Maximum velocity quantum circuits

We consider the long-time limit of out-of-time-order correlators (OTOCs) in two classes of quantum lattice models with time evolution governed by local unitary quantum circuits and maximal butterfly velocity $v_B=1$. Using a transfer matrix approach, we present analytic results for the long-time value of the OTOC on and inside the light cone. First, we consider “dual-unitary” circuits with various levels of ergodicity, including the integrable and nonintegrable kicked Ising model, where we show exponential decay away from the light cone and relate both the decay rate and the long-time value to those of the correlation functions. Second, we consider a class of kicked $\mathit{XY}$ models similar to the integrable kicked Ising model, again satisfying $v_B=1$, highlighting that maximal butterfly velocity is not exclusive to dual-unitary circuits.

Figure 1

A “brick wall” quantum circuit. The qubits are arranged in a horizontal row with two-qubit gates acting alternately on the even and odd links between them. Discrete time runs vertically.

Figure 2

Typical behavior of the OTOC in a maximal velocity circuit (in this case the kicked Ising model) with $v_B=1$, where a maximal value is reached on the light cone and the OTOC decays exponentially away from the light cone.

Figure 3

Evolution of $\langle {\sigma }_{\alpha }(t,t){\sigma }_{\beta }(0,0)\rangle$ for $t>0$, with $q=2$ and ${\sigma }_{\alpha }$ and ${\sigma }_{\beta }$ Pauli matrices, where the legend denotes $\alpha ,\beta$. Exponential decay can be clearly observed after an initial transient regime, where all correlation functions decay at the same rate. The two-qubit gate $U$ is parametrized as in Appendix pp2.

Figure 4

$C_{\alpha \beta }^{+}(x,t)$ (a) and $C_{\alpha \beta }^{-}(x,t)$ (b) for ${\sigma }_{\alpha }={\sigma }_{\beta }={\sigma }_x$ and a random dual-unitary circuit (see Appendix pp2). Only the values inside and on the light cone are shown. In the top plot $(t-x)$ is even, and the long-time values are given by $-1/(q^2-1)=-1/3$ for $x=t$ and 0 otherwise. The bottom plot denotes the evolution for $(t-x)$ odd, where the inset details the (logarithm of the) long-time values of ${lim}_{t\to \infty }\left|C_{\alpha \beta }(t-2n-1,t)\right|$ from Eq. (58) for a larger range of $n$, where the exponential decay can be clearly observed.

Figure 5

Values of $C_{\alpha \beta }^{+}(x,t)$ (a) and $C_{\alpha \beta }^{-}(x,t)$ (b) for ${\sigma }_{\alpha }=\left({\sigma }_x+{\sigma }_z\right)/\sqrt{2}$ and ${\sigma }_{\beta }={\sigma }_y$ for evolution using the KIM with $h_1=0.4$ and $h_2=0.6$. For even $(t-x)$ the long-time values are given by $-1/2$ for $x=t$ and 0 otherwise. The bottom plot denotes the evolution for odd $(t-x)$, where the inset details the (logarithm of the) long-time values of ${lim}_{t\to \infty }\left|C_{\alpha \beta }(t-2n-1,t)\right|$ for a larger range of $n$, where the exponential decay $\propto cos{(h_1+h_2)}^{2n}$ can be clearly observed.

Figure 6

Evolution of the correlation function on the light cone $\langle {\sigma }_{\alpha }(t,t){\sigma }_{\beta }(0,0)\rangle$ for the KIM and ${\sigma }_{\alpha ,\beta }$ parametrized as in Fig. 5, showing exponential decay $\propto cos{(h_1+h_2)}^t$ with prefactor $c_{\alpha \beta }=[{\alpha }_{x}cos\left(h_2\right)-{\alpha }_{y}sin\left(h_2\right)][{\beta }_{x}cos\left(h_1\right)-{\beta }_{y}sin\left(h_1\right)]$.

Figure 7

Values of $C(x,t)$ for $x=t-n, {\sigma }_{\alpha }={\sigma }_x/\sqrt{6}+{\sigma }_y/\sqrt{2}+{\sigma }_z/\sqrt{3}$, and ${\sigma }_{\beta }={\sigma }_x/\sqrt{6}-{\sigma }_y/\sqrt{2}+{\sigma }_z/\sqrt{3}$ for evolution using the kicked $\mathit{XY}$ model with $J_z=\pi /10$. Dotted lines represent the analytic results, and the inset details ${lim}_{(x+t)\to \infty }C(t-n,t)$ for a larger range of $n$. The immediate saturation to a constant value can be clearly observed.

Figure 8

Values of $C_{\alpha \beta }(x,t)$ for $x=t-n, {\sigma }_{\alpha }={\sigma }_x/\sqrt{6}+{\sigma }_y/\sqrt{2}+{\sigma }_z/\sqrt{3}$, and ${\sigma }_{\beta }={\sigma }_x/\sqrt{6}-{\sigma }_y/\sqrt{2}+{\sigma }_z/\sqrt{3}$ for evolution using the kicked $\mathit{XY}$ model with $J_z=\pi /10$. Dotted lines represent the analytic results for $lim(x+t)\to \infty$, and the inset details ${lim}_{(x+t)\to \infty }C(t-n,t)$ for a larger range of $n$.

Figure 9

Evolution of the correlation function on the light cone $\langle {\sigma }_{\alpha }(t,t){\sigma }_{\beta }(0,0)\rangle$ for the kicked $\mathit{XY}$ model and ${\sigma }_{\alpha ,\beta }$ parametrized as in Fig. 8.