- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review A

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Two-dimensional quantum-lattice-gas algorithm for anisotropic Burger-like equations

Niccoló Fonio*

Pierre Sagaut

Giuseppe Di Molfetta

  • *Contact author: niccol.fonio11@gmail.com
PDF

Phys. Rev. A 114, 032426 – Published 10 September, 2026

DOI: https://doi.org/10.1103/34zq-6m8d

Abstract

Building on hybrid quantum lattice gas algorithm, we revisit the possibilities of this quantum lattice model. By deriving a correction to the predicted viscosity, we provide analytical and numerical results that refine original formulation. We introduce a minimal two-dimensional generalization of the algorithm, which allows us to simulate anisotropic Burger-like equations while retaining only two lattice velocities. This approach opens a promising route toward embedding momentum conservation and advancing toward Navier-Stokestwo dynamics in two dimensions, going beyond Frisch, Hasslacher, and Pomeau and lattice Boltzmann method with a quantum native model. We highlight how the presented algorithm is more efficient for full state evolution than other quantum nonlinear solvers, nonetheless we clarify its advantages compared to classical lattice gas methods. Being that this model is between classical and quantum computation, it gives a unique perspective on simulating nonlinearities with quantum computers, confirming quantum lattice gas models as a crucial playground for quantum simulations of nonlinearities.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (53)

  1. G. H. Low and I. L. Chuang, Hamiltonian simulation by qubitization, Quantum 3, 163 (2019).
  2. A. Gilyén, Y. Su, G. H. Low, and N. Wiebe, in Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (Association for Computing Machinery, New York, NY, United States, 2019), pp. 193–204.
  3. S. Jin, N. Liu, and Y. Yu, Quantum simulation of partial differential equations via Schrödingerization, Phys. Rev. Lett. 133, 230602 (2024).
  4. J. Hu, S. Jin, N. Liu, and L. Zhang, Quantum circuits for partial differential equations via Schrödingerisation, Quantum 8, 1563 (2024).
  5. D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, in Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, New York, NY, United States, 2014), pp. 283–292.
  6. D. W. Berry, A. M. Childs, A. Ostrander, and G. Wang, Quantum algorithm for linear differential equations with exponentially improved dependence on precision, Commun. Math. Phys. 356, 1057 (2017).
  7. J.-P. Liu, H. Ø. Kolden, H. K. Krovi, N. F. Loureiro, K. Trivisa, and A. M. Childs, Efficient quantum algorithm for dissipative nonlinear differential equations, Proc. Natl. Acad. Sci. USA 118, e2026805118 (2021).
  8. X. Li, X. Yin, N. Wiebe, J. Chun, G. K. Schenter, M. S. Cheung, and J. Mülmenstädt, Potential quantum advantage for simulation of fluid dynamics, Phys. Rev. Res. 7, 013036 (2025).
  9. W. Itani, K. R. Sreenivasan, and S. Succi, Quantum algorithm for lattice Boltzmann (QALB) simulation of incompressible fluids with a nonlinear collision term, Phys. Fluids 36, 017112 (2024).
  10. C. Sanavio and S. Succi, Lattice Boltzmann–Carleman quantum algorithm and circuit for fluid flows at moderate Reynolds number, AVS Quantum Sci. 6, 023802 (2024).
  11. I. Joseph, Koopman–von Neumann approach to quantum simulation of nonlinear classical dynamics, Phys. Rev. Res. 2, 043102 (2020).
  12. Y. Gan, H. Alipanah, J. Cheng, Z. Wu, G. Li, J. J. Mendoza-Arenas, P. Givi, M. R. Malik, B. J. McDermott, and J. Liu, Provably efficient quantum algorithms for solving nonlinear differential equations using multiple bosonic modes coupled with qubits, arXiv:2511.09939.
  13. D. Jennings, K. Korzekwa, M. Lostaglio, and G. Wang, Quantum Koopman algorithms, arXiv:2605.19054.
  14. S. S. Bharadwaj, B. Nadiga, S. Eidenbenz, and K. R. Sreenivasan, Quantum homotopy algorithm for solving nonlinear PDEs and flow problems, arXiv:2512.21033.
  15. S. S. Bharadwaj and K. R. Sreenivasan, Compact quantum algorithms for time-dependent differential equations, Phys. Rev. Res. 7, 023262 (2025).
  16. D. Jennings, K. Korzekwa, M. Lostaglio, R. Ashworth, E. Marsili, and S. Rolston, An end-to-end quantum algorithm for nonlinear fluid dynamics with bounded quantum advantage, arXiv:2512.03758.
  17. J. Gonzalez-Conde, D. Lewis, S. S. Bharadwaj, and M. Sanz, Quantum Carleman linearization efficiency in nonlinear fluid dynamics, Phys. Rev. Res. 7, 023254 (2025).
  18. D. Lewis, S. Eidenbenz, B. Nadiga, and Y. Subaşı, Limitations for quantum algorithms to solve turbulent and chaotic systems, Quantum 8, 1509 (2024).
  19. D. A. Wolf-Gladrow, Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction (Springer, Berlin, 2004).
  20. T. Kruger, Lattice Boltzmann Method-Principles and Practice (Springer, Berlin, 2016).
  21. S. Succi, The Lattice Boltzmann Equation: For Fluid Dynamics and Beyond (Oxford University Press, Oxford, 2001).
  22. L. Budinski, Quantum algorithm for the advection–diffusion equation simulated with the lattice Boltzmann method, Quantum Info. Proc. 20, 57 (2021).
  23. B. Ljubomir, Quantum algorithm for the Navier-Stokes equations by using the streamfunction-vorticity formulation and the lattice Boltzmann method, Int. J. Quantum Inf. 20, 2150039 (2022).
  24. D. Wawrzyniak, J. Winter, S. Schmidt, T. Indinger, U. Schramm, C. Janßen, and N. A. Adams, Unitary quantum algorithm for the lattice-Boltzmann method, arXiv:2405.13391.
  25. E. D. Kumar and S. H. Frankel, Quantum unitary matrix representation of the lattice Boltzmann model for low Reynolds fluid flow simulation, AVS Quantum Sci. 7, 013802 (2025).
  26. N. Fonio, L. Budinski, V. Lahtinen, and P. Sagaut, Adaptive lattice-gas algorithm: Classical and quantum implementations, Phys. Rev. E 112, 035302 (2025).
  27. A. D. B. Zamora, L. Budinski, V. Lahtinen, and P. Sagaut, Lattice gas automata with floating-point numbers: A connection between molecular dynamics and lattice Boltzmann method for quantum computers, Phys. Rev. E 112, 015305 (2025).
  28. B. Wang, Z. Meng, Y. Zhao, and Y. Yang, Quantum lattice Boltzmann method for simulating nonlinear fluid dynamics, npj Quantum Inf. 11, 196 (2025).
  29. S. Succi, W. Itani, K. Sreenivasan, and R. Steijl, Quantum computing for fluids: Where do we stand? Europhys. Lett. 144, 10001 (2023).
  30. A. A. Zecchi, C. Sanavio, S. Perotto, and S. Succi, Improved amplitude amplification strategies for the quantum simulation of classical transport problems, Quantum Sci. Technol. 10, 035039 (2025).
  31. Y. Xiao, L. Yang, C. Shu, Y. Du, H. Dong, and J. Wu, Quantum lattice kinetic scheme for solving two-dimensional and three-dimensional incompressible flows, arXiv:2505.10883.
  32. L. Xu, M. Li, L. Zhang, H. Sun, and J. Yao, Improved quantum lattice Boltzmann method for advection-diffusion equations with a linear collision model, Phys. Rev. E 111, 045305 (2025).
  33. M. Bediche, M. van Waveren, D. Ricot, and P. Sagaut, Fully Quantum Algorithm for the 1-Dimensional Linear Lattice Boltzmann Method, in Quantum Engineering Sciences and Technologies for Industry and Services. QUEST-IS 2025. Communications in Computer and Information Science, edited by F. Barbaresco and F. Gerin (Springer, Cham, 2025), Vol. 2744.
  34. J. Yepez, Open quantum system model of the one-dimensional Burgers equation with tunable shear viscosity, Phys. Rev. A 74, 042322 (2006).
  35. P. J. Love and B. M. Boghosian, Type II quantum algorithms, Physica A 362, 210 (2006).
  36. P. Love, On quantum extensions of hydrodynamic lattice gas automata, Condens. Matter 4, 48 (2019).
  37. J. Yepez, G. Vahala, and L. Vahala, Vortex-antivortex pair in a Bose-Einstein condensate, Eur. Phys. J.: Spec. Top. 171, 9 (2009).
  38. J. Yepez, G. Vahala, L. Vahala, and M. Soe, Superfluid turbulence from quantum Kelvin wave to classical Kolmogorov cascades, Phys. Rev. Lett. 103, 084501 (2009).
  39. P. Arrighi, G. Di Molfetta, I. Márquez-Martín, and A. Pérez, Dirac equation as a quantum walk over the honeycomb and triangular lattices, Phys. Rev. A 97, 062111 (2018).
  40. G. Di Molfetta, Quantum Walks, Limits, and Transport Equations (IOP Publishing, Bristol, 2024).
  41. https://github.com/niccolofonio/QuantumLatticeBoltzmannGas.
  42. M. M. Micci and J. Yepez, Measurement-based quantum lattice gas model of fluid dynamics in 2+1 dimensions, Phys. Rev. E 92, 033302 (2015).
  43. G. Wissocq, P. Sagaut, and J.-F. Boussuge, An extended spectral analysis of the lattice Boltzmann method: Modal interactions and stability issues, J. Comput. Phys. 380, 311 (2019).
  44. G. Wissocq and P. Sagaut, Hydrodynamic limits and numerical errors of isothermal lattice Boltzmann schemes, J. Comput. Phys. 450, 110858 (2022).
  45. S. A. Hosseini, M. Atif, S. Ansumali, and I. V. Karlin, Entropic lattice Boltzmann methods: A review, Comput. Fluids 259, 105884 (2023).
  46. G. G. McNamara and G. Zanetti, in Lattice Gas Methods For Partial Differential Equations (CRC Press, Boca Raton, 2019), pp. 289–296.
  47. U. Frisch, D. d'Humieres, B. Hasslacher, P. Lallemand, Y. Pomeau, and J.-P. Rivet, in Lattice Gas Methods for Partial Differential Equations (CRC Press, Boca Raton, 2019), pp. 77–136.
  48. B. M. Boghosian and C. D. Levermore, in Lattice Gas Methods For Partial Differential Equations (CRC Press, Boca Raton, 2019), pp. 483–496.
  49. M. A. Pravia, Z. Chen, J. Yepez, and D. G. Cory, Experimental demonstration of quantum lattice gas computation, Quantum Info. Proc. 2, 97 (2003).
  50. G. P. Berman, A. A. Ezhov, D. I. Kamenev, and J. Yepez, Simulation of the diffusion equation on a type-II quantum computer, Phys. Rev. A 66, 012310 (2002).
  51. A. Montanaro, Quantum speedup of Monte Carlo methods, Proc. R. Soc. A 471, 20150301 (2015).
  52. A. W. Harrow, A. Hassidim, and S. Lloyd, Quantum algorithm for linear systems of equations, Phys. Rev. Lett. 103, 150502 (2009).
  53. D. Wawrzyniak, J. Winter, S. Schmidt, T. Indinger, C. F. Janßen, U. Schramm, and N. A. Adams, Linearized quantum lattice-Boltzmann method for the advection-diffusion equation using dynamic circuits, Comput. Phys. Commun. 317 109856 (2025).

Outline

Information

Phys. Rev. A 114, 032426– Published 10 September, 2026
  • Received 3 March 2026
  • Accepted 20 August 2026
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation