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.
- Publication Date:
- Research Org.:
- Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States); Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States); Lawrence Livermore National Laboratory (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. doi: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 = {Tue Jul 16 00:00:00 EDT 2019},
month = {Tue Jul 16 00:00:00 EDT 2019}
}
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https://doi.org/10.1038/s41598-019-46729-0
https://doi.org/10.1038/s41598-019-46729-0
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Cited by: 17 works
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Figures / Tables:
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 »
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Works referenced in this record:
A per-cent-level determination of the nucleon axial coupling from quantum chromodynamics
journal, May 2018
- Chang, C. C.; Nicholson, A. N.; Rinaldi, E.
- Nature, Vol. 558, Issue 7708
A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem
journal, April 2001
- Farhi, E.; Goldstone, J.; Gutmann, S.
- Science, Vol. 292, Issue 5516
Strengths and weaknesses of weak-strong cluster problems: A detailed overview of state-of-the-art classical heuristics versus quantum approaches
journal, August 2016
- Mandrà, Salvatore; Zhu, Zheng; Wang, Wenlong
- Physical Review A, Vol. 94, Issue 2
Thermalization, Freeze-out, and Noise: Deciphering Experimental Quantum Annealers
journal, December 2017
- Marshall, Jeffrey; Rieffel, Eleanor G.; Hen, Itay
- Physical Review Applied, Vol. 8, Issue 6
A deceptive step towards quantum speedup detection
journal, July 2018
- Mandrà, Salvatore; Katzgraber, Helmut G.
- Quantum Science and Technology, Vol. 3, Issue 4
Quantum Algorithm for Linear Systems of Equations
journal, October 2009
- Harrow, Aram W.; Hassidim, Avinatan; Lloyd, Seth
- Physical Review Letters, Vol. 103, Issue 15
Extending the eigCG algorithm to non-symmetric linear systems with multiple right-hand sides
conference, June 2010
- Abdel-Rehim, Abdou M.; Orginos, Kostas; Stathopoulos, Andreas
- Proceedings of The XXVII International Symposium on Lattice Field Theory — PoS(LAT2009)
Thermally assisted quantum annealing of a 16-qubit problem
journal, May 2013
- Dickson, N. G.; Johnson, M. W.; Amin, M. H.
- Nature Communications, Vol. 4, Issue 1
Bond chaos in spin glasses revealed through thermal boundary conditions
journal, June 2016
- Wang, Wenlong; Machta, Jonathan; Katzgraber, Helmut G.
- Physical Review B, Vol. 93, Issue 22
Demonstration of a Scaling Advantage for a Quantum Annealer over Simulated Annealing
journal, July 2018
- Albash, Tameem; Lidar, Daniel A.
- Physical Review X, Vol. 8, Issue 3
Computing and Deflating Eigenvalues While Solving Multiple Right-Hand Side Linear Systems with an Application to Quantum Chromodynamics
journal, January 2010
- Stathopoulos, Andreas; Orginos, Konstantinos
- SIAM Journal on Scientific Computing, Vol. 32, Issue 1
Retrieving the ground state of spin glasses using thermal noise: Performance of quantum annealing at finite temperatures
journal, September 2016
- Nishimura, Kohji; Nishimori, Hidetoshi; Ochoa, Andrew J.
- Physical Review E, Vol. 94, Issue 3
Adiabatic quantum programming: minor embedding with hard faults
journal, November 2013
- Klymko, Christine; Sullivan, Blair D.; Humble, Travis S.
- Quantum Information Processing, Vol. 13, Issue 3
Pseudo-Boolean optimization
journal, November 2002
- Boros, Endre; Hammer, Peter L.
- Discrete Applied Mathematics, Vol. 123, Issue 1-3
An integrated programming and development environment for adiabatic quantum optimization
journal, January 2014
- S. Humble, T.; J. McCaskey, A.; S. Bennink, R.
- Computational Science & Discovery, Vol. 7, Issue 1
Experimental realization of quantum algorithms for a linear system inspired by adiabatic quantum computing
journal, January 2019
- Wen, Jingwei; Kong, Xiangyu; Wei, Shijie
- Physical Review A, Vol. 99, Issue 1
Sherrington-Kirkpatrick model in a transverse field: Absence of replica symmetry breaking due to quantum fluctuations
journal, June 1989
- Ray, P.; Chakrabarti, B. K.; Chakrabarti, Arunava
- Physical Review B, Vol. 39, Issue 16
Quantum annealing in the transverse Ising model
journal, November 1998
- Kadowaki, Tadashi; Nishimori, Hidetoshi
- Physical Review E, Vol. 58, Issue 5
Defining and detecting quantum speedup
journal, June 2014
- Ronnow, T. F.; Wang, Z.; Job, J.
- Science, Vol. 345, Issue 6195
Solving Systems of Linear Equations with a Superconducting Quantum Processor
journal, May 2017
- Zheng, Yarui; Song, Chao; Chen, Ming-Cheng
- Physical Review Letters, Vol. 118, Issue 21
Decoherence in adiabatic quantum computation
journal, February 2009
- Amin, M. H. S.; Averin, Dmitri V.; Nesteroff, James A.
- Physical Review A, Vol. 79, Issue 2
Decoherence in adiabatic quantum computation
journal, June 2015
- Albash, Tameem; Lidar, Daniel A.
- Physical Review A, Vol. 91, Issue 6
Local coherence and deflation of the low quark modes in lattice QCD
journal, July 2007
- Lüscher, Martin
- Journal of High Energy Physics, Vol. 2007, Issue 07
Quantum Algorithms for Systems of Linear Equations Inspired by Adiabatic Quantum Computing
journal, February 2019
- Subaşı, Yiğit; Somma, Rolando D.; Orsucci, Davide
- Physical Review Letters, Vol. 122, Issue 6
Multi-level adaptive solutions to boundary-value problems
journal, May 1977
- Brandt, Achi
- Mathematics of Computation, Vol. 31, Issue 138
Nonperturbative -body to two-body commuting conversion Hamiltonians and embedding problem instances into Ising spins
journal, May 2008
- Biamonte, J. D.
- Physical Review A, Vol. 77, Issue 5
Minor-embedding in adiabatic quantum computation: I. The parameter setting problem
journal, September 2008
- Choi, Vicky
- Quantum Information Processing, Vol. 7, Issue 5
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