Complexity Equals Action

Abstract: We conjecture that the quantum complexity of a holographic state is dual to the action of a certain spacetime region that we call a Wheeler-DeWitt patch. We illustrate and test the conjecture in the context of neutral, charged, and rotating black holes in AdS, as well as black holes perturbed with static shells and with shock waves. This conjecture evolved from a previous conjecture that complexity is dual to spatial volume, but appears to be a major improvement over the original. In light of our results, we discuss the hypothesis that black holes are the fastest computers in nature.

• The paper introduces a novel conjecture linking quantum complexity to the gravitational action within the Wheeler-DeWitt patch, overcoming limitations of volume-based models.
• It analyzes various AdS black hole configurations, including charged and rotating types, to demonstrate a universal computational bound aligned with Lloyd's limit.
• Perturbative evidence from shockwave dynamics reinforces the link between complexity growth and gravitational effects, bridging quantum information and spacetime geometry.

Complexity Equals Action: An Exploration of Holographic Complexity

The paper "Complexity Equals Action" presents a significant conjecture within the framework of holography, specifically relating to the AdS/CFT correspondence. The authors propose that the quantum complexity of a holographic state corresponds to the action of a specific spacetime region known as the Wheeler-DeWitt (WDW) patch. This hypothesis builds on an earlier conjecture suggesting complexity is related to spatial volume, offering a refined perspective that aligns more closely with theoretical expectations.

The authors substantiate their conjecture through detailed analyses involving various configurations of black holes in Anti-de Sitter (AdS) space, including neutral, charged, and rotating black holes, as well as those subjected to static and shockwave perturbations. They argue that this new conjecture overcomes limitations of the previous volume-based conjecture, which required arbitrary length scales for different configurations.

Key Components of the Conjecture

1. Complexity-Action Relationship: The authors propose that the computational complexity of forming a boundary state from a reference state is directly proportional to the action within the WDW patch. Specifically, they frame complexity in terms of a gravitational action, using the Einstein-Hilbert action with additional contributions for regions with charged or rotating components.
2. Black Holes as Computation Models: The paper posits that black holes might represent the ultimate computational platforms, potentially saturating the conjectured limits on the rate of computation, commonly referred to as Lloyd's bound. This conjecture is framed within the broader context of quantum computing and information theory, suggesting black holes could operate at maximum efficiency in processing information.
3. Universality and Generality: According to the paper, the complexity-action conjecture holds for black holes of varying sizes and dimensionalities, showing a universality that was not apparent in previous conjectures which linked complexity to spatial volume. The results suggest a consistent computational bound across different black hole configurations without the need for introducing varying parameters.
4. Perturbative Evidence: The paper strengthens its claims through the analysis of perturbations, including thermal-scale operators manifesting as ingoing null shock waves. The complexity growth, as calculated from the action of these shock waves, aligns with theoretical predictions around chaotic growth and the scrambling effect in quantum systems. This agreement supports the idea of complexity being intimately tied to gravitation and geometry.

Implications and Future Directions

The implications of this conjecture are profound for both theoretical and practical purposes. The hypothesis offers a compelling framework for understanding quantum complexity in holographic systems, potentially bridging gaps between gravitational theories and quantum information. Practically, if black holes can indeed be viewed as optimal computing entities, this could influence how computational processes are conceptualized in high-energy physics and cosmology.

Future research could explore verifying the conjecture across broader classes of boundary theories, including those that introduce higher-derivative terms in the bulk action. Additionally, exploring the ground state complexity and how it informs the dynamics of holographic dualities remains a critical area of interest. The conjecture also encourages examination into the conjecture’s compatibility with less strongly-coupled boundary theories, which may bring new insights into the structure of holographic duality.

This paper invites researchers to further dissect the delicate interplay between complexity, action, and gravity, potentially illuminating deeper principles underlying quantum gravity and spacetime. Such inquiry pushes forward the boundaries of current understanding, suggesting a paradigm where complex computational processes are mirrored in the fundamental fabric of the universe.