Symmetric cluster expansions with tensor networks

Abstract

Cluster expansions for the exponential of local operators are constructed using tensor networks. In contrast to other approaches, the cluster expansion does not break any spatial or internal symmetries and exhibits a very favorable prefactor to the error scaling versus bond dimension. This is illustrated by time evolving a matrix product state using very large time steps, and by constructing a robust algorithm for finding ground states of two-dimensional Hamiltonians using projected entangled pair states as fixed points of two-dimensional transfer matrices.

Tangent-space methods for truncating uniform MPS

“…This approximation is provided in terms of a quantum circuit, or, more generally, an MPO, and can be obtained from e.g. a Suzuki-Trotter decomposition 6,20,21 or a cluster expansion 16,22 . The resulting MPO is then applied to the current state, encoded as MPS, followed by a bond truncation 1 .…”

“…It is based on ideas developed in the context of tangent space methods for uniform matrix product states 12,13 and the variational uniform matrix product state algorithm 14,15 . Our main motivation is the development of efficient MPS algorithms which can deal with time-evolution methods involving MPOs with large bond dimension and of efficient and wellconditioned ways of contracting PEPS 16 . It also overcomes a main limitation of algorithms based on the timedependent variational principle (TDVP) [17][18][19] , where it is difficult to build up entanglement starting from a lowentangled state by allowing large time steps.…”

Efficient higher-order matrix product operators for time evolution

“…Still, these methods all rely on evolving states by taking small time steps, to the effect that some nonequilibrium properties remain difficult to calculate up to the desired precision without investing a tremendous amount of CPU hours. Recently, a new approach [9] based on cluster expansions was introduced to find tensor network approximations of the time evolution operator that are accurate for much larger time steps, but again this approach is limited to short-range interactions.…”

“…[7] or as an extension of the cluster-expansion approach of Ref. [9] to generic Hamiltonians; the form of the MPO reduces to the one of Ref. [7] when considering the firstorder case.…”

Simulating thermal density operators with cluster expansions and tensor networks

“…In Ref. 12, it was indeed realized that such a cluster expansion can be encoded as a tensor network operator (TNO) with moderate bond dimension, in a way that is size-extensive, preserves all spatial and internal symmetries and works in any dimension. It was shown that such a "cluster TNO" is a very efficient numerical tool for (i) simulating simulating the real-time evolution of a global quench in a 1-D spin chain, or (ii) optimizing a ground-state approximation with a projected entangledpair state.…”