# 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.

- Authors:

- Universidad Politencia de Cartagena (Spain)
- Univ. of Massachusetts, Boston, MA (United States); Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1467335

- Report Number(s):
- [LA-UR-18-21985]

[Journal ID: ISSN 1029-8479]

- Grant/Contract Number:
- [AC52-06NA25396]

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- [Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2018; Journal Issue: 8]; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; Atomic and Nuclear Physics

### Citation Formats

```
Molina Vilaplana, J., and del Campo, Adolfo. Complexity functionals and complexity growth limits in continuous MERA circuits. United States: N. p., 2018.
Web. doi:10.1007/JHEP08(2018)012.
```

```
Molina Vilaplana, J., & del Campo, Adolfo. Complexity functionals and complexity growth limits in continuous MERA circuits. United States. doi:10.1007/JHEP08(2018)012.
```

```
Molina Vilaplana, J., and del Campo, Adolfo. Fri .
"Complexity functionals and complexity growth limits in continuous MERA circuits". United States. doi:10.1007/JHEP08(2018)012. https://www.osti.gov/servlets/purl/1467335.
```

```
@article{osti_1467335,
```

title = {Complexity functionals and complexity growth limits in continuous MERA circuits},

author = {Molina Vilaplana, J. and del Campo, Adolfo},

abstractNote = {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.},

doi = {10.1007/JHEP08(2018)012},

journal = {Journal of High Energy Physics (Online)},

number = [8],

volume = [2018],

place = {United States},

year = {2018},

month = {8}

}

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Works referencing / citing this record:

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