Abstract
The numerical resolution of systems of linear equations is an important problem which recurs continously in applied sciences. In particular, it represents an indispensable tool in applied mathematics which can be utilized as a foundation to more complicated problems (e.g. optimization problems, partial differential equations, eigenproblems, etc.). In this work, we introduce a solver for systems of linear equations based on quantum mechanics. More specifically, given a system of linear equations we introduce an equivalent optimization problem which objective function defines an electrostatic potential. Then, we evolve a many-body quantum system immersed in this potential and show that the corresponding Wigner quasi-distribution function converges to the global energy minimum. The simulations are performed by using the time-dependent, ab-initio, many-body Wigner Monte Carlo method. Finally, by numerically emulating the (random) process of measurement, we demonstrate that one can extract the solution of the original mathematical problem. As a proof of concept we solve 3 simple, but different, linear systems with increasing complexity. The outcomes clearly show that our suggested approach is a valid quantum algorithm for the resolution of systems of linear equations.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000)
B.E. Kane, A silicon-based nuclear spin quantum computer. Nature 393, 133–137 (1998)
L.C.L. Hollenberg, A.S. Dzurak, C. Wellard, A.R. Hamilton, D.J. Reilly, G.J. Milburn, R.G. Clark, Charged-based quantum computing using single donors in semiconductors. Phys. Rev. B 69, 113301 (2004)
K. Yokoi, D. Moraru, M. Ligowski, M. Tabe, Single-gated single-electron transfer in nonuniform arrays of quantum dots. Jpn. J. Appl. Phys. 48, 024503 (2009)
E. Hamid, D. Moraru, J.C. Tarido, S. Miki, T. Mizuno, M. Tabe, Single-electron transfer between two donors in nanoscale thin silicon-on-insulator field-effect transistors. Appl. Phys. Lett. 97, 262101 (2010)
D. Moraru, A. Udhiarto, M. Anwar, R. Nowak, R. Jablonski, E. Hamid, J.C. Tarido, T. Mizuno, M. Tabe, Atom devices based on single dopants in silicon nanostructures. Nanoscale Res. Lett. 6, 479 (2011)
M. Simmons, S. Schofield, J. Obrien, N. Curson, L. Oberbeck, T. Hallam, R. Clark, Towards the atomic-scale fabrication of a silicon-based solid state quantum computer. Surf. Sci. 532–535, 1209–1218 (2003)
E. Nielsen, R.W. Young, R.P. Muller, M.S. Carroll, Implications of simultaneous requirements for low-noise exchange gates in double quantum dots. Phys. Rev. B 82, 075319 (2010)
M.W. Johnson, M.H.S. Amin, S. Gildert, T. Lanting, F. Hamze, N. Dickson, R. Harris, A.J. Berkley, J. Johansson, P. Bunyk, E.M. Chapple, C. Enderud, J.P. Hilton, K. Karimi, E. Ladizinsky, N. Ladizinsky, T. Oh, I. Perminov, C. Rich, M.C. Thom, E. Tolkacheva, C.J.S. Truncik, S. Uchaikin, J. Wang, B. Wilson, G. Rose, Quantum annealing with manufactured spins. Nature 473, 194–198 (2011)
T. Kadowaki, H. Nishimori, Quantum annealing in the transverse Ising model. Phys. Rev. E 58, 5355 (1998)
A.B. Finilla, M.A. Gomez, C. Sebenik, D.J. Doll, Quantum annealing: a new method for minimizing multidimensional functions. Chem. Phys. Lett. 219, 343 (1994)
G.E. Santoro, E. Tosatti, Optimization using quantum mechanics: quantum annealing through adiabatic evolution. J. Phys. A 39, R393 (2006)
A. Das, B.K. Chakrabarti, Colloquium: quantum annealing and analog quantum computation. Rev. Mod. Phys. 80, 1061 (2008)
S. Boixo, T.F. Rønnow, S.V. Isakov, Z. Wang, D. Wecker, D.A. Lidar, J.M. Martinis, M. Troyer, Evidence for quantum annealing with more than one hundred qubits. Nat. Phys. 10, 218–224 (2014)
T.F. Rønnow, Z. Wang, J. Job, S. Boixo, S.V. Isakov, D. Wecker, J.M. Martinis, D.A. Lidar, M. Troyer, Defining and detecting quantum speedup. Science 345, 420–426 (2014)
J. Pan, Y. Cao, X. Yao, Z. Li, C. Ju, H. Chen, X. Peng, S. Kais, J. Du, Experimental realization of quantum algorithm for solving linear systems of equations. Phys. Rev. A 89, 022313 (2014)
X.D. Cai, C. Weedbrook, Z.E. Su, M.C. Chen, M. Gu, M.J. Zhu, L. Li, N.L. Liu, C.Y. Lu, J.W. Pan, Experimental quantum computing to solve systems of linear equations. Phy. Rev. Lett. 110, 230501 (2013)
E. Wigner, On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749 (1932)
E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, D. Sorensen, LAPACK Users’ Guide (Society for Industrial and Applied Mathematics, Philadelphia, 1999)
M. Galassi et al., GNU Scientific Library Reference Manual, 3rd edn. (Network Theory Limited, Bristol, 2009)
J.M. Sellier, M. Nedjalkov, I. Dimov, S. Selberherr, A benchmark study of the Wigner Monte-Carlo method. Monte Carlo Methods Appl. De Gruyter (2014). doi:10.1515/mcma-2013-0018
J.M. Sellier, I. Dimov, The many-body Wigner Monte Carlo method for time-dependent Ab-initio quantum simulations. J. Comput. Phys. 273, 589–597 (2014)
J.M. Sellier, M. Nedjalkov, I. Dimov, S. Selberherr, Decoherence and time reversibility: the role of randomness at interfaces. J. Appl. Phys. 114, 174902 (2013)
I. Dimov, Monte Carlo Algorithms for Linear Problems, Pliska (Studia Mathematica Bulgarica), Vol. 13 (2000), 1997, pp. 57–77
I. Dimov, T. Gurov, Monte Carlo Algorithm for Solving Integral Equations with Polynomial Non-linearity. Parallel Implementation, Pliska (Studia Mathematica Bulgarica), Vol. 13 (2000), 1997, pp. 117–132
J.M. Sellier, GNU Archimedes. Available www.gnu.org/software/archimedes. Accessed 08 Dec 2014
J.M. Sellier, Nano-archimedes. www.nano-archimedes.com. Accessed 08 Dec 2014
J.M. Sellier, I. Dimov, Wigner-Boltzmann Monte Carlo method applied to electron transport in the presence of a single dopant. Comput. Phys. Commun. 185, 2427–2435 (2014)
Acknowledgments
This work has been supported by the the project EC AComIn (FP7-REGPOT-2012-2013-1).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Sellier, J.M., Dimov, I. (2016). On a Quantum Algorithm for the Resolution of Systems of Linear Equations. In: Fidanova, S. (eds) Recent Advances in Computational Optimization. Studies in Computational Intelligence, vol 610. Springer, Cham. https://doi.org/10.1007/978-3-319-21133-6_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-21133-6_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-21132-9
Online ISBN: 978-3-319-21133-6
eBook Packages: EngineeringEngineering (R0)