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Incorporating high-accuracy, finite-volume shock stabilization methods into a quantum algorithm for nonlinear partial differential equations

Ryan Vogt1, Harold Berjamin2, Hunter Rouillard3, Edward Raff4,*, Priyanka Ranade1, Torstein Collett5, Roberto D'Angelo-Cosme1, James Holt1, and Frank Gaitan3,†

  • *Present address: CrowdStrike, 206 E. 9th St, Suite 1400, Austin, Texas 78701, USA.
  • †Contact author: fgaitan@lps.umd.edu
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Phys. Rev. A 112, 052431 – Published 17 November, 2025

DOI: https://doi.org/10.1103/9phc-5m7b

Abstract

Conservation laws are of fundamental importance in science and engineering, and so are ubiquitous in applications. For continuum systems such as classical fluids and plasmas, they give rise to coupled systems of nonlinear partial differential equations (PDEs) whose solutions can develop discontinuities such as shocks. Finite-volume (FV) methods are an important tool for finding approximate solutions to such PDEs as they are based on an integral formulation of the conservation laws that can handle the presence of discontinuities. Weighted essentially nonoscillatory (WENO) methods are a class of high-resolution methods that can (i) produce solutions of arbitrarily high-order accuracy in regions where the solution is smooth, while (ii) simultaneously quashing the development of nonphysical oscillations in the vicinity of discontinuities that often cause numerical simulations to fail. In this paper we show how FV and WENO methods can be incorporated into a quantum algorithm for solving systems of nonlinear PDEs (QPDE) introduced by one of the authors. We numerically simulate application of the modified QPDE algorithm to three verification problems. The first has a smooth solution, while the other two are well-known one-dimensional Riemann problems whose exact solutions contain moving shocks, contact discontinuities, and rarefaction waves. For all three problems excellent agreement is found between the results of the QPDE simulations and the exact solutions.

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References (33)

  1. F. Gaitan, Finding flows of a Navier-Stokes fluid through quantum computing, npj Quantum Inf. 6, 61 (2020).
  2. F. Gaitan, Finding solutions of the Navier-Stokes equations through quantum Computing—Recent progress, a generalization, and next steps forward, Adv. Quantum Technol. 4, 2100055 (2021).
  3. F. Oz, R. K. S. S. Vuppala, K. Kara, and F. Gaitan, Solving Burgers' equation with quantum computing, Quantum Inf. Process. 21, 30 (2022).
  4. B. Basu, S. Gurusamy, and F. Gaitan, A quantum algorithm for computing dispersal of submarine volcanic tephra, Phys. Fluids 36, 036607 (2024).
  5. C. J. Myers, N. Gentile, H. Rouillard, R. Vogt, F. Graziani, and F. Gaitan, Quantum simulation of non-equilibrium Marshak waves, J. Plasma Phys. 90, 805900601 (2024).
  6. F. Gaitan, F. Graziani, and M. Porter, Simulating nonlinear radiation diffusion through quantum computing, Int. J. Theor. Phys. 63, 260 (2024).
  7. R. J. Leveque, Finite Volume Methods for Hyperbolic Problems (Cambridge University Press, New York, 2002).
  8. S. K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Sb. Math. 47, 271 (1959).
  9. P. D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation, Commun. Pure Appl. Math. 7, 159 (1954).
  10. P. D. Lax and B. Wendroff, Systems of conservation laws, Commun. Pure Appl. Math. 13, 217 (1960).
  11. E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd ed. (Springer, New York, 2009).
  12. X. D. Liu, S. Osher, and T. Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys. 115, 200 (1994).
  13. G. Jiang and C. W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126, 202 (1996).
  14. C. W. Shu, High order weighted essentially non-oscillatory schemes for convection dominated problems, SIAM Rev. 51, 82 (2009).
  15. G. A. Sod, A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys. 27, 1 (1978).
  16. F. Gaitan, Circuit implementation of oracles used in a quantum algorithm for solving nonlinear partial differential equations, Phys. Rev. A 109, 032604 (2024)
  17. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, New York, 1987).
  18. B. Kacewicz, Almost optimal solution of initial-value problems by randomized and quantum algorithms, J. Complexity 22, 676 (2006).
  19. W. Gautschi, Numerical Analysis (Birkhäuser, New York, 2012).
  20. A. Iserles, A First Course in the Numerical Analysis of Differential Equations (Cambridge University Press, New York, 2009).
  21. G. Brassard, P. Hoyer, M. Mosca, and A. Tapp, Quantum amplitude amplification and estimation, in Quantum Computation and Information, edited by S. J. Lomonaco, Jr. and H. E. Brandt, Contemporary Mathematics (AMS, Providence, RI, 2000), Vol. 305, pp. 53–74.
  22. E. Novak, Quantum complexity of integration, J. Complexity 17, 2 (2001).
  23. Y. Guo, T. Xiong, and Y. Shi, A positivity-preserving high order finite volume compact-Weno scheme for compressible Euler equations, J. Comput. Phys. 274, 505 (2014).
  24. C. Hirsch, Numerical Computation of Internal and External Flows, Vol. II: Computational Methods for Inviscid and Viscous Flows (Wiley, New York, 1990).
  25. B. Basu, A. Staino, and F. Gaitan, On solving for shocks and waves using a quantum algorithm, Comput. Fluids 290, 106559 (2025).
  26. F. Nguyen, J.-P. Laval, P. Kestener, A. Cheskidov, R. Shvydkoy, and B. Dubrulle, Local estimates of Hölder exponents in turbulent vectorfields, Phys. Rev. E 99, 053114 (2019).
  27. F. Nguyen, J.-P. Laval, and B. Dubrulle, Characterizing most irregular small-scale structures in turbulence using local Hölder exponents, Phys. Rev. E 102, 063105 (2020).
  28. B. Dubrulle and J. D. Gibbon, A correspondence between the multifractal model of turbulence and the Navier-Stokes equations, Philos. Trans. R. Soc. A 380, 20210092 (2022).
  29. B. Dubrulle, Multi-fractality, universality, and singularity in turbulence, Fractal Fract. 6, 613 (2022).
  30. S. Heinrich, Quantum integration in Sobolev classes, J. Complexity 19, 19 (2003).
  31. P. A. Davidson, Turbulence, 2nd ed. (Oxford University Press, New York, 2015).
  32. The simulation code used to generate the results appearing in Figs. 1– 9 can be downloaded from https://github.com/DrVogtster/QPDE-FV-WENO.
  33. R. Courant, K. O. Friedrichs, and H. Lewy, Über die partiellen differenzengleichungen der mathematischen physik, Math. Ann. 100, 32 (1928) [English translation: On the partial difference equations of mathematical physics, IBM J. Res. Dev. 11, 215 (1967)].

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Phys. Rev. A 112, 052431– Published 17 November, 2025
  • Received 18 September 2024
  • Revised 26 April 2025
  • Accepted 28 October 2025
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