Classical Physics

Use quantum computers to solve linear systems of equations or simulate nonlinear physical systems.

Arithmetic operations

About the difficulty of performing simple arithmetic operations on a quantum computer. While factoring a number is much easier on a quantum computer, multiplying two real numbers is much more difficult! One algorithm involves the QFT.

Linear Systems of Equations

Nonlinear Problems

  • Lubasch et al., Variational quantum algorithms for nonlinear problems, 2020 [82] arXiv:1907.09032, doi:10.1103/physreva.101.010301
    • “Nonlinear problems including nonlinear partial differential equations can be efficiently solved by variational quantum computing.”

Special applications:

Fluid Dynamics

  • Ray et al., Solving the Navier-Stokes Equation, 2019 [111] - Los Alamos
  • Gaitan, Finding flows of a Navier-Stokes fluid, 2020 [40]
  • Gaitan, Finding Solutions of the Navier-Stokes Equations through Quantum Computing—Recent Progress, a Generalization, and Next Steps Forward, 2021, pdf:wiley, doi:10.1002/qute.202100055.
  • Steijl, book chapters: Quantum Algorithms for Fluid Simulations [122] and Quantum Algorithms for Nonlinear Equations in Fluid Mechanics [121]
  • Steijl, Quantum Algorithms for Nonlinear Equations in Fluid Mechanics, 2020, html:intechopen, doi:10.5772/intechopen.95023.
    • A key contributing factor to the limited progress in algorithms for non-linear problems is the inherent linearity of quantum mechanics. For quantum algorithms encoding information as amplitudes of a quantum state vector, nonlinear (product) terms cannot be obtained by multiplying these amplitudes by themselves, as a result of the no-cloning theorem that prohibits the copying of an arbitrary quantum state.

    • Quantum circuits for squaring floating-point numbers

    • Quantum circuits for multiplication of floating-point numbers

  • Griffin et al., Quantum algorithms for direct numerical simulation, 2020 [51]


Three papers supervised by Givi - Pittsburgh:

  • Sammak et al., Potential for Turbulence Simulations, 2015 [115]

  • Xu et al., Turbulent Mixing Simulation via a Quantum Algorithm, 2018 [137]

  • Xu et al., Reactant conversion rate in homogeneous turbulence, 2019 [138]


  • Brassard et al., Quantum algorithm to approximate the mean, 2011 [20] (Brassard is one of the authors of the BB84 protocol)


Using a quantum computer to solve problems of classical physics faces numerous and fundamental problems:

  • Performing some of the basic arithmetic operations as the multiplication is much more complex and inefficient on a quantum hardware.

  • Nonlinear product terms cannot be obtained easily, as a result of the no-cloning theorem.

  • Most classical physics problems involves spatial and temporal discretization that results in a huge amount of (qu)bits to represent the problem with the required accuracy, posing a serious obstacle to the use of a quantum computer in a foreseeable future.

  • Even using a QPU for subproblems that are “hard” to solve with classical algorithms may require a large number of error-corrected qubits. And these subproblems still have to be identified.

Complements: An Introduction » Quantum Computing » Applications