Variational Monte Carlo with the multiscale entanglement renormalization ansatz

Abstract

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 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.

The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems

“…Noteworthy classes of tensor networks that have been proposed towards the quantum many-body problem are: Projected Entangled Pair States (PEPS) [61,62], weighted graph states [63], the Multiscale Entanglement Renormalization Ansatz (MERA) [64][65][66][67], Branching MERA [68,69], Tree Tensor Networks (TTN) [70][71][72][73][74][75], entangled-plaquette states [76], string-bond states [77], tensor networks with graph enhancement [78,79] and continuous MPS [80]. Additionally, various proposals have been put forward to embed known stochastic variational techniques, such as Monte Carlo, into the tensor network framework [76,[81][82][83][84][85][86], to connect tensor networks to quantum error correction techniques [87], and to construct direct relations between few-body and many-body quantum systems [88]. The numerical tools we will introduce in the next two sections indeed apply to most of these TN classes.…”

Tree tensor networks for generative modeling

“…Naturally, a further development to the current work is by introducing the structure of multi-scale entanglement renormalization ansatz (MERA) 44,45 , another type of tensor network we can expect to have tractable partition function while hopefully being able to preserve better the long range dependences in the data.…”

Perfect sampling with unitary tensor networks

“…(21)- (28). Importantly, all these tensor networks can be contracted with a cost that scales as O(χ 2 ) with the bond dimension χ , and they are therefore computationally less expensive than an exact contraction, which has cost O(χ 3 ).…”

“…(21)- (28) can be found in Ref. 28. A key point is that, for unitary tensor networks such as uMPS, uTTN, and MERA, the computational cost of generating the above sequence of density matrices and probabilities often does not exceed (to leading order in χ and effective size L C ) the cost of a single sweep in Markov chain Monte Carlo.…”