Operator dynamics in k-Markov random circuits

Abstract

We demonstrate that $k$-Markov sequences of unitary gates provide low-cost handles to manipulate the rate and structure of information spreading compared to traditional random, 0-Markov, circuits. For SWAP gates and brickwork circuits, we use graph cover time to demonstrate how $k$-Markov processes can be used to control operator transport. With SWAP gates and the set of Clifford gates that can change operator weight, we show how $k$-Markov sequences can be used to manipulate scrambling time and generate novel structures of spatial-temporal correlations across a qubit network. We show that $k$-Markov circuits constructed from PSWAP gates at fixed angle are equivalent to standard brickwork circuits with PSWAP angle drawn from non-uniform distributions generated by the $k$-Markov process. In those circuits, the time evolution of the average Hamming weight and the space-time correlation structure after equilibrium again vary significantly from the 0-Markov case, depending on the transition probabilities of the process.

Figures (4)

• Figure 1: Top: Mean cover times over system size, $\langle t_c\rangle/N$ for $k$-Markov SWAP circuits as a function of $N$, for several different rules. The effective diffusion constant, $D$, is a function of the transition probabilities and controls the shift from nearly ballistic operator motion for smaller systems to diffusive transport in larger systems. Open symbols are used for cases where less than 90% of runs reached full coverage before the limit $200N$ steps. For the $k=1$ example, only 54% finished, while the $k=2$ example had 87% finish. For those cases the position of the point is a lower bound on the mean cover time, biased low. Shaded regions show the $1\sigma$ variation over 500 trials. Bottom: Scrambling times in Clifford weight-changing circuits, as a function of the fraction of SWAP single-layer (${\bf A}_{\rm SWAP}$), for $k=0$ rules and the shortest times found for $k=1$ and $k=2$, all for $N=512$.
• Figure 2: Panel (a) operator structure for one realization of a $k=0$ rule. The signatures of the SWAP gate transport are most visible before $t_{\rm scr}$, but partially retained even at very late times. (b) The late-time ($t\gg t_{\rm scr}$) correlation structure $C(\Delta x,\Delta t)$ averages over sites and 5 realizations. (c) and (d) The difference between double-layer and single-layer $k=1$ late-time correlations and those of $k=0$ at matched $f_A=0.83$.
• Figure 3: Heatmaps (upper panels, one realization) and late-time spatial-temporal correlation structure (lower panels, averaged over five realizations, all sites, and about 1000 layers well after the network weight has equilibrated past the threshold) for a 30-qubit network, under four different rules for generating run-length distributions as follows (a) Uniformly distributed, with average length $E[R_A]=E[R_B]=80$; (b) Generated by a $k=0$ process for the $A$, $B$ alphabet, where $P(A)=0.75$ sets the geometric distributions $E[R_A]=4$, $E[R_B]=4/3$; (c) Generated by a $k=1$ process where the geometric distributions for $R_A$, $R_B$ are chosen to be the same; (d) Generated by a $k=2$ process with different expectation values for $A$ and $B$ run lengths.
• Figure 4: The time evolution of the weight in the domain-wall sector of operators, for a single realization of each model scenarios shown in Figure \ref{['fig:eight_panel']} and two others. The $k=2$ symmetric case with $E[R_A]=E[R_B]=4$ shows that at small run length (and for the same seed), the difference between the geometric distributions at $k=1$ is small. The $k=2$ symmetric case with $E[R_A]=E[R_B]=32$ shows the different effective filter applied by distributions that contain more long run lengths.