Tensor network states form a variational ansatz class widely used in the study of quantum many-body systems. Geometrically, these states form an algebraic variety of tensors with rich representation theoretic structure. It is known that tensors on the "boundary" of this variety can provide more efficient representations for states of physical interest, but the pathological geometric properties of the boundary make it difficult to extend the classical optimization methods. In recent work, we introduced a new ansatz class which includes states at the boundary of the tensor network variety. I will present some of the geometric features of this class and explain how it can be used in the variational approach. This is based on joint work with M. Christandl, D. Stilck-Franca and A. Werner.

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