Chaos, Complexity, and Random Matrices (1706.05400v2)

Abstract: Chaos and complexity entail an entropic and computational obstruction to describing a system, and thus are intrinsically difficult to characterize. In this paper, we consider time evolution by Gaussian Unitary Ensemble (GUE) Hamiltonians and analytically compute out-of-time-ordered correlation functions (OTOCs) and frame potentials to quantify scrambling, Haar-randomness, and circuit complexity. While our random matrix analysis gives a qualitatively correct prediction of the late-time behavior of chaotic systems, we find unphysical behavior at early times including an $\mathcal{O}(1)$ scrambling time and the apparent breakdown of spatial and temporal locality. The salient feature of GUE Hamiltonians which gives us computational traction is the Haar-invariance of the ensemble, meaning that the ensemble-averaged dynamics look the same in any basis. Motivated by this property of the GUE, we introduce $k$-invariance as a precise definition of what it means for the dynamics of a quantum system to be described by random matrix theory. We envision that the dynamical onset of approximate $k$-invariance will be a useful tool for capturing the transition from early-time chaos, as seen by OTOCs, to late-time chaos, as seen by random matrix theory.

• The paper demonstrates that GUE Hamiltonians induce an O(1) scrambling time through rapid OTOC decay linked to spectral form factors.
• It applies frame potentials to track the evolution toward Haar-randomness, showcasing a transient formation of unitary k-designs.
• The research establishes quantitative bounds on quantum circuit complexity growth, offering practical insights for quantum computation.

Chaos, Complexity, and Random Matrices

The paper "Chaos, Complexity, and Random Matrices" explores the intricate interplay between chaos and complexity within quantum systems. The authors, Cotler et al., rigorously explore the characteristics and implications of chaotic dynamics via Gaussian Unitary Ensemble (GUE) Hamiltonians, utilizing out-of-time-ordered correlation functions (OTOCs) and frame potentials to quantify the phenomena of scrambling, Haar-randomness, and circuit complexity.

Analytical Framework and Findings

The research critically hinges on the use of GUE Hamiltonians, whose Haar-invariance facilitates computational simplicity. The ensemble-averaged dynamics, irrespective of the basis, serve as a backdrop for understanding chaos through several theoretical lenses:

• OTOCs and Spectral Form Factors: The paper demonstrates a connection between the decay of OTOCs and spectral form factors. Specifically, they relate higher-point spectral form factors with averaged OTOCs leading to a broader understanding of chaos as quantified by these correlation functions. One notable discovery is that GUE predicts an OTOC decay at early times, yielding an $\mathcal{O}(1)$ scrambling time, indicating a faster-than-expected scrambling relative to thermalization time scales.
• Frame Potentials and Complexity: The authors track the progression of GUE-evolved unitaries towards Haar-randomness through the calculation of frame potentials. They find that the dynamics initially approach achieving a quantum unitary design before diverging from Haar randomness at later times. This manifests as a transient formation of unitary $k$-designs, which serve as a measure of how closely the ensemble mimics the uniformity of Haar randomness.

Implications of Quantum Dynamics

The theoretical implications extend into both the practical aspects of quantum computation and the fundamental understanding of quantum chaos:

• Quantum Circuit Complexity: By analyzing frame potentials within the GUE, the authors introduce quantitative bounds on the complexity growth of quantum circuits. Their results suggest a quadratic growth in the complexity, implying a steadily increasing demand for quantum computational resources over time.
• Delayed Haar-randomization: As noted in their treatment of $k$-designs, the time scales for achieving approximate Haar-uniformity inform both theoretical and experimental scenarios involving quantum systems striving for randomness and complete information scrambling.

Future Directions

The paper poses several enticing prospects for future research. One avenue is refining how $k$-invariance, which relates ensemble dynamics to Haar-invariance, serves as a metric for understanding the transition between early-time localized phenomena and the late-time universal behavior exhibited by random matrix theory. Additionally, translating these findings to other symmetry classes of random matrices and characterizing such dynamics in real-world quantum systems or black hole analogs could illuminate the universality of these chaotic phenomena.

Moreover, understanding the onset and development of late-time Haar-random behavior in different setups opens possibilities for advancements in both quantum information theory and quantum simulations. These explorations could have contextual implications in robust quantum algorithm design and black hole information paradox studies.

Conclusion

In summation, the work of Cotler et al. presents a solid foundation in differentiating the chaotic and complex behaviors of quantum systems via the lens of random matrix theory, offering critical insights into the dynamical capabilities of quantum models. Their methodological approach not only enriches the theoretical landscape but also equips future research with novel analytical tools to dig deeper into the phenomenology of quantum chaos and complexity.