Replica Tensor Train

TL;DR

The paper introduces Replica Tensor Train, a novel tensor network method combining quantum Monte Carlo, capable of capturing volume-law entanglement, and applicable to ground-state calculations of local Hamiltonians.

Contribution

It presents a new tensor network ansatz that can handle volume-law entanglement and integrates Monte Carlo sampling for observable computation, avoiding gradient descent.

Findings

Successfully applied to a 2D spin model with low computational cost.

Enables algebraic ground-state search without gradient descent.

Extends to Krylov-subspace methods within the variational manifold.

Abstract

We describe a numerical many-body technique that is based on both tensor networks and quantum Monte Carlo. The variational ansatz is a tensor network that can harvest volume-law entanglement. It is constructed from a tensor train to which one applies a set of non-local operators that force several indices of the tensor train to represent the same physical index, hence its name -- replica tensor train (RTT). From the tensor network toolbox, it inherits the possibility to make linear combinations of these states and apply a certain class of operators. We can therefore find the ground-state of a local Hamiltonian in a purely algebraic way as in standard tensor network algorithms -- i.e. without using gradient descent methods. On the other hand, the volume-law structure forbids calculating physical observables directly. In much the same way as on a quantum computer where one can prepare a state but can only sample it at the end, here we have to use Markov Chain Monte Carlo to compute the observables. We further show that the approach can be extended to build Krylov-subspace ground-state methods within the variational manifold. We illustrate the different algorithms on a two-dimensional spin model with a transverse magnetic field, which can be solved by this approach at low computational cost.