Solvable Quantum Circuits in Tree+1 Dimensions

Abstract

We devise tractable models of unitary quantum many-body dynamics on tree graphs, as a first step towards a deeper understanding of dynamics in non-Euclidean spaces. To this end, we first demonstrate how to construct strictly local quantum circuits that preserve the symmetries of trees, such that their dynamical light cones grow isotropically. For trees with coordination number z, such circuits can be built from z-site gates. We then introduce a family of gates for which the dynamics is exactly solvable; these satisfy a set of constraints that we term 'tree-unitarity'. Notably, tree-unitarity reduces to the previously-established notion of dual-unitarity for z = 2, when the tree reduces to a line. Among the unexpected features of tree-unitarity is a trade-off between 'maximum butterfly velocity' dynamics of out-of-time-order correlators and the existence of non-vanishing correlation functions in multiple directions, a tension absent in one-dimensional dual-unitary models and their Euclidean generalizations. We connect the existence of (a wide class of) solvable dynamics with non-maximal butterfly velocity directly to a property of the underlying circuit geometry called $δ$-hyperbolicity, and argue that such dynamics can only arise in non-Euclidean geometries. We give various examples of tree-unitary gates, discuss dynamical correlations, out-of-time-order correlators, and entanglement growth, and show that the kicked Ising model on a tree is a physically-motivated example of maximum-velocity tree-unitary dynamics.