Operator Spreading in Random Unitary Circuits

By A. Nahum, S. Vijay, Jeongwan Haah. 2017

786 citations recorded by the source corpus.

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+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 determines the butterfly speed vB. We find that in 1+1D, the “front” of the OTOC broadens diffusively, with a width scaling in time as t1/2. 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 t1/3 in 2+1D and as t0.240 in 3+1D (exponents of the Kardar-Parisi-Zhang universality class). We support our analytic argument with simulations in 2+1D. 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+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 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+1D circuits.

Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/a6a5598a9e8d26c29598d1e073e4215ff9e24374

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Citations in the corpus

Random Quantum Circuits (2022, methodology): “First, for small numbers of layers circuit averages can be mapped to classical Markov processes (110, 129, 111, 130, 112, 131, 49, 50, 92), describing the incoherent (dephased) evolution of quantum operators (Sec.”
Localization from Hilbert space shattering: From theory to physical realizations (2020, background): “This is a model of quantum circuit dynamics [46–50] constrained to conserve both a U(1) charge and its dipole moment.”
Fracton phases of matter (2020, background): “[103] Adam Nahum, Sagar Vijay, and Jeongwan Haah, “Operator spreading in random unitary circuits,” Physical Review X 8 (2018), 10.”
Many-Body Quantum Magic (2020, background): “[73] Adam Nahum, Sagar Vijay, and Jeongwan Haah, “Operator spreading in random unitary circuits,” Phys.”
Dynamical Purification Phase Transition Induced by Quantum Measurements (2019, background): “[70] A.”
Velocity-dependent Lyapunov exponents in many-body quantum, semiclassical, and classical chaos (2018, methodology): “x has been incorporated inside a growing cluster in a fictitious stochastic growth process [43].”
Statistical mechanics of quantum error correcting codes (2020, background): “[25–28] for random Haar unitary circuits without measurements, where it was first pointed out that the entanglement entropy can be viewed as free energies of “entanglement domain walls”.”
Measurement-driven entanglement transition in hybrid quantum circuits (2019, methodology): “[52] Adam Nahum, Sagar Vijay, and Jeongwan Haah, “Operator Spreading in Random Unitary Circuits,” Physical Review X 8, 021014 (2018), arXiv:1705.”