Can asymptotically solvable quantum circuits capture generic chaotic dynamics?

Solvability constrains correlations, not chaos

Exactly solvable quantum circuits have been central to understanding non-equilibrium quantum matter, but their tractability comes from simplifications. The paper states the tradeoff directly: solvability does not prevent chaotic dynamics, yet it imposes non-trivial constraints on the correlations that are generated. Dual-unitary circuits, which are dual under exchanging space and time, are the standard example of a constrained family that is still generically ergodic.

The question the authors pose follows from that tradeoff. If constraints must restrict correlations, can one still learn something about the generic, hard-to-access case? Their answer is to weaken the constraint rather than remove it: instead of imposing space-time duality everywhere, they impose it only after a tuneable number of steps. That is the design principle behind the family they call asymptotically solvable.

Special gates set a tuneable length scale

The circuits are brickwork circuits built from local gates acting on a one-dimensional chain of qudits, with each gate carrying a position-dependent parameter. Most gates are generic; at selected positions the parameter is set to zero, producing "special" gates. The only requirement on the distribution of these zeros is that the distance between two of them does not scale with system size.

That spacing is the tuneable threshold. The distance between the two special gates enclosing a local subsystem determines the timescale after which the subsystem's dynamics become exactly accessible. For shorter times, the same spacing acts as an effective system size, which is what makes efficient numerical evaluation possible. The authors also note an unavoidable degree of spatial inhomogeneity: the distinctive properties require special gates at distances that do not scale with system size.

Correlations fill a strip, not a few lines

For asymptotically solvable circuits, the authors show that correlations inside the causal light cone are exactly confined to a vertical strip, with the only leakage allowed along the line where the spatial separation equals the time. Between two operators located between different pairs of zeros, the correlation is identically zero unless they are separated by a light-cone distance. Otherwise, the correlation is determined by repeatedly applying a fixed channel built from the gates between the nearest zeros.

This is a sharper statement than it may first appear. In dual-unitary circuits, correlations reduce to a few lines of the causal light cone. Here they are non-trivial within a full area of the light cone whose size is controlled by the distance between the nearest special gates. The authors emphasize that the special gates do not simply cut the system into disconnected parts: correlations moving at the speed of light still exist, so the gates act as a filter on which correlations propagate to large scales. A worked numerical example uses gates with a small homogeneous longitudinal field and special gates placed at two bonds, showing non-trivial correlations at all times inside the light cone when the two operators are close enough.

Ergodic in general, exactly solvable at one point

The family is generically ergodic, but it contains a non-interacting point. The authors use that point to obtain exact analytical results for the early-time regime, which is precisely the regime that is not solvable in general. These analytical results complement numerical experiments on the non-solvable early-time behaviour.

The free-point analysis is technically detailed: the quench problem separates into two homogeneous free-fermionic parts, and the entanglement entropies become a sum of two contributions, one from each set of degrees of freedom. That separation is what allows the familiar quasiparticle picture to be applied. The important point for a reader is the role this point plays in the argument: it supplies exact benchmarks for the generic-looking short-time dynamics rather than serving as the main object of study.

Quench dynamics need compatible initial states

The second test is thermalisation after a quantum quench. Here the authors must restrict the initial states: they consider quenches from compatible, asymptotically solvable non-equilibrium states. This is a real scope condition, not a technical afterthought. The solvability of the post-quench dynamics depends on the initial state belonging to the compatible class, so the results do not automatically extend to arbitrary initial states.

Within that class, the physical picture found for dynamical correlations reappears for post-quench correlations and entanglement dynamics. The authors also discuss the regime where the spacing between special gates is much larger than time, where the circuit undergoes generic dynamics and is therefore difficult to describe. Entanglement growth illustrates why the long-time limit is not immediate: if the cut is not centred on a special gate, the growth cannot be computed from the coarse-grained dual-unitary structure alone and only becomes asymptotically equivalent to the dual-unitary result when time greatly exceeds the spacing.

Periodic spacing flows to dual-unitary circuits

When the special gates are arranged periodically, the circuit is translationally invariant and the authors show an exact renormalisation group flow: coarse-grained blocks of gates satisfy the DU2 condition, a recently introduced generalisation of dual unitarity. Infinite, periodically spaced asymptotically solvable circuits approach DU2 circuits in one step and remain there, since DU2 is a fixed point. This holds for arbitrary block size and regardless of the free parameters in the other gates.

The authors are careful about what this does not imply. The flow does not mean the dynamics immediately reduces to dual-unitary behaviour, because non-equilibrium dynamics involves non-trivial time boundaries. They also note that the argument cannot be applied without strict translation symmetry: when the separation between special gates varies across the circuit, dual-unitary physics still eventually emerges, but the timescale differs at different spatial points and is hard to characterise by simple coarse graining.

Fixed-point entanglement bounds the solvable points

The authors test solvability directly by probing the entanglement structure of the fixed points, which does not depend on the initial state. They ask whether the fixed-point tensor, viewed as a state of many qudits, can be approximated by a low-entangled matrix product state as time grows. Entanglement growing linearly with time implies that an efficient matrix product state representation is not possible, which is a strong indicator that the circuit is not solvable.

The numerical results show that the fixed-point entanglement does not stay bounded for any parameter value except the already known solvable points at the two ends of the parameter range. Near one of those endpoints, the delay before the curves reach their asymptotic slope appears to grow as the gates approach the solvable point, suggesting that the order of the two limits may matter. The authors state this uncertainty is not core to their claim; what they can state clearly is the boundedness result at the known solvable points only.

A filter between solvable and generic dynamics

The paper's distinctive contribution is not a claim that generic chaotic dynamics have been solved. It is a construction in which the distance between special gates becomes an explicit control parameter separating two regimes: below the threshold, dynamics look generic and are accessible only numerically or through the free point; above it, dynamics become exactly solvable and flow toward dual-unitary behaviour under coarse graining. The correlation structure is correspondingly richer than in dual-unitary or DU2 circuits, filling an area of the light cone rather than a few lines.

The authors point to several open directions, including whether large-scale solvability can be attained without special gates at all, whether conservation laws with diffusive or super-diffusive transport can be embedded, and whether spectral properties and out-of-time-order correlators inherit the same threshold structure. For a reader deciding how to use this result, the practical split is the spacing: it sets both the timescale of exact accessibility and the effective system size for short-time computation, while the compatible-initial-state requirement limits which quench questions the solvable long-time results can answer.

Sources used in this answer