Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT

Abstract

Abstract:We propose an optimization procedure for Euclidean path-integrals that evaluate CFT wave functionals in arbitrary dimensions. The optimization is performed by minimizing certain functional, which can be interpreted as a measure of computational complexity, with respect to background metrics for the path-integrals. In two dimensional CFTs, this functional is given by the Liouville action. We also formulate the optimization for higher dimensional CFTs and, in various examples, find that the optimized hyperbolic metrics coincide with the time slices of expected gravity duals. Moreover, if we optimize a reduced density matrix, the geometry becomes two copies of the entanglement wedge and reproduces the holographic entanglement entropy. Our approach resembles a continuous tensor network renormalization and provides a concrete realization of the proposed interpretation of AdS/CFT as tensor networks. The present paper is an extended version of our earlier report arXiv:1703.00456 and includes many new results such as evaluations of complexity functionals, energy stress tensor, higher dimensional extensions and time evolutions of thermofield double states.

Topological Complexity in AdS3/CFT2

“…Interesting questions for the future include generalizing this approach to higher dimensions; understanding subregion complexity using the optimization approach of []; relating our approach with the holographic renormalization properties of the different proposals for complexity; and studying subregion complexity in time‐dependent systems …”

“…Interesting questions for the future include generalizing this approach to higher dimensions; understanding subregion complexity using the optimization approach of [38,[46][47][48]; relating our approach with the holographic renormalization properties of the different proposals for complexity; [11,49] and studying subregion complexity in time-dependent systems. [12] Turning to tensor network states, we proposed that their subregion complexity should be understood as the number of local tensors required to build the map embedding the the Hilbert space cut by the RT surface in the Hilbert space of the entangling region A.…”

Holographic complexity in Vaidya spacetimes. Part I

“…Ultimately, one would like to establish a concrete translation of the new observables in the bulk to a specific quantity in the boundary theory, but this probably requires new insights into how complexity can be formulated in quantum field theories. Some interesting progress in this direction can be found in [39][40][41].…”

Complexity is simple!

“…More precisely, we will define a gate to act simply on the state |ψ if ψ|G i |ψ ∼ 1−O( ), where is the tolerance. 4 Although this definition is slightly outside the original framework described in [38], it is nevertheless in harmony with recent attempts to represent a quantum circuit by taking the continuum limit of the tensor network preparing the state [43]. From the tensor network perspective, each node in the network contributes a gate like e vO , where O is a hermitian operator and 'v' is roughly the infinitesimal volume associated with the gate.…”