Asymptotically Solvable Quantum Circuits

The gist The discovery of chaotic quantum circuits with (partially) solvable dynamics has played a key role in our understanding of non-equilibrium quantum matter and, at the same time, has helped the development of concrete platforms for quantum computation

Asymptotically Solvable Circuits

The paper introduces a family of ‘asymptotically solvable’ quantum circuits where solvability constraints only affect correlations on length scales beyond a tuneable threshold, meaning their dynamics are only solvable for long enough times, and their behaviour for shorter time scales is not affected by solvability constraints This approach contrasts with previous methods that relied on averaging under random unitary operations or imposing constraints, which inevitably limited the generality of the observed phenomenology The authors propose a more effective route to lift the space-time duality constraint by imposing ‘unitarity’ only after a tuneable number of steps This implies that the circuits are only solvable for long enough time scales while their behaviour for shorter time scales is not affected by solvability constraints and departs substantially from that of solvable cases

Solvability Via Space-Time Duality

The analysis focuses on local quantum circuits evolved by a unitary operator U which can be written in the brickwork form U = UoUe Solvability via space-time duality is identified by exchanging the roles of space and time, considering evolution in the spatial direction rather than temporal one Solvable circuits are characterized when the influence matrices take simple forms, which occurs when the infinite temperature state is invariant under space evolution, leading to ‘dual-unitary’ (DU) gates The hierarchy of constraints involves conditions like DUn, where the amplitude of non-identity operators vanishes after n − 1 spatial steps provided they do not interact with other operator strings

Correlation Functions and Dynamics

For asymptotically solvable circuits, dynamical correlations inside the causal light cone take a ‘dagger shape’ where they are exactly confined to a vertical strip with the only leakage allowed along the x − y = t line These correlations can be described by the repeated application of a fixed channel The transfer matrix Tl functions as a unitary evolution U ⊗ U∗ that has been projected onto the sub-sector where the environment both starts and ends in the infinite temperature state, i.e., Tl = 1/d squared (⟨hashtag⊗1l−2⊗ ⟨hashtag)U⊗U∗ (hashtag⟩⊗1l−2⊗ hashtag⟩)

Long-Time Entanglement Velocity

For large times t, the entanglement velocity becomes independent of the Rényi index and is determined by the DU2 gates The analysis shows that for long enough times, even when separating DU2 gates by many generic s non-DU2 gates, the entanglement shows the same growth and even the same slope/velocity The line tension gives us the expected results for the other quantities in the system

Early Time Dynamics

The early time dynamics can be qualitatively different to the long-time solvable dynamics that we have been able to characterise so far, leading to the conclusion that these dynamics are in no sense solvable and that in the general case only the long-time dynamics may be characterised analytically In the homogeneous case, for d = 2 and longitudinal fields zero, the time evolution operator can be written as a product of three mutually commuting terms The central conclusion to draw is that in the homogeneous case the dynamics at the free point does not appear to be similar to the DU2 circuit in any sense

Discussion

Asymptotically solvable circuits involve an unavoidable degree of spatial inhomogeneity, and their distinctive properties require the presence of ‘special’ ergodic gates applied at distances that are not scaling with the system size These special gates act to ‘filter’ the correlations allowed to propagate to large scales The work prompts many fascinating questions for future research, such as the possibility of attaining large-scale solvability without ever requiring special gates

How it works

1. The authors introduce local relations of the form 1 = Us̸=0 ⊗ U∗s̸=0, which produce a generalisation of the DU condition that is not part of the hierarchy of Ref. [23]

2. The circuit structure can be solved by a family of gates Us that are dependent on a continuous parameter s from 0 to 1, interpolating between DU2 and DU gates

3. In inhomogeneous circuits built of gates fulfilling Eq. (14), dynamical correlations are supported both on the light cone edge and within a region of a configurable width, giving them what we will refer to as a “dagger shape”

4. The entanglement velocity becomes bounded from below by that of the DU2 circuit, suggesting that for long enough times the entanglement shows the same growth and even the same slope/velocity

5. In the non-interacting limit (hx = 0), the time evolution operator can be written as a product of three mutually commuting terms U = e iπ/2 Px Xx U1U2 This decomposition explains why the entanglement growth is bounded from below by the DU2 circuit

6. The analysis of the fixed point shows that for generic choices of s, the entanglement velocity does not settle into n-independent values, indicating that again the Schmidt spectrum of the wave function is becoming increasingly non-flat under these dynamics

7. The line tension gives us the expected results for the other quantities in the system, and its behavior reflects only what is attained for t >> l

8. The analysis of eigenvectors shows that a necessary condition for the existence of a non-trivial uni-modular eigenvector is that it must have non-trivial support at the edges, specifically h1 2, h1 ∈ [0, π/4] and symmetrically hl− 1 2, hl ∈ [0, π/4]

9. The upper bound on the spectral radius of T(n)l is shown to be ρ(T(n)l) ≤ 1/d 2n = 1/(n(1-n))Λ This upper bound is indeed the eigenvalue corresponding to the eigenvectors in Eq. (69)

The paper establishes that asymptotically solvable circuits exhibit a rich pattern of spatiotemporal correlations that are not reduced to a few lines of the causal light cone but are non-trivial within a full area of the light cone whose size is controlled by the distance between the nearest special gates The key finding is that for long enough times, these circuits approach DU2 dynamics, and their entanglement velocity becomes independent of the Rényi index

The paper's findings are significant because they provide a concrete framework for understanding generic non-equilibrium quantum many-body dynamics by introducing a class of systems where exact analytical results can be obtained for long times.

1. Improved understanding of non-equilibrium quantum matter dynamics: The system can now characterise dynamics by analyzing dynamical correlations on different length scales, as solvability constraints only affect correlations on length scales beyond a tuneable threshold.

2. Enhanced simulation of quantum quenches: The system allows for the characterization of thermalisation dynamics following quantum quenches from compatible (asymptotically solvable) non-equilibrium initial states, providing a more accurate description of how systems evolve out of equilibrium.

3. Exact analytical results for early-time regimes: The inclusion of a non-interacting point provides an avenue to complement those of numerical experiments, on the non-solvable early time regime, allowing for exact analytical results where generic dynamics are otherwise intractable.

4. Characterization of correlation support structure: The system predicts that dynamical correlations inside the causal light cone take the dagger shape where they are exactly confined to a vertical strip with the only leakage allowed along the x - y = t line.

5. Prediction of long-time entanglement velocity: For large times, it is shown that the slope of entanglement growth divided by log d — the ‘entanglement velocity’ becomes independent of the Rényi index and is determined by the DU2 gates, allowing for a more robust prediction of late-time scrambling behavior.

6. Modeling complex transport phenomena: The analysis suggests that removing the special gates should allow to study cases where the conserved charges display diffusive or even super-diffusive [82–87] transport, enabling the modeling of non-trivial charge transport in interacting circuits.