Quantum chaos and the complexity of spread of states (2202.06957v2)

Abstract: We propose a measure of quantum state complexity defined by minimizing the spread of the wave-function over all choices of basis. Our measure is controlled by the "survival amplitude" for a state to remain unchanged, and can be efficiently computed in theories with discrete spectra. For continuous Hamiltonian evolution, it generalizes Krylov operator complexity to quantum states. We apply our methods to the harmonic and inverted oscillators, particles on group manifolds, the Schwarzian theory, the SYK model, and random matrix models. For time-evolved thermofield double states in chaotic systems our measure shows four regimes: a linear "ramp" up to a "peak" that is exponential in the entropy, followed by a "slope" down to a "plateau". These regimes arise in the same physics producing the slope-dip-ramp-plateau structure of the Spectral Form Factor. Specifically, the complexity slope arises from spectral rigidity, distinguishing different random matrix ensembles.

• The paper introduces a novel complexity measure by minimizing the spread of a quantum state over orthonormal bases using the Lanczos algorithm.
• It validates the approach numerically across systems like the Schwarzian theory, SYK model, and random matrices, revealing linear growth, overshoot, and plateau dynamics.
• The study links complexity evolution to spectral rigidity and energy variance, offering actionable insights into chaotic dynamics in quantum systems.

Quantum Chaos and the Complexity of the Spread of States

The paper, "Quantum chaos and the complexity of spread of states" by Balasubramanian, Caputa, Magan, and Wu, presents a distinctive approach to quantifying the complexity of quantum states. This research innovatively suggests that complexity can be defined by minimizing the spread of a quantum state over various orthonormal bases. Utilizing concepts from quantum chaos and information theory, this measure is analytically and numerically explored in multiple quantum systems, offering insights into the dynamics of state complexity.

The authors propose defining quantum state complexity by finding the basis that minimizes the spread of the state wavefunction, measured as the survival amplitude—the probability of the wavefunction remaining unchanged. This definition circumvents the usual basis ambiguity by minimizing over all basis choices, a task achievable through the Lanczos algorithm, which generates a recast Hamiltonian in a tri-diagonal form within the Krylov basis. This approach builds on foundational ideas by Kolmogorov and Rissanen, extending them to the quantum domain.

The complexity measure is validated through application to a range of systems, including the Schwarzian theory, random matrix models, and the Sachdev-Ye-Kitaev (SYK) model. The complexity in chaotic systems typically grows linearly for times exponentially long in terms of the system's entropy before reaching a saturation plateau, aligning with prior conjectures about the behavior of quantum complexity in such settings. However, this paper adds nuance by revealing an overshoot into a peak, followed by a descent—termed a "slope"—before settling into the plateau. This slope resembles the ramp in the spectral form factor, driven by eigenvalue spectral rigidity, a universal feature of quantum chaotic systems and random matrix theory.

The examination of various systems emphasizes that spread complexity, beyond just confirming the expected linear growth, exposes the intricate dynamics of quantum state evolution. For instance, the analysis of the Schwarzian theory showed a close relation between state complexity and energy variance, reinforced by matching transitions in the chaotic regime characterized by random matrix statistics. This allows the authors to conjecture about the universality of these dynamics across quantum chaotic models.

Furthermore, the paper connects the Lanczos coefficients' growth patterns to the predicted complexity dynamics. Through numerical simulations involving the SYK model and random matrix ensembles, the results illustrate the presence of the four distinct complexity regimes, further reinforcing the linkage between spectral rigidity and complexity slope after the peak.

This research not only strengthens our understanding of the relationship between quantum chaos and complexity but also sets the stage for exploring the implications in other complex quantum systems, potentially including quantum gravity and black hole physics. The results suggest deeper inquiries into how quantum state complexity might illuminate the inner workings of quantum fields in chaotic and integrable regimes, highlighting potential bridges to quantum geometry and thermodynamics.