Einstein's Equations from Varying Complexity

Abstract

A recent proposal equates the circuit complexity of a quantum gravity state with the gravitational action of a certain patch of spacetime. Since Einstein's equations follow from varying the action, it should be possible to derive them by varying complexity. I present such a derivation for vacuum solutions of pure Einstein gravity in three-dimensional asymptotically anti-de Sitter space. The argument relies on known facts about holography and on properties of Tensor Network Renormalization, an algorithm for coarse-graining (and optimizing) tensor networks.

4 pages, 2 figures

References in corpus (22)

• The Large N Limit of Superconformal Field Theories and Supergravity
• Holographic Derivation of Entanglement Entropy from AdS/CFT
• Geometry of the 2+1 Black Hole
• Entanglement renormalization
• A class of quantum many-body states that can be efficiently simulated
• Complexity Equals Action
• Entanglement Renormalization and Holography
• Complexity, action, and black holes
• Tensor Network Renormalization
• Three-dimensional quantum geometry and black holes
• Anti-de Sitter Space from Optimization of Path Integrals in Conformal Field Theories
• Entanglement renormalization for quantum fields
• A Stereoscopic Look into the Bulk
• Warped Conformal Field Theory
• A hole-ographic spacetime
• Entanglement, Holography and Causal Diamonds
• Tensor network renormalization yields the multi-scale entanglement renormalization ansatz
• Tensor Networks from Kinematic Space
• From Path Integrals to Tensor Networks for AdS/CFT
• Causal structure of the entanglement renormalization ansatz
• A tensor network quotient takes the vacuum to the thermal state
• Local scale transformations on the lattice with tensor network renormalization

Cited by in corpus (109)

• Towards Complexity for Quantum Field Theory States
• Quantum chaos and the complexity of spread of states
• Complexity and entanglement for thermofield double states
• Quantum Computation as Gravity
• Circuit complexity in interacting QFTs and RG flows
• Holographic Complexity in Vaidya Spacetimes II
• Quantum Computational Complexity -- From Quantum Information to Black Holes and Back
• Circuit Complexity for Coherent States
• Complexity as a novel probe of quantum quenches: universal scalings and purifications
• Path-Integral Complexity for Perturbed CFTs
• Time Evolution of Complexity: A Critique of Three Methods
• Chaos and Complexity in Quantum Mechanics
• Holographic Complexity in Vaidya Spacetimes I
• Quantum Complexity of Time Evolution with Chaotic Hamiltonians
• Quantum Dynamics in Krylov Space: Methods and Applications
• Complexity of Mixed States in QFT and Holography
• Complexity and the bulk volume, a new York time story
• Holographic Complexity Equals Which Action?
• The First Law of Complexity
• Path integral optimization as circuit complexity
• Holographic Spacetimes as Quantum Circuits of Path-Integrations
• Holographic Subregion Complexity from Kinematic Space
• Quantum Information in Holographic Duality
• Holographic Entanglement of Purification from Conformal Field Theories
• Complexity for Conformal Field Theories in General Dimensions
• Renormalized Circuit Complexity
• Cosmological Complexity
• The Multi-faceted Inverted Harmonic Oscillator: Chaos and Complexity
• Holographic subregion complexity under a thermal quench
• Holographic complexity and non-commutative gauge theory
• Complexity change under conformal transformations in AdS${}_3$/CFT${}_2$
• Krylov complexity of density matrix operators
• Sewing spacetime with Lorentzian threads: complexity and the emergence of time in quantum gravity
• Action growth rate for a higher curvature gravitational theory
• Complexity of Holographic Superconductors
• Holographic Complexity in dS${}_{d+1}$
• Lorentzian threads as 'gatelines' and holographic complexity
• On complexity growth in massive gravity theories, the effects of chirality and more
• WdW-patches in AdS${}_3$ and complexity change under conformal transformations II
• Spread complexity in saddle-dominated scrambling
• From CFTs to theories with Bondi-Metzner-Sachs symmetries: Complexity and out-of-time-ordered correlators
• Spread and Spectral Complexity in Quantum Spin Chains: from Integrability to Chaos
• Quantum complexity in gravity, quantum field theory, and quantum information science
• Quantum state complexity meets many-body scars
• Complexity from the Reduced Density Matrix: a new Diagnostic for Chaos
• Path-Integral Optimization from Hartle-Hawking Wave Function
• Circuit complexity for generalised coherent states in thermal field dynamics
• Binding Complexity and Multiparty Entanglement
• Conformal bootstrap to Rényi entropy in 2D Liouville and super-Liouville CFTs
• Cutoff Dependence and Complexity of the CFT${}_2$ Ground State
• Holographic Path-Integral Optimization
• Krylov Complexity and Spectral Form Factor for Noisy Random Matrix Models
• Geometry and Complexity of Path Integrals in Inhomogeneous CFTs
• Holographic Subregion Complexity in Einstein-Born-Infeld theory
• Complexity growth of rotating black holes with a probe string
• Krylov spread complexity as holographic complexity beyond JT gravity
• Investigating two counting methods of the holographic complexity
• The Early Universe as an Open Quantum System: Complexity and Decoherence
• Subsystem complexity after a global quantum quench
• Holographic complexity in charged Vaidya black hole
• Spacetime as a quantum circuit
• Complexity for Open Quantum System
• Quantum entanglement in cosmology
• Complexity and Emergence of Warped ${\text{AdS}}_3$ Space-time from Chiral Liouville Action
• Decoherence, Entanglement Negativity and Circuit Complexity for Open Quantum System
• Computing spacetime
• Kasner interiors from analytic hairy black holes
• Emergent Einstein Equation in p-adic CFT Tensor Networks
• Complexity of operators generated by quantum mechanical Hamiltonians
• A non-perturbative field theory approach for the Kondo effect: Emergence of an extra dimension and its implication for the holographic duality conjecture
• Path Integral Complexity and Kasner singularities
• Holographic complexity for black branes with momentum relaxation
• Revisit on holographic complexity in two-dimensional gravity
• Continuous tensor network renormalization for quantum fields
• Bulk reconstruction of metrics inside black holes by complexity
• Brickwall One-Loop Determinant: Spectral Statistics & Krylov Complexity
• Q-curvature and Path Integral Complexity
• Tensor networks as conformal transformations
• Chaos and multifold complexity for an inverted harmonic oscillator
• Emergent geometry in recursive renormalization group transformations
• Complexity growth in Gubser-Rocha models with momentum relaxation
• Quantum computational complexity, Einstein's equations and accelerated expansion of the Universe
• Dynamics of monitored SSH Model in Krylov Space: From Complexity to Quantum Fisher Information
• Subregion complexity in holographic thermalization with dS boundary
• Holographic complexity of the disk subregion in (2+1)-dimensional gapped systems
• Bending the Bruhat-Tits Tree I:Tensor Network and Emergent Einstein Equations
• Geometric encoding of renormalization group $\beta -$functions in an emergent holographic dual description
• To be unitary-invariant or not?: a simple but non-trivial proposal for the complexity between states in quantum mechanics/field theory
• Subregion complexity and confinement-deconfinement transition in a holographic QCD model
• Sectional curvatures distribution of complexity geometry
• Emergent dual holographic description as a non-perturbative generalization of the Wilsonian renormalization group
• The entanglement properties of holographic QCD model with a critical end point
• Holographic Complexity for disentangled states
• Entanglement transfer from quantum matter to classical geometry in an emergent holographic dual description of a scalar field theory
• Spacetime as the optimal generative network of quantum states: a roadmap to QM=GR?
• Complexity growth for topological black holes by holographic method
• Weighting gates in circuit complexity and holography
• Holographic complexity for nonlinearly charged Lifshitz black holes
• Towards a Fisher-information description of complexity in de Sitter universe
• Holographic Complexity from the Crofton's Formula in Lorentzian AdS${}_3$
• What kind of "complexity" is dual to holographic complexity?
• Gravity dualities of quantum distances
• Probing the self-coherence of primordial quantum fluctuations with complexity
• Purification Complexity without Purifications
• Holographic and QFT Complexity with angular momentum
• Circuit Complexity for Coherent-Thermal States in Bosonic String Theory
• Holographic entanglement renormalisation for fermionic quantum matter
• Coherent States and Particle Scattering in Loop Quantum Gravity
• Comparison between renormalization group derived emergent dual holography and string theory based holographic duality