Optimization at the boundary of the tensor network variety

Tensor network states form a variational ansatz class widely used, both analytically and numerically, in the study of quantum many-body systems. It is known that if the underlying graph contains a cycle, e.g., as in projected entangled pair states, then the set of tensor network states of given bond dimension is not closed. Its closure is the tensor network variety. Recent work has shown that states on the boundary of this variety can yield more efficient representations for states of physical interest, but it remained unclear how to systematically find and optimize over such representations. We address this issue by defining an ansatz class of states that includes states at the boundary of the tensor network variety of given bond dimension. We show how to optimize over this class in order to find ground states of local Hamiltonians by only slightly modifying standard algorithms and code for tensor networks. We apply this method to different models and observe favorable energies and runtimes when compared with standard tensor network methods.

Automated summary

Tensor network states are a widely used variational ansatz class in the study of quantum many-body systems. Recent work shows that states on the boundary of the tensor network variety can provide more efficient representations for states of interest. By defining a class of states that includes boundary states, it is possible to optimize over this class to find ground states of local Hamiltonians with favorable energies and runtimes.