Complexity Functionals and Complexity Growth Limits in Continuous MERA Circuits

TL;DR

This paper introduces a new way to measure the complexity of quantum field theory states using cMERA tensor networks, linking it to Liouville theory and quantum gravity, and analyzing its growth limits.

Contribution

It establishes an operational complexity measure for cMERA circuits, connecting complexity to Liouville field theory and quantum gravity, and explores growth bounds in quantum dynamics.

Findings

Complexity equals least action in cMERA circuits.

Liouville theory describes the complexity functional.

Complexity growth saturates quantum speed limits.

Abstract

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.