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 $L$ sites, we sample over configurations of an effective lattice, which is made of just $\mathrm{ln}(L)$ 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.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 $L$ sites, we sample over configurations of an effective lattice, which is made of just $\mathrm{ln}(L)$ 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.

arXiv:1201.3975Physical Review BApr 30, 201219 citations

Survey of Tensor Networks

Samuel Desrosiers, Glen Evenbly, Thomas E. Baker

Bulletin of the American Physical SocietyMar 5, 20190 citations

Perfect sampling with unitary tensor networks

Andrew J. Ferris, Guifré Vidal

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.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.

arXiv:1201.3974Physical Review BApr 30, 2012138 citations

Scaling of contraction costs for entanglement renormalization algorithms including tensor Trotterization and variational Monte Carlo

Thomas Barthel, Qiang Miao

Strongly correlated quantum matter, such as the insufficiently understood high- temperature superconductors, are difficult to simulate numerically. The multiscale entanglement renormalization ansatz (MERA) is a hierarchical class of tensor networks that efficiently captures the typical ground-state entanglement structures. Motivated by recent advances to employ MERA in quantum-classical algorithms, the authors take a fresh look here, analyzing new variations of the approach including tensor Trotterization and importance sampling. Optimal contraction sequences and algorithmic phase diagrams show that these quantum-inspired methods can substantially reduce computational costs.Strongly correlated quantum matter, such as the insufficiently understood high- temperature superconductors, are difficult to simulate numerically. The multiscale entanglement renormalization ansatz (MERA) is a hierarchical class of tensor networks that efficiently captures the typical ground-state entanglement structures. Motivated by recent advances to employ MERA in quantum-classical algorithms, the authors take a fresh look here, analyzing new variations of the approach including tensor Trotterization and importance sampling. Optimal contraction sequences and algorithmic phase diagrams show that these quantum-inspired methods can substantially reduce computational costs.

arXiv:2407.21006Physical review. B./Physical review. BJan 2, 20252 citations

L

sites, we sample over configurations of an effective lattice, which is made of just

\mathrm{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.