# Quantum annealing for systems of polynomial equations

## Abstract

Numerous scientific and engineering applications require numerically solving systems of equations. Classically solving a general set of polynomial equations requires iterative solvers, while linear equations may be solved either by direct matrix inversion or iteratively with judicious preconditioning. However, the convergence of iterative algorithms is highly variable and depends, in part, on the condition number. We present a direct method for solving general systems of polynomial equations based on quantum annealing, and we validate this method using a system of second-order polynomial equations solved on a commercially available quantum annealer. We then demonstrate applications for linear regression, and discuss in more detail the scaling behavior for general systems of linear equations with respect to problem size, condition number, and search precision. Finally, we define an iterative annealing process and demonstrate its efficacy in solving a linear system to a tolerance of 10^{–8}.

- Authors:

- Publication Date:

- Research Org.:
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States); Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC); USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1619570

- Alternate Identifier(s):
- OSTI ID: 1543218; OSTI ID: 1561954; OSTI ID: 1771026

- Report Number(s):
- LLNL-JRNL-761230

Journal ID: ISSN 2045-2322; 10258; PII: 46729

- Grant/Contract Number:
- AC52-07NA27344; AC0500OR22725; AC05-00OR22725; AC02-05CH11231

- Resource Type:
- Published Article

- Journal Name:
- Scientific Reports

- Additional Journal Information:
- Journal Name: Scientific Reports Journal Volume: 9 Journal Issue: 1; Journal ID: ISSN 2045-2322

- Publisher:
- Nature Publishing Group

- Country of Publication:
- United Kingdom

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; 73 NUCLEAR PHYSICS AND RADIATION PHYSICS; Quantum information; Quantum simulation

### Citation Formats

```
Chang, Chia Cheng, Gambhir, Arjun, Humble, Travis S., and Sota, Shigetoshi. Quantum annealing for systems of polynomial equations. United Kingdom: N. p., 2019.
Web. https://doi.org/10.1038/s41598-019-46729-0.
```

```
Chang, Chia Cheng, Gambhir, Arjun, Humble, Travis S., & Sota, Shigetoshi. Quantum annealing for systems of polynomial equations. United Kingdom. https://doi.org/10.1038/s41598-019-46729-0
```

```
Chang, Chia Cheng, Gambhir, Arjun, Humble, Travis S., and Sota, Shigetoshi. Tue .
"Quantum annealing for systems of polynomial equations". United Kingdom. https://doi.org/10.1038/s41598-019-46729-0.
```

```
@article{osti_1619570,
```

title = {Quantum annealing for systems of polynomial equations},

author = {Chang, Chia Cheng and Gambhir, Arjun and Humble, Travis S. and Sota, Shigetoshi},

abstractNote = {Numerous scientific and engineering applications require numerically solving systems of equations. Classically solving a general set of polynomial equations requires iterative solvers, while linear equations may be solved either by direct matrix inversion or iteratively with judicious preconditioning. However, the convergence of iterative algorithms is highly variable and depends, in part, on the condition number. We present a direct method for solving general systems of polynomial equations based on quantum annealing, and we validate this method using a system of second-order polynomial equations solved on a commercially available quantum annealer. We then demonstrate applications for linear regression, and discuss in more detail the scaling behavior for general systems of linear equations with respect to problem size, condition number, and search precision. Finally, we define an iterative annealing process and demonstrate its efficacy in solving a linear system to a tolerance of 10–8.},

doi = {10.1038/s41598-019-46729-0},

journal = {Scientific Reports},

number = 1,

volume = 9,

place = {United Kingdom},

year = {2019},

month = {7}

}

https://doi.org/10.1038/s41598-019-46729-0

*Citation information provided by*

Web of Science

Web of Science

#### Figures / Tables:

^{−6}for the norm of the relative residual. All matrices are rank 12, with smaller eigenvalues as κ(P

^{(1)}) increases, but identical eigenvectors. Thee same right-hand side is solved for allmore »

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Figures / Tables found in this record:

*Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.*