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arXiv · 2311.13433

State Diagrams to determine Tree Tensor Network Operators

Richard M. Milbradt·Qunsheng Huang·Christian B. Mendl
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Abstract

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 $3$.

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Richard M. Milbradt, Qunsheng Huang, Christian B. Mendl. 2023-11-22. State Diagrams to determine Tree Tensor Network Operators. https://doi.org/10.21468/scipostphyscore.7.2.036

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