Complexity Functionals and Complexity Growth Limits in Continuous MERA Circuits

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.

26 pages, improved presentation, new section on real time evolution, references added

References in corpus (5)

• Quantum speed limit for physical processes
• Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT
• On the Time Dependence of Holographic Complexity
• Complexity is Simple
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Cited by in corpus (26)

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• Circuit complexity in interacting QFTs and RG flows
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• Time Evolution of Complexity: A Critique of Three Methods
• Universal chaotic dynamics from Krylov space
• Quantum Complexity of Time Evolution with Chaotic Hamiltonians
• Quantum Dynamics in Krylov Space: Methods and Applications
• Ultimate Speed Limits to the Growth of Operator Complexity
• Developments in the Tensor Network -- from Statistical Mechanics to Quantum Entanglement
• Holographic Spacetimes as Quantum Circuits of Path-Integrations
• Aspects of The First Law of Complexity
• Quantum speed limits on operator flows and correlation functions
• Geometric Operator Quantum Speed Limit, Wegner Hamiltonian Flow and Operator Growth
• Binding Complexity and Multiparty Entanglement
• Surface/State correspondence and $T\overline{T}$ deformation
• Holographic Path-Integral Optimization
• Decay of a Thermofield-Double State in Chaotic Quantum Systems
• Geometry and Complexity of Path Integrals in Inhomogeneous CFTs
• Complexity growth of rotating black holes with a probe string
• Generalised quantum speed limit for arbitrary time-continuous evolution
• Holographic Complexity of LST and Single Trace $T\bar{T}$
• Complexity growth of massive black hole with a probe string
• Holographic Complexity of LST and Single Trace $T\bar{T}$, $J\bar{T}$ and $T\bar{J}$ deformations
• Complexity growth of BTZ black hole in massive gravity with a null string
• Quantum fluctuation on the worldsheet of probe string in BTZ black hole
• Holographic and QFT Complexity with angular momentum