Complexity functionals and complexity growth limits in continuous MERA circuits

• Published in: The journal of high energy physics Vol. 2018; no. 8; pp. 1 - 25
• Main Authors: Molina-Vilaplana, J., del Campo, A.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2018; Springer Nature B.V; Springer Berlin; SpringerOpen
• Subjects: 2D Gravity; Atomic and Nuclear Physics; Circuits; Classical and Quantum Gravitation; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Complexity; CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; Continuity (mathematics); Elementary Particles; Field Theories in Lower Dimensions; Field theory; Functionals; High energy physics; Holography and condensed matter physics (AdS/CMT); Liouville equations; Parameter identification; Physics; Physics and Astronomy; Quantum entanglement; Quantum Field Theories; Quantum Field Theory; Quantum gravity; Quantum Physics; Quantum theory; Regular Article - Theoretical Physics; Relativity Theory; Renormalization Group; Speed limits; String Theory; Field Theories in Lower Dimensions; Renormalization Group; 2D Gravity; Holography and condensed matter physics (AdS/CMT)
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP08(2018)012

A bstract Using the path integral associated to a cMERA tensor network, we provide an operational definition for the complexity of a cMERA circuit/state which is relevant to investigate the complexity of states in quantum field theory. In this framework, it is possible to explicitly establish the correspondence (Minimal) Complexity = (Least) Action. Remarkably, it is also shown how the cMERA complexity action functional can be seen as the action of a Liouville field theory, thus establishing a connection with two dimensional quantum gravity. Concretely, the Liouville mode is identified with the variational parameter defining the cMERA circuit. The rate of complexity growth along the cMERA renormalization group flow is obtained and shown to saturate limits which are in close resemblance to the fundamental bounds to the speed of evolution in unitary quantum dynamics, known as quantum speed limits. We also show that the complexity of a cMERA circuit measured through these complexity functionals, can be cast in terms of the variationally-optimized amount of left-right entanglement created along the cMERA renormalization flow. Our results suggest that the patterns of entanglement in states of a QFT could determine their dual gravitational descriptions through a principle of least complexity.