Asymptotically Solvable Quantum Circuits

Abstract

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. It was shown that solvability does not prevent the generation of chaotic dynamics, however, it imposes non-trivial constraints on the generated correlations. A natural question is then whether it is possible to gain insight into the generic case despite the latter being very hard to access. To address this question here we introduce a family of 'asymptotically solvable' quantum circuits where the solvability constraints only affect correlations on length scales beyond a tuneable threshold. This means that their dynamics are only solvable for long enough times: for times shorter than the threshold they are generic. We show this by computing both their dynamical correlations on the equilibrium (infinite temperature) state and their thermalisation dynamics following quantum quenches from compatible (asymptotically solvable) non-equilibrium initial states. The class of systems we introduce is generically ergodic but contains a non-interacting point, which we use to provide exact analytical results, complementing those of numerical experiments, on the non-solvable early time regime.

Figures (12)

• Figure 1: Example of the tensor network resulting from a dynamical correlation function. Shaded in blue and red are respectively the left and right fixed points of the space evolution (influence matrices). These objects encode the effects of the surrounding system upon the local region containing the two operators.
• Figure 2: The types of fixed points/influence matrices seen in solvable and non-solvable circuits. a) the fixed points seen in generic gates, with no simplification the growing width with time implies there is no efficient representation of the state. b) the infinite temperature product state that comes from dual-unitarity (DU). c) the pair-correlated state the results from DU2. d) the fixed point resulting from DU3 that still has a width that scales with time. e) the proposal in this paper, to insert inhomogeneities into an otherwise generic circuit of type (a) that truncate the fixed point to always have finite width.
• Figure 3: Example of the structure of the circuit. Pictured here is $(\mathbb{U}\otimes\mathbb{U}^*)^t$ the folded evolution operator applied $t=3$ times. The evolution operator acts on $2L=12$ qubits positioned on the half integers. The light blue gates represent positions at which $s_x=0$. Here we placed them at $x=0, 2, 5$ but our arguments hold for arbitrary positions of these inhomogeneities.
• Figure 4: Schematic of the dynamical correlations $c_{ij}(x,0,t)$ seen in the different types of circuits. a) Generic non-solvable dynamics are only constrained by the unitarity of the gates giving a simple lightcone support to the correlations. b) Partially restricted yet non-solvable dynamics such as those seen in DU3 circuits where the correlations exist on the $|x|=t$ line and within a steeper interior lightcone. c) An example of a solvable circuit where the correlations are restricted to individual lines, in this case $|x|=0, t$ as seen in DU2 circuits. d) The support of correlations seen in the asymptotically solvable circuits seen in this paper. Correlations spread in a generic manner up until the inhomogeneities of the circuit restrict them. The only correlations outside of these regions exist along the $|x|=t$ lines.
• Figure 5: A correlation function between two identical random traceless Hermitian operators supported on 4 sites. The position of the second operator is fixed with its leftmost spin at $2y=15$. All gates use a small homogeneous longitudinal field $h=\pi/800$. Gates with $s_j=0$ act on the $13$th and $19$th bonds in the system, all other gates use $s_j = \pi/8$. As a result, inside the light cone we see non-trivial correlations at all times when the two operators $|x-y|\leq4$.