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arXiv:1201.3974 · Physical Review B

Perfect sampling with unitary tensor networks

Andrew J. Ferris, Guifré Vidal

Published Apr 30, 2012• 138 citations
arXivHTMLPDF

Abstract

Tensor network states are powerful variational Ans\"atze for many-body ground states of quantum lattice models. The use of Monte Carlo sampling techniques in tensor network approaches significantly reduces the cost of tensor contractions, potentially leading to a substantial increase in computational efficiency. Previous proposals are based on a Markov chain Monte Carlo scheme generated by locally updating configurations and, as such, must deal with equilibration and autocorrelation times, which result in a reduction of efficiency. Here we propose perfect sampling schemes, with vanishing equilibration and autocorrelation times, for unitary tensor networks, namely, tensor networks based on efficiently contractible, unitary quantum circuits, such as unitary versions of the matrix product state (MPS) and tree tensor network (TTN), and the multiscale entanglement renormalization Ansatz (MERA). Configurations are directly sampled according to their probabilities in the wave function, without resorting to a Markov chain process. We consider both complete sampling, involving all the relevant sites of the system, and incomplete sampling, which only involves a subset of those sites and which can result in a dramatic (basis-dependent) reduction of sampling error.

DOI
10.1103/physrevb.85.165146

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Tensor network states are powerful variational Ans"atze for many-body ground states of quantum lattice models. The use of Monte Carlo sampling techniques in tensor network approaches significantly reduces the cost of tensor contractions, potentially leading to a substantial increase in computational efficiency. Previous proposals are based on a Markov chain Monte Carlo scheme generated by locally updating configurations and, as such, must deal with equilibration and autocorrelation times, which result in a reduction of efficiency. Here we propose perfect sampling schemes, with vanishing equilibration and autocorrelation times, for unitary tensor networks, namely, tensor networks based on efficiently contractible, unitary quantum circuits, such as unitary versions of the matrix product state (MPS) and tree tensor network (TTN), and the multiscale entanglement renormalization Ansatz (MERA). Configurations are directly sampled according to their probabilities in the wave function, without resorting to a Markov chain process. We consider both complete sampling, involving all the relevant sites of the system, and incomplete sampling, which only involves a subset of those sites and which can result in a dramatic (basis-dependent) reduction of sampling error.
Monte Carlo sampling techniques have been proposed as a strategy to reduce the computational cost of contractions in tensor network approaches to solving many-body systems. Here, we put forward a variational Monte Carlo approach for the multiscale entanglement renormalization ansatz (MERA), which is a unitary tensor network. Two major adjustments are required compared to previous proposals with nonunitary tensor networks. First, instead of sampling over configurations of the original lattice, made of LLL sites, we sample over configurations of an effective lattice, which is made of just ln(L)\mathrm{ln}(L)ln(L) sites. Second, the optimization of unitary tensors must account for their unitary character while being robust to statistical noise, which we accomplish with a modified steepest descent method within the set of unitary tensors. We demonstrate the performance of the variational Monte Carlo MERA approach in the relatively simple context of a finite quantum spin chain at criticality, and discuss future, more challenging applications, including two-dimensional systems.
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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.
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