Numerical Algorithms

, Volume 82, Issue 3, pp 1009–1045 | Cite as

Simple digital quantum algorithm for symmetric first-order linear hyperbolic systems

  • F. Fillion-GourdeauEmail author
  • E. Lorin
Original Paper


This paper is devoted to the derivation of a digital quantum algorithm for the Cauchy problem for symmetric first-order linear hyperbolic systems, thanks to the reservoir technique. The reservoir technique is a method designed to avoid artificial diffusion generated by first-order finite volume methods approximating hyperbolic systems of conservation laws. For some class of hyperbolic systems, namely, those with constant matrices in several dimensions, we show that the combination of (i) the reservoir method and (ii) the alternate direction iteration operator splitting approximation allows for the derivation of algorithms only based on simple unitary transformations, thus being perfectly suitable for an implementation on a quantum computer. The same approach can also be adapted to scalar one-dimensional systems with non-constant velocity by combining with a non-uniform mesh. The asymptotic computational complexity for the time evolution is determined and it is demonstrated that the quantum algorithm is more efficient than the classical version. However, in the quantum case, the solution is encoded in probability amplitudes of the quantum register. As a consequence, as with other similar quantum algorithms, a post-processing mechanism has to be used to obtain general properties of the solution because a direct reading cannot be performed as efficiently as the time evolution.


First-order hyperbolic systems Quantum algorithms Quantum information theory Reservoir method 


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  1. 1.
    Abrams, D.S., Lloyd, S.: Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors. Phys. Rev. Lett. 83, 5162–5165 (1999)Google Scholar
  2. 2.
    Aharonov, D., Ta-Shma, A.: Adiabatic quantum state generation and statistical zero knowledge. In: Proceedings of the thirty-fifth annual ACM symposium on theory of computing, pp. 20–29, ACM (2003)Google Scholar
  3. 3.
    Alouges, F., De Vuyst, F., Le Coq, G., Lorin, E.: A process of reduction of the numerical diffusion of usual order one flux difference schemes for nonlinear hyperbolic systems [un procédé de réduction de la diffusion numérique des schémas à différence de flux d’ordre un pour les systèmes hyperboliques non linéaires]. C.R. Math. 335(7), 627–632 (2002)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Alouges, F., De Vuyst, F., Le Coq, G., Lorin, E.: The reservoir scheme for systems of conservation laws. In: Finite volumes for complex applications, III (Porquerolles, 2002), pp. 247–254. Hermes Sci. Publ., Paris (2002)Google Scholar
  5. 5.
    Alouges, F., De Vuyst, F., Le Coq, G., Lorin, E.: The reservoir technique: a way to make Godunov-type schemes zero or very low diffuse. application to Colella-Glaz solver. Eur. J. Mech. B. Fluids 27(6), 643–664 (2008)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Alouges, F., Le Coq, G., Lorin, E.: Two-dimensional extension of the reservoir technique for some linear advection systems. J. of Sc. Comput. 31(3), 419–458 (2007)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Arrighi, P., Nesme, V., Forets, M.: The Dirac equation as a quantum walk: higher dimensions, observational convergence. J. Phys. A Math. Theor. 47(46), 465302 (2014)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Aspuru-Guzik, A., Dutoi, A.D., Love, P.J., Head-Gordon, M.: Simulated quantum computation of molecular energies. Science 309(5741), 1704–1707 (2005)Google Scholar
  9. 9.
    Barenco, A. , Bennett, C.H., Cleve, R., Divincenzo, D.P., Margolus, N., Shor, P., Sleator, T., Smolin, J.A., Weinfurter, H.: Elementary gates for quantum computation. Phys. Rev. A 52(5), 3457–3467 (1995)Google Scholar
  10. 10.
    Barends, R., Lamata, L., Kelly, J., García-Álvarez, L., Fowler, A.G., Megrant, A., Jeffrey, E., White, T.C., Sank, D., Mutus, J.Y., et al.: Digital quantum simulation of fermionic models with a superconducting circuit. Nat. Commun. 6(7654) (2015)Google Scholar
  11. 11.
    Barends, R., Shabani, A., Lamata, L., Kelly, J., Mezzacapo, A., Las Heras, U., Babbush, R., Fowler, A.G., Campbell, B., Chen, Y., et al.: Digitized adiabatic quantum computing with a superconducting circuit. Nature 534 (7606), 222–226 (2016)Google Scholar
  12. 12.
    Benenti, G., Strini, G.: Quantum simulation of the single-particle Schroedinger equation. Am. J. Phys. 76(7), 657–662 (2008)Google Scholar
  13. 13.
    Bergholm, V., Vartiainen, J.J., Moettoenen, M., Salomaa, M.M.: Quantum circuits with uniformly controlled one-qubit gates. Phys. Rev. A 71, 052330 (2005)Google Scholar
  14. 14.
    Berry, D.W.: High-order quantum algorithm for solving linear differential equations. J. Phys. A Math. Theor. 47(10), 105301 (2014)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Berry, D.W., Ahokas, G., Cleve, R., Sanders, B.C.: Efficient quantum algorithms for simulating sparse Hamiltonians. Commun. Math. Phys. 270 (2), 359–371 (2007)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Blass, A., Gurevich, Y.: Ancilla-approximable quantum state transformations. J. Math. Phys. 56(4), 042201 (2015)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Boghosian, B.M., Taylor, W.: Simulating quantum mechanics on a quantum computer. Physica D: Nonlinear Phenomena 120(1), 30–42 (1998)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Brown, K.L., Munro, W.J., Kendon, V.M.: Using quantum computers for quantum simulation. Entropy 12(11), 2268 (2010)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Cao, Y., Papageorgiou, A., Petras, I., Traub, J., Kais, S.: Quantum algorithm and circuit design solving the Poisson equation. New J. Phys. 15(1), 013021 (2013)MathSciNetGoogle Scholar
  20. 20.
    Cramer, M., Plenio, M.B., Flammia, S.T., Somma, R., Gross, D., Bartlett, S.D., Landon-Cardinal, O., Poulin, D., Liu, Y.-K.: Efficient quantum state tomography. Nat. Commun. 1, 149 (2010)Google Scholar
  21. 21.
    D’Ariano, G.M., Paris, M.G.A., Sacchi, M.F.: Quantum tomography. Advances in Imaging and Electron Physics 128, 206–309 (2003)Google Scholar
  22. 22.
    Deutsch, D.: Quantum theory, the Church-Turing principle and the universal quantum computer. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 400(1818), 97–117 (1985)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Douglas, B.L., Wang, J.B.: Efficient quantum circuit implementation of quantum walks. Phys. Rev. A 79, 052335 (2009)Google Scholar
  24. 24.
    Feynman, R.P.: Simulating physics with computers. Int. J. Theor. Phys. 21 (6), 467–488 (1982)MathSciNetGoogle Scholar
  25. 25.
    Fillion-Gourdeau, F., Lorin, E., Bandrauk, A.D.: Numerical solution of the time-dependent Dirac equation in coordinate space without fermion-doubling. Comput. Phys. Comm. 183(7), 1403–1415 (2012)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Fillion-Gourdeau, F., Lorin, E., Bandrauk, A.D.: Resonantly enhanced pair production in a simple diatomic model. Phys. Rev. Lett. 110(1), 013002 (2013)Google Scholar
  27. 27.
    Fillion-Gourdeau, F., MacLean, S., Laflamme, R.: Algorithm for the solution of the dirac equation on digital quantum computers. Phys. Rev. A 95, 042343 (2017)Google Scholar
  28. 28.
    Georgescu, I.M., Ashhab, S., Nori, F.: Quantum simulation. Rev. Mod. Phys. 86, 153–185 (2014)Google Scholar
  29. 29.
    Godlewski, E., Raviart, P.-A.: Hyperbolic Systems of Conservation Laws, vol. 3/4 of mathématiques & Applications (Paris) [Mathematics and Applications]. Ellipses, Paris (1991)zbMATHGoogle Scholar
  30. 30.
    Godlewski, E., Raviart, P.-A.: Numerical Approximation of Hyperbolic Systems of Conservation Laws, vol. 118 of Applied Mathematical Sciences. Springer, New York (1996)zbMATHGoogle Scholar
  31. 31.
    Green, A.S., Lumsdaine, P.L., Ross, N.J., Selinger, P., Valiron, B.: An introduction to quantum programming in quipper. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 7948 LNCS: 110–124 (2013)Google Scholar
  32. 32.
    Green, A.S., Lumsdaine, P.L., Ross, N.J., Selinger, P., Valiron, B.: Quipper: A scalable quantum programming language. In: Proceedings of the ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI), pp. 333–342 (2013)Google Scholar
  33. 33.
    Grover, L., Rudolph, T.: Creating superpositions that correspond to efficiently integrable probability distributions. arXiv:quant-ph/0208112 (2002)
  34. 34.
    Harrow, A.W., Hassidim, A., Lloyd, S.: Quantum algorithm for linear systems of equations. Phys. Rev. Lett. 103(15), 150502,4 (2009)MathSciNetGoogle Scholar
  35. 35.
    Jordan, S.P., Lee, K.S.M., Preskill, John: Quantum algorithms for quantum field theories. Science 336(6085), 1130–1133 (2012)Google Scholar
  36. 36.
    Kassal, I., Jordan, S.P., Love, P.J., Mohseni, M., Aspuru-Guzik, A.: Polynomial-time quantum algorithm for the simulation of chemical dynamics. Proc. Natl. Acad. Sci. 105(48), 18681–18686 (2008)Google Scholar
  37. 37.
    Kassal, I., Whitfield, J.D., Perdomo-Ortiz, A., Yung, M.-H., Aspuru-Guzik, A.: Simulating chemistry using quantum computers. Annu. Rev. Phys. Chem. 62, 185207 (2011)Google Scholar
  38. 38.
    Kaye, P., Mosca, M.: Quantum networks for generating arbitrary quantum states. arXiv:quant-ph/0407102 quant-ph/0407102(2004)
  39. 39.
    Julian Kelly, R., Barends, A.G., Fowler, A., Megrant, E., Jeffrey, T.C., White, D., Sank, J.Y., Mutus, B., Campbell, Y, et al.: Chen State preservation by repetitive error detection in a superconducting quantum circuit. Nature 519(7541), 66–69 (2015)Google Scholar
  40. 40.
    Labbé, S., Lorin, E.: On the reservoir technique convergence for nonlinear hyperbolic conservation laws. I. J. Math. Anal. Appl. 356(2), 477–497 (2009)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Lanyon, B.P., Hempel, C., Nigg, D., Müller, M., Gerritsma, R., Zähringer, F., Schindler, P., Barreiro, J.T., Rambach, M., Kirchmair, G., Hennrich, M., Zoller, P., Blatt, R., Roos, C.F.: Universal digital quantum simulation with trapped ions. Science 334(6052), 57–61 (2011)Google Scholar
  42. 42.
    Leveque, R.J.: Finite Volume Methods for Hyperbolic Problems, vol. 31. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  43. 43.
    Lloyd, S.: Universal quantum simulators. Science 273, 1073–1078 (1996)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Meyer, D.A.: Quantum computing classical physics. Philosophical Transactions of the Royal Society of London A: Mathematical, Phys. Eng. Sci. 360(1792), 395–405 (2002)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Mezzacapo, A., Sanz, M., Lamata, L., Egusquiza, I.L., Succi, S., Solano, E.: Quantum simulator for transport phenomena in fluid flows. Sci. Rep. 5(13153) (2015)Google Scholar
  46. 46.
    Negrevergne, C., Mahesh, T.S., Ryan, C.A., Ditty, M., Cyr-Racine, F., Power, W., Boulant, N., Havel, T., Cory, D.G., Laflamme, R.: Benchmarking quantum control methods on a 12-qubit system. Phys. Rev. Lett. 96, 170501 (2006)Google Scholar
  47. 47.
    Nielsen, M.A, Chuang, I.L.: Quantum computation and quantum information. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  48. 48.
    Papageorgiou, A., Traub, J.F.: Measures of quantum computing speedup. Phys. Rev. A 88, 022316 (2013)Google Scholar
  49. 49.
    Rønnow, T.F., Wang, Z., Job, J., Boixo, S., Isakov, S.V., Wecker, D., Martinis, J.M., Lidar, D.A., Troyer, M.: Defining and detecting quantum speedup. Science 345(6195), 420–424 (2014)Google Scholar
  50. 50.
    Salathé, Y., Mondal, M., Oppliger, M., Heinsoo, J., Kurpiers, P., Potočnik, A., Mezzacapo, A., Las Heras, U., Lamata, U., Solano, E., Filipp, S., Wallraff, A.: Digital quantum simulation of spin models with circuit quantum electrodynamics. Phys. Rev. X 5, 021027 (2015)Google Scholar
  51. 51.
    Serre, D.: Systémes de lois de conservation. I. Fondations. [Foundations]. Diderot Editeur, Paris. Hyperbolicité, entropies, ondes de choc. [Hyperbolicity, entropies, shock waves] (1996)Google Scholar
  52. 52.
    Shor, P.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Sinha, S., Russer, P.: Quantum computing algorithm for electromagnetic field simulation. Quantum Inf. Process 9(3), 385–404 (2010)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Smoller, J.: Shock waves and reaction-diffusion equations, vol. 258 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science]. Springer, New York-Berlin (1983)Google Scholar
  55. 55.
    Somma, R., Ortiz, G., Gubernatis, J.E., Knill, E., Laflamme, R.: Simulating physical phenomena by quantum networks. Phys. Rev. A 65, 042323 (2002)Google Scholar
  56. 56.
    Steane, A.: Quantum computing. Rep. Prog. Phys. 61(2), 117 (1998)MathSciNetGoogle Scholar
  57. 57.
    Strikwerda, J.C.: Finite Difference Schemes and Partial Differential Equations, 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2004)zbMATHGoogle Scholar
  58. 58.
    Vartiainen, J.J., Moetioenen, M., Salomaa, M.M.: Efficient decomposition of quantum gates. Phys. Rev. Lett. 92(17), 177902–1 (2004)Google Scholar
  59. 59.
    Wang, Xi-Lin, Chen, Luo-Kan, Li, W., Huang, H.-L., Liu, C., Chen, C., Luo, Y.-H., Su, Z.-E., Wu, D., Li, Z.-D., Lu, H., Hu, Y., Jiang, X., Peng, C.-Z., Li, L., Liu, N.-L., Chen, Y.-A., Lu, C.-Y., Pan, J.-W.: Experimental ten-photon entanglement. Phys. Rev. Lett. 117, 210502 (2016)Google Scholar
  60. 60.
    Wiebe, N., Berry, D., Hoyer, P., Sanders, B.C.: Higher order decompositions of ordered operator exponentials. J. Phys. A Math. Theor. 43(6), 065203 (2010)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Wiesner, S.: Simulations of many-body quantum systems by a quantum computer. arXiv:quant-ph/9603028 quant-ph/9603028
  62. 62.
    Yung, Man-Hong, Nagaj, Daniel, Whitfield, James D., Aspuru-Guzik, A.: Simulation of classical thermal states on a quantum computer A transfer-matrix approach. Phys. Rev. A 82, 060302 (2010)Google Scholar
  63. 63.
    Yung, M.-H., Whitfield, J.D., Boixo, S., Tempel, D.G., Aspuru-Guzik, A.: Introduction to Quantum Algorithms for Physics and Chemistry, pp 67–106. Wiley, Hoboken (2014)zbMATHGoogle Scholar
  64. 64.
    Zalka, C.: Efficient simulation of quantum systems by quantum computers. Fortschritte der Physik 46(6-8), 877–879 (1998)MathSciNetGoogle Scholar
  65. 65.
    Zalka, C.: Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London A: Mathematical, Phys. Eng. Sci. 454(1969), 313–322 (1998)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.INRS-Énergie, Matériaux et TélécommunicationsUniversité du QuébecVarennesCanada
  2. 2.Institute for Quantum ComputingUniversity of WaterlooWaterlooCanada
  3. 3.Centre de Recherches MathématiquesUniversité de MontréalMontréalCanada
  4. 4.School of Mathematics and StatisticsCarleton UniversityOttawaCanada

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