Citation: Yu, Xie-Hang, et al. "Dual-isometric Projected Entangled Pair States." arXiv preprint arXiv:2404.16783 (2024).
Web: https://arxiv.org/abs/2404.16783
Tags: Computer-scientific, Tensor-networks
This paper presents a new subclass of product entangled pair states (PEPS) known as dual-isometric PEPS. This subclass further restricts the previously defined class of isometric PEPS. A dual isometric PEPS satisfies the property that its local 5-index tensors are isometries when considered as maps from the bottom-left/bottom-middle indices to the other three indices, and when considered as maps from the bottom-right/bottom-middle indices to the other three indices. This imposes an additional condition when compared to isometric PEPS, whose tensors are only required to be isometries when considered as maps from the bottom-right/bottom-middle to the other three indices.
The point of the dual-isometric condition is that it facilitates the computation of expectation values of local observables. There is an efficient algorithm for computing expectation values of local observables in dual-isometric PEPS, whereas computing expectation values of local observables is BQP-complete for PEPS.
The argument this paper makes is that dual-isometric PEPS facilitate many computations and that they are expressive enough to capture all of the phenomena we want to model by tensor networks. For instance, they can be used to model string-net phases and phase transitions between them.
Maybe it's not very useful because I can't draw pictures, but here is my summary of why local observables can be efficiently computed in dual-isometric PEPS:
The expectation value of a local observable can be viewed graphically as stacking tensors. On the bottom is the dual-isometric PEPS, in the middle is the local observable, and on the top is the complex conjugate of the dual-isometric PEPS. A bit of thought shows that the dual-isometric condition exactly says that the tensors in the top-right and top-left corners can be contracted. Thus, the tensors on the right and left of the local observable can all be contracted away starting from the corners. The expectation value of the local observable on the whole PEPS is then equal to the expectation of the local observable on the 1D sub-tensor of the peps living below the inserted local observable. Expectation values of networks in 1D can be efficiently computed, and thus the problem is solved.