Skip to main content
SpringerLink
Log in

Quantum algorithm for the advection–diffusion equation simulated with the lattice Boltzmann method

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

A novel quantum algorithm for solving advection–diffusion equation by the lattice Boltzmann method is proposed. The presented quantum algorithm is composed of two major segments. In the first segment, equilibrium distribution function, presented in the form of a non-unitary diagonal matrix, is quantum circuit implemented by using a standard-form encoding approach. For the second segment, the quantum walk procedure as a method for implementing the propagation step is applied. The constructed algorithm is presented as a series of single- and two-qubit gates, as well as multiple-input controlled-NOT gates. In order to demonstrate the validity of the proposed quantum algorithm, the unsteady one-dimensional (1D) and two-dimensional (2D) advection–diffusion equations are solved by using the IBM’s quantum computing software development framework Qiskit, while the analytic solution and the classic code are used for verification. Finally, the complexity analysis and directions for future work are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Grover, L.: A fast quantum mechanical algorithm for database search. In: Proceedings, 28th Annual ACM Symposium on the Theory of Computing, p. 212 (1996)

  2. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information (Repr. ed.). Cambridge Univ. Press (2001)

  3. Coppersmith, D.: An Approximate Fourier Transform Useful in Quantum Factoring. Technical Report. IBM, New York (1994)

    Google Scholar 

  4. Shor, P.W.: Algorithms for quantum computation: discrete logarithms and factoring. In: Proceedings 35th Annual Symposium on Foundations of Computer Science, pp. 124–134 (1994)

  5. Harrow, A.W., Hassidim, A., Lloyd, S.: Quantum algorithm for linear systems of equations. Phys. Rev. Lett. 103(15) (2009). ISSN 1079-7114. https://doi.org/10.1103/physrevlett.103.150502

  6. Ambainis, A.: Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations (2010)

  7. Qian, P., Huang, W.-C., Long, G.-L.: A quantum algorithm for solving systems of nonlinear algebraic equations (2019)

  8. Wang, H., Xiang, H.: Quantum algorithm for total least squares data fitting. Phys. Lett. A 383(19), 2235–2240 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  9. Doronin, S.I., Feldman, E.B., Zenchuk, A.I.: Solving systems of linear algebraic equations via unitary transformations on quantum processor of IBM quantum experience. Quant. Inform. Process. 19(68) (2020)

  10. Cao, Y., Romero, J., Olson, J.P., Degroote, M., Johnson, P.D., Kieferov, M., Kivlichan, I.D., Menke, T., Peropadre, B., Sawaya, N.P.D., Sim, S., Veis, L., Aspuru-Guzik, A.: Quantum chemistry in the age of quantum computing. Chem. Rev. 119(19), 10856–10915 (2019)

    Article  Google Scholar 

  11. Kerenidis, I.S., Prakash, A.: Quantum gradient descent for linear systems and least squares. Phys. Rev. A 101(2), 022316 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  12. Farhi, E., Goldstone, J., Gutmann, S.: A quantum approximate optimization algorithm (2014)

  13. Schuld, M., Petruccione, F.: Supervised Learning with Quantum Computers. Springer Nature, Switzerland (2018)

    Book  Google Scholar 

  14. Garg, S., Ramakrishnan, G.: Advances in quantum deep learning: an overview (2020)

  15. Sharma, S.: Qeml (quantum enhanced machine learning): using quantum computing to enhance ml classifiers and feature spaces (2020)

  16. Farhi, E., Neven, H.: Classification with quantum neural networks on near term processors (2018)

  17. Schuld, M., Bocharov, A., Svore, K.M., Wiebe, N.: Circuit-centric quantum classifiers. Phys. Rev. A 101(3), 2020. ISSN 2469-9934. https://doi.org/10.1103/physreva.101.032308

  18. Killoran, N., Bromley, T.R., Arrazola, J.M., Schuld, M., Quesada, N., Lloyd, S.: Continuous-variable quantum neural networks. Phys. Rev. Res. 1(3), 2019. ISSN 2643-1564. https://doi.org/10.1103/physrevresearch.1.033063

  19. Mari, A., Bromley, T.R., Izaac, J., Schuld, M., Killoran, N.: Transfer learning in hybrid classical-quantum neural networks (2019)

  20. Peruzzo, A., McClean, J., Shadbolt, P., Yung, M.-H., Zhou, X.-Q., Love, P.J., Aspuru-Guzik, A., OBrien, J.L.: A variational eigenvalue solver on a photonic quantum processor. Nature Commun. 5 (2014). https://doi.org/10.1038/ncomms5213

  21. Prieto, C.B., LaRose, R., Cerezo, M., Subasi, Y., Cincio, L., Coles, P.J.: Variational quantum linear solver (2019)

  22. Huang, H.Y., Bharti, K., Rebentrost, P.: Near-term quantum algorithms for linear systems of equations (2019)

  23. Stokes, J., Izaac, J., Killoran, N., Carleo, G.: Quantum natural gradient. Quantum 4, 269, 2020. ISSN 2521-327X. https://doi.org/10.22331/q-2020-05-25-269

  24. Sweke, R., Wilde, F., Meyer, J., Schuld, M., Fhrmann, P. K., Piganeau, B. M., Eisert, J.: Stochastic gradient descent for hybrid quantum-classical optimization, (2019)

  25. Yamamoto, N.: On the natural gradient for variational quantum eigensolver (2019)

  26. Berry, D.W.: High-order quantum algorithm for solving linear differential equations. J. Phys. A: Math. Theor. 47(10), 105301 (2014). https://doi.org/10.1088/1751-8113/47/10/105301. 10.1088%2F1751-8113%2F47%2F10%2F105301

  27. Childs, A.M., Liu, J.P., Ostrander, A.: High-precision quantum algorithms for partial differential equations (2020)

  28. Costa, P.C.S., Jordan, S., Ostrander, A.: Quantum algorithm for simulating the wave equation. Phys. Rev. A 99, 012323 (2019). https://doi.org/10.1103/PhysRevA.99.012323

  29. 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). https://doi.org/10.1088/1367-2630/15/1/013021

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Wang, S., Wang, Z., Li, W., Fan, L., Wei, Z., Yongjian, G.: Quantum fast Poisson solver: the algorithm and complete and modular circuit design. Quant. Inform. Process. (2020). https://doi.org/10.1007/s11128-020-02669-7

    Article  MathSciNet  Google Scholar 

  31. Berry, D.W., Childs, A.M., Ostrander, A., Wang, G.: Quantum algorithm for linear differential equations with exponentially improved dependence on precision. Commun. Math. Phys. 356, 1057–1081 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  32. Xin, T., Wei, S., Cui, J., Xiao, J., Arrazola, I., Lamata, L., Kong, X., Dawei, L., Solano, E., Long, G.: Quantum algorithm for solving linear differential equations: theory and experiment. Phys. Rev. A 101, 032307 (2020). https://doi.org/10.1103/PhysRevA.101.032307

    Article  ADS  MathSciNet  Google Scholar 

  33. Leyton, S.K., Osborne, T.J.: A quantum algorithm to solve nonlinear differential equations (2008)

  34. Rivet, J.P., Boon, J.P.: Lattice Gas Hydrodynamics. Cambridge University Press, London (2001)

    Book  Google Scholar 

  35. Rothman, D.H., Zaleski, S.: Lattice-Gas Cellular Automata Simple Models of Complex Hydrodynamics. Cambridge University Press, London (1996)

    MATH  Google Scholar 

  36. Chen, S., Doolen, G.D.: Lattice Boltzmann method for fluid flows. Ann. Rev. Fluid Mech. 30, 329–364 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  37. Mohamad, A.A.: Lattice Boltzmann Method–Fundamentals and Engineering Applications with Computer Codes. Springer, London (2011)

    MATH  Google Scholar 

  38. Yepez, J.: Quantum lattice-gas model for computational fluid dynamics. Phys. Rev. E 63, 046702 (2001)

    Article  ADS  Google Scholar 

  39. Berman, G.P., Ezhov, A.A., Kamenev, D.I., Yepez, J.: Simulation of the diffusion equation on a type-ii quantum computer. Phys. Rev. E 66, 012310 (2002). https://doi.org/10.1103/PhysRevA.66.012310

    Article  ADS  Google Scholar 

  40. Micci, M.M., Yepez, J.: Measurement-based quantum lattice gas model of fluid dynamics in 2+1 dimensions. Phys. Rev. E 92, 033302 (2015). https://doi.org/10.1103/PhysRevE.92.033302

    Article  ADS  Google Scholar 

  41. Yepez, J.: Type-ii quantum computers. Int. J. Mod. Phys. C 12(09), 1273–1284 (2001). https://doi.org/10.1142/S0129183101002668

    Article  ADS  MathSciNet  Google Scholar 

  42. Todorova, B.N., Steijl, R.: Quantum algorithm for the collisionless Boltzmann equation. J. Comput. Phys. 409, (2020)

  43. Gaitan, F.: Finding flows of a Navier–Stokes fluid through quantum computing. NPJ Quant. Inform. (2020). https://doi.org/10.1038/s41534-020-00291-0

    Article  Google Scholar 

  44. Low, G.H., Chuang, I.L.: Hamiltonian simulation by qubitization. Quantum 3, 163 (2019). ISSN 2521-327X. https://doi.org/10.22331/q-2019-07-12-163

  45. Abraham, H., Offei, A., Akhalwaya, I.Y., Aleksandrowicz, G., Alexander, T., Arbel, E., Asfaw, A., Azaustre, C., Ngoueya, A., Barkoutsos, P., Barron, G., Bello, L., Ben-Haim, Y., Bevenius, D., Bishop, L.S., Bolos, S., Bosch, S., Bravyi, S., Bucher, D., Burov, A., Cabrera, F., Calpin, P., Capelluto, L., Carballo, J., Carrascal, G., Chen, A., Chen, C.-F., Chen, R., Chow, J.M., Claus, C., Cocking, R., Cross, A.J., Cross, A.W., Cross, S., Cruz-Benito, J., Culver, C., Córcoles-Gonzales, A.D., Dague, S., El Dandachi, T., Dartiailh, M., Frr, D., Davila, A.R., Dekusar, A., Ding, D., Doi, J., Drechsler, E., Drew, Dumitrescu, E., Dumon, K., Duran, I., EL-Safty, K., Eastman, E., Eendebak, P., Egger, D., Everitt, M., Fernández, P.M., Ferrera, A.H., Chevallier, F., Frisch, A., Fuhrer, A., George, M., Gacon, J., Gago, B.G., Gambella, C., Gambetta, J.M., Gammanpila, A., Garcia, L., Garion, S., Gilliam, A., Gomez-Mosquera, J., de la Puente González, S., Gorzinski, J., Gould, I., Greenberg, D., Grinko, D., Guan, W., Gunnels, J.A., Haglund, M., Haide, I., Hamamura, I., Hamido, O.C., Havlicek, V., Hellmers, J., Herok, L., Hillmich, S., Horii, H., Howington, C., Hu, S., Hu, W., Imai, H., Imamichi, T., Ishizaki, K., Iten, R., Itoko, T., Seaward, J., Javadi, A., Jessica, Jivrajani, M., Johns, K., Jonathan-Shoemaker, Kachmann, T., Kanazawa, N., Kang-Bae, Karazeev, A., Kassebaum, P., King, S., Knabberjoe, Kobayashi, Y., Kovyrshin, A., Krishnakumar, R., Krishnan, V., Krsulich, K., Kus, G., LaRose, R., Lacal, E., Lambert, R., Latone, J., Lawrence, S., Li, G., Liu, D., Liu, P., Maeng, Y., Malyshev, A., Manela, J., Marecek, J., Marques, M., Maslov, D., Mathews, D., Matsuo, A., McClure, D.T., McGarry, C., McKay, D., McPherson, D., Meesala, S., Metcalfe, T., Mevissen, M., Mezzacapo, A., Midha, R., Minev, Z., Mitchell, A., Moll, N., Mooring, M.D., Morales, R., Moran, N., MrF, Murali, P., Müggenburg, J., Nadlinger, D., Nakanishi, K., Nannicini, G., Nation, P., Navarro, E., Naveh, Y., Neagle, S.W., Neuweiler, P., Niroula, P., Norlen, H., O’Riordan, L.J., Ogunbayo, O., Ollitrault, P., Oud, S., Padilha, D., Paik, H., Perriello, S., Phan, A., Piro, F., Pistoia, M., Piveteau, C., Pozas-iKerstjens, A., Prutyanov, V., Puzzuoli, D., Pérez, J., Quintiii, Ramagiri, N., Rao, A., Raymond, R., Martín-Cuevas Redondo, R., Reuter, M., Rice, J., Rodríguez, D.M., Karur, R., Rossmannek, M., Ryu, M., Tharrmashastha, S.A.P.V., Ferracin, S., Sandberg, M., Sargsyan, H., Sathaye, N., Schmitt, B., Schnabel, C., Schoenfeld, Z., Scholten, T.L., Schoute, E., Schwarm, J., Sertage, I.F., Setia, K., Shammah, N., Shi, Y., Silva, A., Simonetto, A., Singstock, N., Siraichi, Y., Sitdikov, I., Sivarajah, S., Sletfjerding, M.B., Smolin, J.A., Soeken, M., Sokolov, I.O., Thomas, S., Starfish, Steenken, D., Stypulkoski, M., Sun, S., Sung, K.J., Takahashi, H., Tavernelli, I., Taylor, C., Taylour, P., Thomas, S., Tillet, M., Tod, M., Tomasik, M., de la Torre, E., Trabing, K., Treinish, M., Pe, T., Turner, W., Vaknin, Y., Valcarce, C.R., Varchon, F., Vazquez, A.C., Villar, V., Vogt-Lee, D., Vuillot, C., Weaver, J., Wieczorek, R., Wildstrom, J.A., Winston, E., Woehr, J.J., Woerner, S., Woo, R., Wood, C.J., Wood, R., Wood, S., Wood, S., Wootton, J., Yeralin, D., Yonge-Mallo, D., Young, R., Yu, J., Zachow, C., Zdanski, L., Zhang, H., Zoufal, C., Zoufalc, a matsuo, adekusar drl, bcamorrison, brandhsn, chlorophyll zz, dekel.meirom, dekool, dime10, drholmie, dtrenev, elfrocampeador, faisaldebouni, fanizzamarco, gadial, gruu, jagunther, jliu45, kanejess, klinvill, kurarrr, lerongil, ma5x, merav aharoni, michelle4654, ordmoj, rmoyard, saswati qiskit, sethmerkel, strickroman, sumitpuri, tigerjack, toural, vvilpas, welien, willhbang, yang.luh, yotamvakninibm, and Mantas Čepulkovskis. Qiskit: An open-source framework for quantum computing (2019)

  46. Bhatnagar, P.L., Gross, E.P., Krook, M.: A model for collision processes in gases. i: Small amplitude processes in charged and neutral one-component system. Phys. Rev. 94, 511–525 (1954)

    Article  ADS  Google Scholar 

  47. Zhou, J.G.: Macroscopic lattice Boltzmann method (maclab) (2019)

  48. Shende, V.V., Bullock, S.S., Markov, I.L.: Synthesis of quantum-logic circuits. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 25(6), 1000–1010 (2006). ISSN 1937-4151.https://doi.org/10.1109/TCAD.2005.855930

  49. Kay, A.: Tutorial on the quantikz package (2020)

  50. Childs, A.M.: Universal computation by quantum walk. Phys. Rev. Lett. 102(18) (2009). ISSN 1079-7114. https://doi.org/10.1103/PhysRevLett.102.180501

  51. Shakeel, A.: Efficient and scalable quantum walk algorithms via the quantum Fourier transform (2019)

  52. Hahn, X.: Fortran for visual studio code (2015). https://marketplace.visualstudio.com/items?itemName=Gimly81.fortran

Download references

Author information

Authors and Affiliations

  1. Faculty of Technical Sciences, University of Novi Sad, Trg Dositeja Obradovića 6, Novi Sad, 21000, Serbia

    Ljubomir Budinski

Authors
  1. Ljubomir Budinski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ljubomir Budinski.

Ethics declarations

Conflicts of interest

Not applicable.

Funding

Not applicable.

Availability of data and material

Not applicable.

Code availability

Not applicable.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Budinski, L. Quantum algorithm for the advection–diffusion equation simulated with the lattice Boltzmann method. Quantum Inf Process 20, 57 (2021). https://doi.org/10.1007/s11128-021-02996-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-021-02996-3

Keywords

Search

Navigation