Operator dynamics in a Brownian quantum circuit

We view the operator spreading in chaotic evolution as a stochastic process of height growth. The height of an operator represents the size of its support and chaotic evolution increases the height. We consider $N$-spin models with all two-body interactions and embody the height picture in a random model. The exact solution shows that the mean height, being proportional to the squared commutator, grows exponentially within $logN$ scrambling time and saturates in a manner of logistic function. We propose that the temperature dependence of the chaos bound could be due to initial height biased toward high operators, which has a smaller Lyapunov exponent.

Figure 1

Height of a Pauli string. Left: A Pauli string consisting of tensor products of Pauli matrices. The height is the number of Pauli matrices. Right: The commutator with an overlapping interaction term can increase the height by 1.

Figure 2

Comparison of the mean height with logistic function (mean field) and analytic solution in Eq. (21). The large fluctuation in the intermediate time makes the dynamics slower than the logistic function.

Figure 3

The height distribution and mean height evolved from a simple operator localized at $h=1$. (a) Early time: the profile looks like a collapsing sand pile. It is exponentially decreasing in space and its exponent is exponentially decreasing in time. Inset: numerically measured exponent is linearly proportional to $t$ on the semilogarithmic scale with the slope $-2.96$, close to $-{\lambda }_{q=2}=3$. (b) Late time: the height distribution is exponentially decaying to the steady state of a random operator. Inset: The numerical verification of Eq. (26). (c) Intermediate time: the height distribution has appreciable value over almost the whole range of height. The height fluctuation is of order $N$.

Figure 4

Suppression of the growth rate for the operator with initial $\langle h\rangle \gg 1$. (a) Growth exponent for operators initially localized at $h_0=aN$ with $a>0$. Inset: linear dependence of the exponent with respect to $a$, saturating the bound around $t=0$. (b) Growth exponent for power law decreasing initial condition $f_h\propto \frac{1}{h^{\alpha }}$. $\alpha =\infty$ is the exponential decreasing initial condition. Inset: growth exponent linearly depends on $\alpha$ and saturates to ${\lambda }_q$ when $\alpha >2$.