Lorentzian threads as 'gatelines' and holographic 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'.

5 pages + Supplemental Material. v3: version accepted for publication in PRL

References in corpus (13)

• A class of quantum many-body states that can be efficiently simulated
• Complexity and Shock Wave Geometries
• Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT
• Entwinement and the emergence of spacetime
• Einstein's Equations from Varying Complexity
• Path integral optimization as circuit complexity
• Entanglement equilibrium for higher order gravity
• Path-Integral Optimization from Hartle-Hawking Wave Function
• Holographic Path-Integral Optimization
• Spacetime as a quantum circuit
• A Generalized Momentum/Complexity Correspondence
• Holographic excited states in AdS Black Holes
• From sources to initial data and back again: on bulk singularities in Euclidean AdS/CFT

Cited by in corpus (27)

• Krylov complexity of density matrix operators
• Black holes in dS${}_3$
• Holographic Complexity and de Sitter Space
• Sewing spacetime with Lorentzian threads: complexity and the emergence of time in quantum gravity
• Edge modes as reference frames and boundary actions from post-selection
• Bath deformations, islands and holographic complexity
• Holographic Complexity in dS${}_{d+1}$
• Spread complexity in saddle-dominated scrambling
• Complexity=Anything: Singularity Probes
• Quantum Kerr-de Sitter black holes in three dimensions
• The PEE aspects of entanglement islands from bit threads
• Computing spacetime
• Kasner interiors from analytic hairy black holes
• Holographic complexity of Jackiw-Teitelboim gravity from Karch-Randall braneworld
• Brickwall One-Loop Determinant: Spectral Statistics & Krylov Complexity
• Chaos and multifold complexity for an inverted harmonic oscillator
• Quantum bit threads and holographic entanglement
• Correlations of mixed systems in confining backgrounds
• Thread/State correspondence: from bit threads to qubit threads
• Holographic thermal entropy from geodesic bit threads
• Diving inside holographic metals
• Improved proof-by-contraction method and relative homologous entropy inequalities
• Toward Krylov-based holography in double-scaled SYK
• Towards bit threads in general gravitational spacetimes
• The landscape of complexity measures in 2D gravity
• Non-linear equation of motion for higher curvature semiclassical gravity
• Cost of holographic path integrals