Ternary unitary quantum lattice models and circuits in $2 + 1$ dimensions

TL;DR

This paper introduces ternary unitary quantum gates for 2+1 dimensional lattice models, revealing light-ray structures in correlations and extending solvable matrix product states to higher dimensions with efficient tensor network algorithms.

Contribution

It generalizes dual unitary gates to 2+1 dimensions, constructs ternary unitary four-particle gates, and develops tensor network methods for correlation functions in higher-dimensional quantum models.

Findings

Dynamical correlation functions show light-ray structures.

Ternary unitary gates cancel out in the tensor network for correlations.

Numerical algorithms for computing correlations in 2+1D models are developed.

Abstract

We extend the concept of dual unitary quantum gates to quantum lattice models in $2 + 1$ dimensions, by introducing and studying ternary unitary four-particle gates, which are unitary in time and both spatial dimensions. When used as building blocks of lattice models with periodic boundary conditions in time and space (corresponding to infinite temperature states), dynamical correlation functions exhibit a light-ray structure. We also generalize solvable MPS to two spatial dimensions with cylindrical boundary conditions, by showing that the analogous solvable PEPS can be identified with matrix product unitaries. In the resulting tensor network for evaluating equal-time correlation functions, the bulk ternary unitary gates cancel out. We delineate and implement a numerical algorithm for computing such correlations by contracting the remaining tensors.