# Local random quantum circuits: Ensemble completely positive maps and swap algebras

## Abstract

We define different classes of local random quantum circuits (L-RQC) and show that (a) statistical properties of L-RQC are encoded into an associated family of completely positive maps and (b) average purity dynamics can be described by the action of these maps on operator algebras of permutations (swap algebras). An exactly solvable one-dimensional case is analyzed to illustrate the power of the swap algebra formalism. More in general, we prove short time area-law bounds on average purity for uncorrelated L-RQC and infinite time results for both the uncorrelated and correlated cases.

- Authors:

- Department of Physics and Astronomy, and Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, California 90089-0484, USA and Centre for Quantum Technologies, National University of Singapore, 2 Science Drive 3, Singapore 117542 (Singapore)

- Publication Date:

- OSTI Identifier:
- 22306089

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 8; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; EXACT SOLUTIONS; ONE-DIMENSIONAL CALCULATIONS; QUANTUM MECHANICS; RANDOMNESS; STATISTICS

### Citation Formats

```
Zanardi, Paolo.
```*Local random quantum circuits: Ensemble completely positive maps and swap algebras*. United States: N. p., 2014.
Web. doi:10.1063/1.4891604.

```
Zanardi, Paolo.
```*Local random quantum circuits: Ensemble completely positive maps and swap algebras*. United States. doi:10.1063/1.4891604.

```
Zanardi, Paolo. Fri .
"Local random quantum circuits: Ensemble completely positive maps and swap algebras". United States.
doi:10.1063/1.4891604.
```

```
@article{osti_22306089,
```

title = {Local random quantum circuits: Ensemble completely positive maps and swap algebras},

author = {Zanardi, Paolo},

abstractNote = {We define different classes of local random quantum circuits (L-RQC) and show that (a) statistical properties of L-RQC are encoded into an associated family of completely positive maps and (b) average purity dynamics can be described by the action of these maps on operator algebras of permutations (swap algebras). An exactly solvable one-dimensional case is analyzed to illustrate the power of the swap algebra formalism. More in general, we prove short time area-law bounds on average purity for uncorrelated L-RQC and infinite time results for both the uncorrelated and correlated cases.},

doi = {10.1063/1.4891604},

journal = {Journal of Mathematical Physics},

number = 8,

volume = 55,

place = {United States},

year = {Fri Aug 15 00:00:00 EDT 2014},

month = {Fri Aug 15 00:00:00 EDT 2014}

}

DOI: 10.1063/1.4891604

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