Unifying time evolution and optimization with matrix product states

TL;DR

This paper introduces a new unifying framework for time-evolution and optimization in matrix product states using the time-dependent variational principle, enabling efficient simulation of arbitrary Hamiltonians.

Contribution

A novel integration scheme based on splitting the projector onto the tangent space, unifying time evolution and optimization methods for matrix product states.

Findings

Compatible with arbitrary Hamiltonians, including long-range interactions.

Resembles DMRG algorithm, inheriting its stability and efficiency.

Can be implemented with minimal code changes from existing DMRG implementations.

Abstract

We show that the time-dependent variational principle provides a unifying framework for time-evolution methods and optimisation methods in the context of matrix product states. In particular, we introduce a new integration scheme for studying time-evolution, which can cope with arbitrary Hamiltonians, including those with long-range interactions. Rather than a Suzuki-Trotter splitting of the Hamiltonian, which is the idea behind the adaptive time-dependent density matrix renormalization group method or time-evolving block decimation, our method is based on splitting the projector onto the matrix product state tangent space as it appears in the Dirac-Frenkel time-dependent variational principle. We discuss how the resulting algorithm resembles the density matrix renormalization group (DMRG) algorithm for finding ground states so closely that it can be implemented by changing just a few lines of code and it inherits the same stability and efficiency. In particular, our method is compatible with any Hamiltonian for which DMRG can be implemented efficiently and DMRG is obtained as a special case of imaginary time evolution with infinite time step.