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Tensor network states form a variational ansatz class widely used, bothanalytically and numerically, in the study of quantum many-body systems. It isknown that if the underlying graph contains a cycle, e.g. as in projectedentangled pair states (PEPS), then the set of tensor network states of givenbond dimension is not closed. Its closure is the tensor network variety. Recentwork has shown that states on the boundary of this variety can yield moreefficient representations for states of physical interest, but it remainedunclear how to systematically find and optimize over such representations. Weaddress this issue by defining a new ansatz class of states that includesstates 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 localHamiltonians by only slightly modifying standard algorithms and code for tensornetworks. We apply this new method to a different of models and observefavorable energies and runtimes when compared with standard tensor networkmethods.

Optimization at the boundary of the tensor network variety