Lorentzian threads as 'gatelines' and holographic complexity

TL;DR

This paper reformulates the holographic complexity conjecture using Lorentzian flows, introducing a thread-based interpretation and a refined measure that connects tensor networks, spacetime perturbations, and Einstein's equations.

Contribution

It introduces Lorentzian flow-based reformulation of holographic complexity, linking tensor networks, spacetime dynamics, and a new ensemble measure of complexity.

Findings

Complexity equals minimal flux of Lorentzian flows.

Nested flows bound the rate of complexity growth.

Bulk Einstein equations relate to the holographic first law of complexity.

Abstract

The continuous min flow-max cut principle is used to reformulate the 'complexity=volume' conjecture using Lorentzian flows -- divergenceless norm-bounded timelike vector fields whose minimum flux through a boundary subregion is equal to the volume of the homologous maximal bulk Cauchy slice. The nesting property is used to show the rate of complexity is bounded below by "conditional complexity", describing a multi-step optimization with intermediate and final target states. Conceptually, discretized Lorentzian flows are interpreted in terms of threads or gatelines such that complexity is equal to the minimum number of gatelines used to prepare a CFT state by an optimal tensor network (TN) discretizing the state. We propose a refined measure of complexity, capturing the role of suboptimal TNs, as an ensemble average. The bulk symplectic potential provides a 'canonical' thread configuration characterizing perturbations around arbitrary CFT states. Its consistency requires the bulk to obey linearized Einstein's equations, which are shown to be equivalent to the holographic first law of complexity, thereby advocating a notion of 'spacetime complexity'.