Operator Spreading in Random Unitary Circuits

TL;DR

This work investigates universal features of operator spreading in chaotic quantum dynamics using Haar-random quantum circuits as a minimal model. It derives a biased diffusion description for the 1+1D averaged OTOC with a finite butterfly speed and diffusive front broadening, and it maps higher-dimensional spreading to KPZ-class classical growth with Tracy-Widom scaling in 2+1D. The authors provide exact spacetime calculations via an Ising-domain-wall representation and reveal a rich structure for entanglement growth, including exact results and bounds on the ratio between entanglement and operator-spreading speeds. They also show that fluctuations between random circuit realizations are subleading to front broadening, and they illustrate how lattice anisotropy shapes the late-time operator profile. The work suggests that these scaling forms and mappings may apply broadly to generic nonintegrable systems, offering a concrete framework for understanding scrambling and entanglement in many-body dynamics.

Abstract

Random quantum circuits yield minimally structured models for chaotic quantum dynamics, 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+1D and higher dimensions, and argue that the coarse-grained pictures carry over to operator spreading in generic many-body systems. In 1+1D, 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 the butterfly speed $v_{B}$. We find that in 1+1D 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. Turning to higher D, we show that the averaged OTOC can be understood exactly via a remarkable correspondence with a classical droplet growth problem. This implies that the width of the front of the averaged OTOC scales as $t^{1/3}$ in 2+1D and $t^{0.24}$ in 3+1D (KPZ exponents). We support our analytic argument with simulations in 2+1D. We point out that, in a lattice model, the late time shape of the spreading operator is in general not spherical. However when full spatial rotational symmetry is present in 2+1D, 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+1D, we map it to the partition function of an Ising-like 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+1D circuits.

Figures (16)

• Figure 1: Left: Random unitary circuit in 1+1D. 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 $\overline{\mathcal{C}}(\boldsymbol{x},t)$ (where the average is over the local unitaries in the quantum circuit) has a front which 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: 2+1D 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: allowed updates in the corresponding stochastic process.
• Figure 5: Geometry of $2+1$D Haar-random circuit.

Theorems & Definitions (1)

• proof : Proof of Eq. \ref{['eq:HaarAvg']}