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  • Open Access

Solvable Quantum Circuits in Tree+1 Dimensions

Oliver Breach1,*, Benedikt Placke1, Pieter W. Claeys2, and S.A. Parameswaran1

  • *Contact author: oliver.breach@physics.ox.ac.uk
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PRX Quantum 6, 040316 – Published 23 October, 2025

DOI: https://doi.org/10.1103/fbgh-rq3l

Abstract

We devise tractable models of unitary quantum many-body dynamics on tree graphs, as a first step toward 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 are 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 nonvanishing 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 nonmaximal 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.

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Popular Summary

Quantum circuits, where qubits evolve in discrete time steps through local unitary gates, have emerged as an important framework for studying the dynamics of many-body quantum systems. In a special class of these, called “dual-unitary” circuits, a symmetry between space and time permits derivation of exact results on how information and entanglement spread, strikingly, even in regimes where the dynamics are chaotic. Motivated by recent advances in experimental quantum platforms to realize long-range connectivity, we investigate circuit dynamics on tree graphs, which serve as a simple test bed for non-Euclidean geometries more generally. Exact results in this setting are of particular importance, since the exponential growth in lattice sites with diameter often makes it challenging to obtain conclusive results from numerical approaches.

Despite the lack of clear space-time symmetry, we develop a new set of constraints generalizing dual-unitarity by instead using isometries in the spatial directions. Such “tree-unitary” circuits retain exact solvability, allowing us to calculate quantities such as correlation functions and entanglement growth, and they reveal dynamics qualitatively distinct from those of their Euclidean counterparts. In particular, operator spreading is slower than the light-cone velocity, which we show cannot be the case for a wide class of solvable circuits in Euclidean geometries.

Our work establishes the first family of exactly solvable quantum circuits in non-Euclidean geometries, with aspects of our results extending beyond the tree lattice to, e.g., hyperbolic lattices. This paves the way for a broader exploration of non-Euclidean quantum dynamics, with implications for both foundational questions in nonequilibrium physics and emerging quantum platforms.

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PRX Quantum 6, 040316– Published 23 October, 2025
  • Received 11 April 2025
  • Revised 16 July 2025
  • Accepted 2 September 2025
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