SciPost logo

SciPost Submission Page

State Diagrams to determine Tree Tensor Network Operators

by Richard M. Milbradt, Qunsheng Huang, Christian B. Mendl

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Richard Milbradt
Submission information
Preprint Link:  (pdf)
Date accepted: 2024-05-30
Date submitted: 2024-05-22 10:22
Submitted by: Milbradt, Richard
Submitted to: SciPost Physics Core
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Computational
  • Quantum Physics
Approaches: Theoretical, Computational


This work is concerned with tree tensor network operators (TTNOs) for representing quantum Hamiltonians. We first establish a mathematical framework connecting tree topologies with state diagrams. Based on these, we devise an algorithm for constructing a TTNO given a Hamiltonian. The algorithm exploits the tensor product structure of the Hamiltonian to add paths to a state diagram, while combining local operators if possible. We test the capabilities of our algorithm on random Hamiltonians for a given tree structure. Additionally, we construct explicit TTNOs for nearest neighbour interactions on a tree topology. Furthermore, we derive a bound on the bond dimension of tensor operators representing arbitrary interactions on trees. Finally, we consider an open quantum system in the form of a Heisenberg spin chain coupled to bosonic bath sites as a concrete example. We find that tree structures allow for lower bond dimensions of the Hamiltonian tensor network representation compared to a matrix product operator structure. This reduction is large enough to reduce the number of total tensor elements required as soon as the number of baths per spin reaches $3$.

Author comments upon resubmission

We have fixed the points brought up by the referee and are thankful for the feedback given.

List of changes

* Removed a short paragraph and an reference to the virtual vertices at the beginning and end of the state diagrams. After reading through the paper, we found that these concepts did not help the understanding and added to the already convoluted assortment of newly defined objects.
* Improved the explanation of equations (17), (18), and (19). Specifically we added a definition for the |*| notation.
* General proofreading and typo corrections.

Published as SciPost Phys. Core 7, 036 (2024)

Login to report or comment