Operator Spreading in Random Unitary Circuits (1705.08975v2)

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.

• The paper demonstrates that operator spreading follows a biased diffusion model in 1+1D, with the front broadening diffusively as t^(1/2).
• It reveals that in higher dimensions the OTOC corresponds to classical droplet growth, adhering to KPZ universality with scaling exponents of t^(1/3) in 2+1D and t^(0.240) in 3+1D.
• The study establishes a theoretical framework linking quantum chaos and thermalization to classical growth dynamics, paving the way for broader applications in many-body systems.

Insights on "Operator Spreading in Random Unitary Circuits"

The paper "Operator Spreading in Random Unitary Circuits" investigates the propagation of quantum operators under chaotic quantum dynamics, specifically using random quantum circuits as models. The main focus is on understanding the universal properties of such dynamics, including entanglement growth, through the lens of operator spreading measured by the out-of-time-order correlator (OTOC). The paper provides both exact results and coarse-grained models across different dimensions, utilizing Haar-random unitary circuits as the primary dynamic substrate. Profound insights are derived, especially in dimensions 1+1 and higher, as the paper connects operator spreading to classical growth dynamics, opening avenues for future research in both quantum and classical domains.

Key Findings and Numerical Results

A central result of the paper is the establishment of exact solutions and coarse-grained models illustrating how operators spread in random circuits. In one-dimensional (1+1D) settings, the authors show that the OTOC can be described accurately by a biased diffusion equation. The butterfly speed, $v_B$, a significant parameter, determines the speed at which the operator front propagates. The paper finds that in 1+1D, this 'front' broadens diffusively, with its width scaling as $t^{1/2}$. They illustrate that fluctuations in the OTOC across different realizations of the random circuit are minimal compared to the intrinsic broadening of the front. This strongly suggests that the coarse-grained models developed can potentially describe operator spreading in generic many-body systems, beyond the confines of random circuits.

When the analysis is extended to higher dimensions (2+1D or 3+1D), the paper uncovers a remarkable correspondence between the OTOC and a classical droplet growth problem. The averaged OTOC behaves similarly to classical growth models described by the Kardar-Parisi-Zhang (KPZ) universality class, which suggests a scaling of the front's width as $t^{1/3}$ in 2+1D, and as $t^{0.240}$ in 3+1D. Importantly, the shape of the spreading operator at late times is nontrivial: it is influenced by underlying lattice symmetries and typically deviates from spherical symmetry, aligning instead with velocity-dependent asymmetries.

Theoretical Implications and Future Directions

The upshot of these findings is that random quantum circuits serve as an effective theoretical framework for understanding universal behaviors in quantum dynamics, notably chaos and thermalization. The connection made between operator spreading in quantum circuits and classical models such as KPZ provides a fresh perspective and could have profound implications for theoretical understandings of quantum thermalization processes.

Moreover, the results reveal a complexity in how finite-dimensional quantum systems, modeled through tools like the OTOC, converge towards equilibrated states, suggesting potential extensions to non-random systems. This work also paves the way for future research to explore the deeper implications of these correspondences, particularly in determining the role of underlying symmetries and dimensional constraints in other quantum frameworks. Additionally, extending these analyses to include the impact of symmetries and conservation laws intrinsic to Hamiltonian systems poses an eagerly awaited challenge.

Conclusion

The paper makes a significant contribution to the paper of quantum dynamics by bridging concepts from quantum information, many-body physics, and classical statistical mechanics. While addressing technical aspects with rigor, it opens questions about how these insights can be applied beyond theoretical models, to real-world quantum systems. This work thus forms a pivotal foundation in integrating classical and quantum perspectives, enabling a more enriched understanding of entropy, information scrambling, and quantum chaos.