# Turbulent Mixing Simulation via a Quantum Algorithm

## Abstract

Probability density function (PDF) methods have been very useful in describing many physical aspects of turbulent mixing. In applications of these methods, modeled PDF transport equations are commonly simulated via classical Monte Carlo techniques, which provide estimates of moments of the PDF at arbitrary accuracy. In this paper, recently developed techniques in quantum computing and quantum enhanced measurements (quantum metrology) are used to construct a quantum algorithm that accelerates the computation of such estimates. This quantum algorithm provides a quadratic speedup over classical Monte Carlo methods in terms of the number of repetitions needed to achieve the desired precision. This paper illustrates the power of this algorithm by considering a binary scalar mixing process modeled by means of the coalescence/dispersion (C/D) closure. The equation is first simulated using classical Monte Carlo methods, where error estimates for the computation of central moments are provided. Then the quantum algorithm for this problem is simulated by sampling from the same probability distribution as that of the output of a quantum computer, and it is shown that significantly fewer resources are required to achieve the same precision. Finally, the results demonstrate potential applications of future quantum computers for simulation of turbulent mixing, and largemore »

- Authors:

- Univ. of Strathclyde, Glasgow, Scotland (United Kingdom). Dept. of Physics. SUPA
- Univ. of Pittsburgh, PA (United States). Mechanical Engineering and Petroleum Engineering
- 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; US Air Force Office of Scientific Research (AFOSR); Engineering and Physical Sciences Research Council (EPSRC)

- OSTI Identifier:
- 1458951

- Report Number(s):
- LA-UR-16-25360

Journal ID: ISSN 0001-1452

- Grant/Contract Number:
- AC52-06NA25396; FA9550-12-1-0057; EP/K000586/1

- Resource Type:
- Accepted Manuscript

- Journal Name:
- AIAA Journal

- Additional Journal Information:
- Journal Volume: 56; Journal Issue: 2; Journal ID: ISSN 0001-1452

- Publisher:
- AIAA

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Computer Science; Information Science; Quantum algorithms, turbulent mixing

### Citation Formats

```
Xu, Guanglei, Daley, Andrew J., Givi, Peyman, and Somma, Rolando D. Turbulent Mixing Simulation via a Quantum Algorithm. United States: N. p., 2017.
Web. doi:10.2514/1.J055896.
```

```
Xu, Guanglei, Daley, Andrew J., Givi, Peyman, & Somma, Rolando D. Turbulent Mixing Simulation via a Quantum Algorithm. United States. doi:10.2514/1.J055896.
```

```
Xu, Guanglei, Daley, Andrew J., Givi, Peyman, and Somma, Rolando D. Thu .
"Turbulent Mixing Simulation via a Quantum Algorithm". United States. doi:10.2514/1.J055896. https://www.osti.gov/servlets/purl/1458951.
```

```
@article{osti_1458951,
```

title = {Turbulent Mixing Simulation via a Quantum Algorithm},

author = {Xu, Guanglei and Daley, Andrew J. and Givi, Peyman and Somma, Rolando D.},

abstractNote = {Probability density function (PDF) methods have been very useful in describing many physical aspects of turbulent mixing. In applications of these methods, modeled PDF transport equations are commonly simulated via classical Monte Carlo techniques, which provide estimates of moments of the PDF at arbitrary accuracy. In this paper, recently developed techniques in quantum computing and quantum enhanced measurements (quantum metrology) are used to construct a quantum algorithm that accelerates the computation of such estimates. This quantum algorithm provides a quadratic speedup over classical Monte Carlo methods in terms of the number of repetitions needed to achieve the desired precision. This paper illustrates the power of this algorithm by considering a binary scalar mixing process modeled by means of the coalescence/dispersion (C/D) closure. The equation is first simulated using classical Monte Carlo methods, where error estimates for the computation of central moments are provided. Then the quantum algorithm for this problem is simulated by sampling from the same probability distribution as that of the output of a quantum computer, and it is shown that significantly fewer resources are required to achieve the same precision. Finally, the results demonstrate potential applications of future quantum computers for simulation of turbulent mixing, and large classes of related problems.},

doi = {10.2514/1.J055896},

journal = {AIAA Journal},

number = 2,

volume = 56,

place = {United States},

year = {2017},

month = {11}

}

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