Onset of Random Matrix Behavior in Scrambling Systems (1803.08050v4)

Abstract: The fine grained energy spectrum of quantum chaotic systems is widely believed to be described by random matrix statistics. A basic scale in such a system is the energy range over which this behavior persists. We define the corresponding time scale by the time at which the linearly growing ramp region in the spectral form factor begins. We call this time $t_{\rm ramp}$. The purpose of this paper is to study this scale in many-body quantum systems that display strong chaos, sometimes called scrambling systems. We focus on randomly coupled qubit systems, both local and $k$-local (all-to-all interactions) and the Sachdev--Ye--Kitaev (SYK) model. Using numerical results for Hamiltonian systems and analytic estimates for random quantum circuits we find the following results. For geometrically local systems with a conservation law we find $t_{\rm ramp}$ is determined by the diffusion time across the system, order $N^2$ for a 1D chain of $N$ qubits. This is analogous to the behavior found for local one-body chaotic systems. For a $k$-local system with conservation law the time is order $\log N$ but with a different prefactor and a different mechanism than the scrambling time. In the absence of any conservation laws, as in a generic random quantum circuit, we find $t_{\rm ramp} \sim \log N$, independent of connectivity.

• The paper examines the time scale ( t _ramp) when random matrix theory statistics appear in strongly chaotic quantum systems.
• It finds t _ramp depends significantly on system type (local vs. k-local) and the presence or absence of conservation laws.
• Understanding this time scale is crucial for quantum information applications and theoretical insights into quantum chaos and thermalization.

Summary of "Onset of Random Matrix Behavior in Scrambling Systems"

The paper "Onset of Random Matrix Behavior in Scrambling Systems" examines the emergence of random matrix theory (RMT) behavior in strongly chaotic quantum systems, also referred to as scrambling systems. The authors explore the time scale over which RMT level statistics, such as those given by the Wigner-Dyson distribution, become applicable in these many-body systems. This time scale is denoted as $t_{\text{ramp}}$.

Key Results and Methodology

The paper focuses on several quantum systems including randomly coupled qubit systems, both local and $k$-local configurations, and the Sachdev-Ye-Kitaev (SYK) model. The systems are analyzed using both numerical simulations for Hamiltonian systems and analytical estimates derived from random quantum circuits.

1. Time Scales and Dynamics: The paper delineates different time scales in many-body quantum chaotic systems:
   • Dissipation time ($t_{\text{diss}}$): Characterized by the decay of two-point functions of typical operators.
   • Scrambling time ($t_{\text{scr}}$): The time taken for an initial localized perturbation to affect the entire system.
   • Ramp time ($t_{\text{ramp}}$): The onset of RMT behavior and spectral rigidity.
2. Numerical Results for Local and Non-local Systems:
   • For geometrically local systems with a conservation law, $t_{\text{ramp}}$ correlates with the diffusion time across the system, proportional to $N^2$ for a 1D chain of $N$ qubits.
   • For $k$-local systems, $t_{\text{ramp}}$ is of the order $\log N$, but distinct from scrambling times.
   • Random quantum circuits without conservation laws show $t_{\text{ramp}} \sim \log N$, independent of system connectivity.
3. Role of Conservation Laws: The presence of conservation laws significantly affects the time scales for the onset of RMT behavior. Systems conserving additional quantities (e.g., spin) exhibit enhanced diffusion timescales, which influence spectral statistics.

Implications

• Practical Implications: Understanding $t_{\text{ramp}}$ in many-body quantum systems is crucial for applications in quantum information, specifically for optimizing quantum algorithms that exploit chaotic dynamics for tasks such as state randomization and approximate unitary designs.
• Theoretical Implications: The results provide insights into the connection between quantum chaos and RMT universality by explicitly establishing the conditions under which RMT behavior manifests. This deepens our understanding of quantum thermalization and the complex transition from quantum coherence to statistical regularity.
• Future Prospects: Future work could explore higher-dimensional scrambling systems, systems with more complex conservation laws, and further develop the analytical frameworks to predict $t_{\text{ramp}}$ more accurately. Additionally, elucidating the role of conservation laws in detail may offer new pathways to control and predict quantum dynamics in engineered quantum systems.

Conclusion

The paper contributes to the understanding of non-equilibrium dynamics of quantum systems, framing the transition to random matrix universality as a fundamental time scale dependent on system parameters and conservation laws. This work holds significant potential in bridging quantum chaotic dynamics with practical applications in quantum computing and information processing.