Solvable Quantum Circuits from Spacetime Lattices

TL;DR

The paper develops completely reducible circuits as a unifying framework for exactly solvable non-integrable quantum dynamics by organizing dual-unitary gates on spacetime lattices. It shows that entanglement growth is captured by the entanglement line tension, which is piecewise-linear with kinks at discrete information-flow directions and exhibits a flat operator entanglement spectrum, with the entanglement velocity set by the base gate's Schmidt rank. A novel knot-theoretic perspective ties solvability to the unlink property and the Kauffman bracket, linking tensor contractions to knot invariants. Numerical studies of spectral form factors across several geometries support chaotic behavior consistent with random-matrix theory and a vanishing Thouless time, illustrating maximal quantum chaos within these solvable non-integrable models. The work also outlines broad future directions, including biunitary connections, measurements, higher dimensions, and more general random spacetime geometries, highlighting the potential for experimental realization.

Abstract

In recent years dual-unitary circuits and their multi-unitary generalizations have emerged as exactly solvable yet chaotic models of quantum many-body dynamics. However, a systematic picture for the solvability of multi-unitary dynamics remains missing. We present a framework encompassing a large class of such non-integrable models with exactly solvable dynamics, which we term \emph{completely reducible} circuits. In these circuits, the entanglement membrane determining operator growth and entanglement dynamics can be characterized analytically. Completely reducible circuits extend the notion of space-time symmetry to more general lattice geometries, breaking dual-unitarity globally but not locally, and allow for a rich phenomenology going beyond dual-unitarity. As example, we introduce circuits that support four and five directions of information flow. We derive a general expression for the entanglement line tension in terms of the pattern of information flow in spacetime. The solvability is shown to be related to the absence of knots of this information flow, connecting entanglement dynamics to the Kauffman bracket as knot invariant. Building on these results, we propose that in general non-integrable dynamics the curvature of the entanglement line tension can be interpreted as a density of information transport. Our results provide a new and unified framework for exactly solvable models of many-body quantum chaos, encompassing and extending known constructions.

Figures (8)

• Figure 1: Examples of spacetime lattice circuits of dual-unitary gates (as represented by blue squres). The horizontal and vertical directions correspond to space and time respectively. (a) A square lattice returns a dual-unitary brickwork circuit, (b) A Kagome lattice of dual-unitary interactions returns triunitary dynamics, (c) a generic lattice which does not enable any simplifications of tensor network diagrams representing observables, (d) a lattice which gives rise to completely reducible dynamics.
• Figure 2: Illustration of the spacetime lattice generated from a base gate \ref{['eq:basegate']}. The horizontal and vertical directions correspond to space and time respectively. While the local interactions are dual-unitary, the full circuit breaks space-time duality.
• Figure 3: Entanglement line tension (ELT) $\mathcal{E}(v)$ of classes of completely reducible circuits. (a) DU and DU2 circuits [Eqs. \ref{['eq:du_elt']}, \ref{['eq:du2_elt']}, and \ref{['eq:du2_elt_2']}]. (b) 4-pyramid lattice [Eq. \ref{['eq:4pyr_elt']}] and larger unit cell generalizations [Eq. \ref{['eq:family_1']}]. (c) Lattice with five directions of information flow [Eq. \ref{['eq:5ray_elt']}].
• Figure 4: Two spacetime lattice circuits with highlighted (a) non-crossing $v=0$ worldline and (b) crossing $v=0$ worldline. The former leads to completely reducible dynamics, whereas the latter does not.
• Figure 5: Illustration how in completely reducible circuits worldlines transporting information along a ray of velocity $v_i$ in spacetime contribute to the ELT. Generally, a worldline along $v_i$ entangles the interval $[x_1,x_2]$ at $t=0$ with the interval $[x_1+v_it,x_2+v_it]$ at the final time $t$. (a) For $v_i<v$ this corresponds to the contribution coming from the initial region $[0,(v-v_i)t]$. (b) For $v_i>v$ the constribution comes from $[-(v_i-v)t,0]$.