Complexity Growth in Integrable and Chaotic Models (2101.02209v2)

Abstract: We use the SYK family of models with $N$ Majorana fermions to study the complexity of time evolution, formulated as the shortest geodesic length on the unitary group manifold between the identity and the time evolution operator, in free, integrable, and chaotic systems. Initially, the shortest geodesic follows the time evolution trajectory, and hence complexity grows linearly in time. We study how this linear growth is eventually truncated by the appearance and accumulation of conjugate points, which signal the presence of shorter geodesics intersecting the time evolution trajectory. By explicitly locating such "shortcuts" through analytical and numerical methods, we demonstrate that: (a) in the free theory, time evolution encounters conjugate points at a polynomial time; consequently complexity growth truncates at $O(\sqrt{N})$, and we find an explicit operator which "fast-forwards" the free $N$-fermion time evolution with this complexity, (b) in a class of interacting integrable theories, the complexity is upper bounded by $O({\rm poly}(N))$, and (c) in chaotic theories, we argue that conjugate points do not occur until exponential times $O(e^N)$, after which it becomes possible to find infinitesimally nearby geodesics which approximate the time evolution operator. Finally, we explore the notion of eigenstate complexity in free, integrable, and chaotic models.

• The paper demonstrates that conjugate points act as critical markers, truncating complexity growth at polynomial scales in free and integrable models and at exponential scales in chaotic systems.
• It employs both analytical and numerical methods to map the evolution of quantum complexity through the lens of the SYK family, highlighting differences in time evolution across different system dynamics.
• The study’s findings offer practical implications for quantum simulations and error correction by uncovering potential shortcuts that optimize the simulation of complex quantum systems.

Analyzing Complexity Growth in Integrable and Chaotic Models

The paper "Complexity Growth in Integrable and Chaotic Models" provides a detailed exploration of how quantum complexity evolves through time in systems with Majorana fermions by studying models within the Sachdev-Ye-Kitaev (SYK) family. The research focuses on understanding complexity as the shortest geodesic length on the unitary group manifold, contrasting free, integrable, and chaotic systems. Specifically, it analyzes how the growth of complexity is curtailed by the emergence of conjugate points that indicate possible shorter geodesics intersecting the trajectory of time evolution.

Key Findings and Methodology

The research employs both analytical and numerical techniques to identify and understand the conjugate points - places where geodesic paths meet and suggest shortcuts. The major observations are as follows:

1. Free Theory Complexity: In free theories, the behavior of complexity is constrained by conjugate points appearing at polynomial times, leading to a truncation in the complexity growth at $O(\sqrt{N})$. A specific operator was identified that can "fast-forward" the evolution in such systems by exploiting this reduced complexity.
2. Integrable Systems: Interacting integrable theories show complexity bounded by $O({\rm poly}(N))$. In these systems, conjugate points are delayed as the integrable interaction is introduced, which results in longer linear complexity growth phases before saturation.
3. Chaotic Models: For chaotic systems, conjugate points appear only after exponential times, specifically at $O(e^N)$, significantly delaying the saturation of complexity. By using principles from random matrix theory and concepts like the Eigenstate Thermalization Hypothesis (ETH), the paper demonstrates why local conjugate points, which arise at polynomial times in free theories, do not affect chaotic theories until much later.

Implications

The paper's findings suggest a hierarchical pattern in complexity growth across different systems:

• Free Systems: Complexity growth is limited quickly by conjugate points, leading to potential optimizations in simulating time evolution.
• Integrable Systems: These can serve as bridges between free and chaotic behaviors, where complexity evolves longer before plateauing.
• Chaotic Systems: The delayed onset of conjugate points aligns with the hypothesis that such systems can continue to grow in complexity exponentially, implicating richer behaviors in phenomena like quantum gravity and holography.

Future Directions

This work opens avenues for further exploration:

• Understanding Conjugate Points in Greater Depth: There is potential for developing more efficient algorithms that capture the role of conjugate points in other sophisticated models, expanding the current understanding.
• Complexity in Quantum Error Correction: There may be parallels and implications for quantum error correction codes; understanding them could enrich the discussions related to bulk-boundary maps within the AdS/CFT context.
• Quantum Computational Applications: By translating these complexity analyses into quantum computational terms, future studies might leverage unexpected shortcuts for practical quantum simulations.

This paper provides significant insights into quantum complexity, offering a robust pathway to understanding how complexity scales and transforms across various quantum theories. As research continues, both the theoretical intuition and numerical evaluation presented here will likely contribute towards more profound applications and interpretations within quantum computation and quantum gravity.