Complexity Functionals and Complexity Growth Limits in Continuous MERA Circuits

TL;DR

The paper defines a path-integral-based, operational notion of complexity for continuous MERA (cMERA) circuits in QFT by mapping the RG flow to a coherence-path integral with action A[Φ]. It shows a deep link between the complexity action and Liouville action, enabling a Liouville-field interpretation (φL) and a geometric hyperbolic structure, leading to the principle of Least Action = Minimal Complexity. It introduces Complexity Growth Limits (CGL) of Margolus-Levitin and Mandelstam-Tamm type, tying the rate of complexity increase to the mean and fluctuations of the entanglement (K) Hamiltonian along the cMERA flow, and demonstrates saturation of these bounds in certain scenarios. The framework also connects complexity to entanglement measures (LREE) and explores perturbations via Liouville vertex operators, extensions to interacting theories, and time-dependent quenches, suggesting a holography-like link between quantum complexity and emergent gravitational descriptions without assuming holographic duality a priori. Overall, the work provides a quantitative, variational, and potentially holographic perspective on how entanglement structures in QFTs determine circuit complexity and emergent geometric descriptions.

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.