampling shows huge advantages in dealing with finite projected entangled pair states, which allows significantly enlarged system size and improves the accuracy of tensor network simulation. We demonstrate our method on the square-lattice antiferromagnetic Heisenberg model up to $32\ifmmode\times\else\texttimes\fi{}32$ sites, as well as a highly frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ model up to $24\ifmmode\times\else\texttimes\fi{}24$ sites. The results, including ground state energy and spin correlations, are in excellent agreement with those of the available quantum Monte Carlo or density matrix renormalization group methods. Therefore, our method substantially advances the calculation of 2D tensor networks for finite systems, and potentially opens a new door towards resolving many challenging strongly correlated quantum many-body problems.
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.
Projected entangled pair states (PEPS) methods have been proven to be powerful tools to solve strongly correlated quantum many-body problems in two dimensions. However, due to the high computational scaling with the virtual bond dimension DDD, in a practical application, PEPS are often limited to rather small bond dimensions, which may not be large enough for some highly entangled systems, for instance, frustrated systems. Optimization of the ground state using the imaginary time evolution method with a simple update scheme may go to a larger bond dimension. However, the accuracy of the rough approximation to the environment of the local tensors is questionable. Here, we demonstrate that by combining the imaginary time evolution method with a simple update, Monte Carlo sampling techniques and gradient optimization will offer an efficient method to calculate the PEPS ground state. By taking advantage of massive parallel computing, we can study quantum systems with larger bond dimensions up to D=10D=10D=10 without resorting to any symmetry. Benchmark tests of the method on the ${J}_{1}\ensuremath{-}{J}_{2}$ model give impressive accuracy compared with exact results.
Quantum annealing with the D-Wave Advantage system in the random Ising model on a cubic lattice is simulated using a three-dimensional (3D) tensor network. The Hamiltonian is driven across a quantum phase transition from a paramagnetic phase to a spin-glass phase. The network is represented as a tensor product state, also known-particularly in two dimensions-as a projected entangled-pair state (PEPS). The annealing procedure is repeated for a range of annealing times in order to test the Kibble-Zurek (KZ) power law governing the residual energy at the end of the annealing ramp. For an infinite lattice with periodic nearest-neighbor random Ising couplings, the final energy is evaluated using a deterministic method. For a finite lattice with open boundaries, we introduce a more efficient Monte Carlo sampling approach. In both cases, the residual energy as a function of annealing time approaches the KZ power law as the annealing time increases.
Abstract In recent years, variational Monte Carlo (VMC) calculations of projected entangled pair states (PEPS) have emerged as a competitive method for computing the ground states of many-body quantum systems. This approach is particularly important for fermionic systems, where sign problems are prevalent. We derive and explain the algorithms for VMC calculations of fermionic PEPS within the swap-gates formulation. As a separate key result, we prove the detailed balance for sequential sampling of tensor networks.
Projected entangled pair states (PEPS) provide an efficient variational ansatz for two-dimensional quantum phases, but computing observables remains challenging because PEPS contraction is generally costly. Here, we parameterize two-dimensional quantum states using variational PEPS subject to isometric constraints and map the resulting ansatz onto monitored quantum circuits, replacing tensor-network contraction with circuit sampling. For infinite cylinders, the transfer matrix defines a quantum channel on the virtual boundary. We use a fixed-point treatment and a monitored-circuit unraveling of this channel to evaluate observables efficiently. Using a constant number of variational parameters and a number of qubits that scales only with the cylinder width, our method yields a phase diagram for the J1J_1J1​-J2J_2J2​ model in qualitative agreement with DMRG results. Because the monitored circuits are compatible with near-term quantum hardware, this approach provides a hybrid quantum-classical framework for simulating two-dimensional quantum many-body systems.