# Convergence conditions for random quantum circuits

## Abstract

Efficient methods for generating pseudorandomly distributed unitary operators are needed for the practical application of Haar-distributed random operators in quantum communication and noise estimation protocols. We develop a theoretical framework for analyzing pseudorandom ensembles generated through a random circuit composition. We prove that the measure over random circuits converges exponentially (with increasing circuit length) to the uniform (Haar) measure on the unitary group, though the rate for uniform convergence must decrease exponentially with the number of qubits. We describe how the rate of convergence for test functions associated with specific randomization tasks leads to weaker convergence conditions that may allow efficient random circuit constructions.

- Authors:

- Institute for Quantum Computing and Department of Applied Math, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1 (Canada)
- (Canada)
- Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, N2L 2Y5 (Canada)
- Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139 (United States)

- Publication Date:

- OSTI Identifier:
- 20786237

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. A; Journal Volume: 72; Journal Issue: 6; Other Information: DOI: 10.1103/PhysRevA.72.060302; (c) 2005 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS; CONVERGENCE; DATA TRANSMISSION; NOISE; QUANTUM COMPUTERS; QUANTUM MECHANICS; QUBITS; RANDOMNESS

### Citation Formats

```
Emerson, Joseph, Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, N2L 2Y5, Livine, Etera, and Lloyd, Seth.
```*Convergence conditions for random quantum circuits*. United States: N. p., 2005.
Web. doi:10.1103/PHYSREVA.72.0.

```
Emerson, Joseph, Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, N2L 2Y5, Livine, Etera, & Lloyd, Seth.
```*Convergence conditions for random quantum circuits*. United States. doi:10.1103/PHYSREVA.72.0.

```
Emerson, Joseph, Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, N2L 2Y5, Livine, Etera, and Lloyd, Seth. Thu .
"Convergence conditions for random quantum circuits". United States.
doi:10.1103/PHYSREVA.72.0.
```

```
@article{osti_20786237,
```

title = {Convergence conditions for random quantum circuits},

author = {Emerson, Joseph and Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, N2L 2Y5 and Livine, Etera and Lloyd, Seth},

abstractNote = {Efficient methods for generating pseudorandomly distributed unitary operators are needed for the practical application of Haar-distributed random operators in quantum communication and noise estimation protocols. We develop a theoretical framework for analyzing pseudorandom ensembles generated through a random circuit composition. We prove that the measure over random circuits converges exponentially (with increasing circuit length) to the uniform (Haar) measure on the unitary group, though the rate for uniform convergence must decrease exponentially with the number of qubits. We describe how the rate of convergence for test functions associated with specific randomization tasks leads to weaker convergence conditions that may allow efficient random circuit constructions.},

doi = {10.1103/PHYSREVA.72.0},

journal = {Physical Review. A},

number = 6,

volume = 72,

place = {United States},

year = {Thu Dec 15 00:00:00 EST 2005},

month = {Thu Dec 15 00:00:00 EST 2005}

}