Operator Spreading in Random Unitary Circuits

Random quantum circuits yield minimally structured models for chaotic quantum dynamics, which are able to capture, for example, universal properties of entanglement growth. We provide exact results and coarse-grained models for the spreading of operators by quantum circuits made of Haar-random unitaries. We study both $1+1\mathrm{D}$ and higher dimensions and argue that the coarse-grained pictures carry over to operator spreading in generic many-body systems. In $1+1\mathrm{D}$, we demonstrate that the out-of-time-order correlator (OTOC) satisfies a biased diffusion equation, which gives exact results for the spatial profile of the OTOC and determines the butterfly speed $v_B$. We find that in $1+1\mathrm{D}$, the “front” of the OTOC broadens diffusively, with a width scaling in time as $t^{1/2}$. We address fluctuations in the OTOC between different realizations of the random circuit, arguing that they are negligible in comparison to the broadening of the front within a realization. Turning to higher dimensions, we show that the averaged OTOC can be understood exactly via a remarkable correspondence with a purely classical droplet growth problem. This implies that the width of the front of the averaged OTOC scales as $t^{1/3}$ in $2+1\mathrm{D}$ and as $t^{0.240}$ in $3+1\mathrm{D}$ (exponents of the Kardar-Parisi-Zhang universality class). We support our analytic argument with simulations in $2+1\mathrm{D}$. We point out that, in two or higher spatial dimensions, the shape of the spreading operator at late times is affected by underlying lattice symmetries and, in general, is not spherical. However, when full spatial rotational symmetry is present in $2+1\mathrm{D}$, our mapping implies an exact asymptotic form for the OTOC, in terms of the Tracy-Widom distribution. For an alternative perspective on the OTOC in $1+1\mathrm{D}$, we map it to the partition function of an Ising-like statistical mechanics model. As a result of special structure arising from unitarity, this partition function reduces to a random walk calculation which can be performed exactly. We also use this mapping to give exact results for entanglement growth in $1+1\mathrm{D}$ circuits.

Popular Summary

A key challenge in the physics of many-body systems is to identify features of nonequilibrium dynamics that are universal, that is, shared between many different systems. In this paper, we reveal new universal theoretical structures governing how signals or disturbances spread through a many-body system (the quantum “butterfly effect”). These structures give new insights into chaotic behavior in many-body quantum systems and into how the dynamics of a quantum system tend to make initially accessible quantum information very difficult to retrieve. This work also reveals fundamental new connections between the quantum butterfly effect and classical statistical mechanics.

We study the butterfly effect using random quantum circuits. These are minimal models that retain fundamental features of chaotic many-body systems (e.g., locality of the interactions between particles), while remaining tractable for theoretical study. From these models, we derive new hydrodynamic equations for the spreading of a quantum operator (roughly speaking, the spreading of a disturbance to the quantum system) in both one and higher spatial dimensions. We also provide universal scaling forms for the structure of a time-evolved operator, which we conjecture are generic for chaotic systems. Finally, we connect entanglement growth and operator spreading with effective statistical mechanics problems for “random walks” in spacetime.

In the future, this framework will be useful for studying a wide range of observables in chaotic many-body systems.

Figure 1

Left: Random unitary circuit in $1+1\mathrm{D}$. Each brick represents an independently Haar-random unitary, acting on the Hilbert space of two adjacent “spins” of local Hilbert space dimension $q$.

Figure 2

Schematic behavior of the average OTOC: We find that the average OTOC $\overline{\mathcal{C}}(\mathit{x},t)$ (where the average is over the local unitaries in the quantum circuit) has a front that broadens as $t^{\alpha }$, with the indicated exponents in various spatial dimensions $d$.

Figure 3

Cartoon for the form proposed here for the OTOC in two spatial dimensions, when lattice anisotropy can be neglected. The functional form is given by the Tracy-Widom distribution $F_2$.

Figure 4

Top row: $2+1\mathrm{D}$ Haar-random quantum circuit. We consider unitary dynamics in which two-site Haar-random unitaries are applied on the bonds of a two-dimensional square lattice, in the columnar dimer configurations shown in (1)–(4). Bottom row: Allowed updates in the corresponding stochastic process.

Figure 5

Geometry of $2+1\mathrm{D}$ Haar-random circuit.

Figure 6

Growth of a classical droplet and the OTOC: We relate the behavior of the OTOC (averaged over the unitaries in the circuit) to a classical stochastic process for the growth of a droplet in two spatial dimensions. A given configuration of the classical droplet is specified by a binary occupation number $n(\mathit{x},t)$ as shown on the left. Remarkably, the average droplet profile $⟨n(\mathit{x},t)⟩$ precisely reproduces the averaged OTOC.

Figure 7

Growth of a 2D cluster ($q=2$): We determine the behavior of the averaged OTOC by simulating the stochastic growth of a two-dimensional cluster over $M=2\times {10}^3$ realizations, with local updates applied at each time step, as described in the text. The average occupation number for the cluster $⟨n(\mathit{x},t)⟩$ is shown for the indicated times in the evolution as it approaches its asymptotic shape.

Figure 8

Fluctuation exponent $\beta$: We fit the profile of the evolving droplet for $q=2$ along the $\theta =\pm \pi /4$ directions (top panel) to extract the mean operator size and magnitude of the fluctuations about the mean. The fluctuations exhibit power-law growth with exponent $\beta =0.3305\pm 0.0269$, consistent with the KPZ value $\beta =1/3$. When fitting the profile along $\theta =0$ (bottom panel), we observe no appreciable growth of the fluctuations; we argue in Appendix pp5 that this occurs for sufficiently large $q$ when the front’s local normal vector is precisely aligned with a lattice axis (as a result of the specific circuit geometry).

Figure 9

The OTOC in $(2+1)\mathrm{D}$: Plot of the front of the averaged OTOC $\overline{\mathcal{C}(r,t)}$ in two spatial dimensions and in the absence of lattice anisotropy, as determined from the exact expression in terms of the Tracy-Widom distribution in the main text.

Figure 10

“Faceting” of the cluster: Shown are the cluster shapes at fixed time $t={10}^3$, for the indicated values of $q$. When $q$ is sufficiently large (third panel), the cluster develops facets along the $\theta =0$, $\pi$ directions, where the normal growth speed is the maximum possible given the circuit geometry. The region shown is the naive light cone.

Figure 11

Anisotropy in the cluster profile: Numerically determined anisotropy in the average shape of the 2D cluster $R(\theta ,t)/t$, at the indicated times. The anisotropy in the cluster shape grows in time and appears to asymptote to a nontrivial steady-state shape.

Figure 12

The calculation of the OTOC in the spacetime picture leads to a partition function for two nonintersecting directed random walks (domain walls in an effective Ising model). These walks have a bulk energy cost [Eq. (69)] and a boundary potential [Eq. (70)] that biases them to a positive and negative velocity, respectively. Directions of null coordinates $u$, $v$ used in the text are indicated.

Figure 13

Elementary tensor for the computation of $\overline{F}$. The boundary conditions in the top-left figure are for ${\ell }_u={\ell }_v=1$.

Figure 14

Weights due to the interaction between adjacent Ising variables arising from the same unitary (top diagram) and from unitaries at adjacent time steps (bottom diagram). After integrating out the “bra” Ising variable $s^{\prime }$, we obtain the weights shown in Fig. 15.

Figure 15

Weights for the three-body interaction that arises after integrating out half of the Ising variables (the bra variables).

Figure 16

A section of one of the domain walls in the bulk (double line). The two configurations shown have equal weight.

Figure 17

The average entanglement purity $\overline{\mathcal{P}}$ maps to the partition function for a directed random walk. At the top boundary, the walk terminates at the position of the entanglement cut between subsystems $A$ and $B$. At the bottom boundary, the end point is free.

Figure 18

Random circuit built from “staircase” unitaries: We use “left” and “right” staircases—built from random two-site unitary operators as shown, and extending over $\ell$ bonds—as the building blocks for a random quantum circuit in which the ratio of the entanglement and butterfly velocities $v_E/v_B$ may be made arbitrarily small.