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Title: 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:
ORCiD logo; ; ;
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}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
https://doi.org/10.1038/s41598-019-46729-0

Citation Metrics:
Cited by: 2 works
Citation information provided by
Web of Science

Figures / Tables:

Figure 1 Figure 1: The number of conjugate gradient iterations grows slightly worse than $\sqrt{κ(P^{(1)})}$. The stopping criterion is a tolerance of 10−6 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 » cases.« less

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    Works referencing / citing this record:

    Hybrid classical-quantum linear solver using Noisy Intermediate-Scale Quantum machines
    journal, November 2019


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