Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Computers & Fluids

On solving for shocks and travelling waves using a quantum algorithm

Biswajit Basu, Andrea Staino, Frank Gaitan

Published Jan 30, 2025• 4 citations
Publisher

Abstract

In this paper, we solve for shocks and travelling waves in advection , inviscid Burgers’ and Burgers’ equations by implementing a recently established quantum algorithm in the literature. The quantum algorithm has been successful in solving Navier–Stokes, flow generated by Burgers’ and submarine tephra flow equations under certain initial and boundary conditions . Here, we further study the efficacy of the quantum algorithm by extending the application to advection , inviscid Burgers’ and Burgers’ equations under different kinds of initial and boundary conditions. In addition to central differencing and upwinding, Lax–Wendroff discretization scheme has also been introduced in the quantum algorithm to observe how numerical dissipation and dispersion are affected. We recover known travelling waves, and shocks with rarefaction and expansion.

DOI
10.1016/j.compfluid.2025.106559

Related papers

Quantum Implementation of Numerical Methods for Convection-Diffusion Equations: Toward Computational Fluid Dynamics

Bofeng Liu, Lixing Zhu, Zixuan Yang +1 more

We present quantum numerical methods for the typical initial boundary value problems (IBVPs) of convection-diffusion equations in fluid dynamics. The IBVP is discretized into a series of linear systems via finite difference methods and explicit time marching schemes. To solve these discrete systems in quantum computers, we design a series of quantum circuits, including four stages of encoding, amplification, adding source terms, and incorporating boundary conditions. In the encoding stage, the initial condition is encoded in the amplitudes of quantum registers as a state vector to take advantage of quantum algorithms in space complexity. In the following three stages, the discrete differential operators in classical computing are converted into unitary evolutions to satisfy the postulate in quantum systems. The related arithmetic calculations in quantum amplitudes are also realized to sum up the increments from these stages. The proposed quantum algorithm is implemented within the open-source quantum computing framework Qiskit [2]. By simulating one-dimensional transient problems, including the Helmholtz equation, the Burgers’ equation, and Navier-Stokes equations, we demonstrate the capability of quantum computers in fluid dynamics.

Communications in Computational Physics•Mar 12, 2023•16 citations

Quantum algorithm for solving the advection equation using Hamiltonian simulation

Peter Brearley, Sylvain Laizet

A quantum algorithm for solving the advection equation by embedding the discrete time-marching operator into Hamiltonian simulations is presented. One-dimensional advection can be simulated directly since the central finite-difference operator for first-order derivatives is anti-Hermitian. Here this is extended to industrially relevant multidimensional flows with realistic boundary conditions and arbitrary finite-difference stencils. A single copy of the initial quantum state is required and the circuit depth grows linearly with the required number of time steps, the sparsity of the time-marching operator, and the inverse of the allowable error. State-vector simulations of a scalar transported in a two-dimensional channel flow and lid-driven cavity configuration are presented as a proof of concept of the proposed approach. Published by the American Physical Society 2024

arXiv:2312.09784•Physical Review A•Jul 9, 2024•28 citations

Hamiltonian Simulation for Solving the Advection Equation With Arbitrary Velocity Field

Niladri Gomes, Gautam Sharma, Jay Pathak

We present a novel approach to solving the advection equation under arbitrary velocity fields using a quantuminspired ‘Schrödingerisation’ technique for Hamiltonian simulation. Although numerous methods exist for solving partial differential equations (PDEs), Hamiltonian simulation remains a relatively underexplored yet promising direction-particularly in the context of long-term, fault-tolerant quantum computing. Building on this potential, our quantum algorithm is designed to accommodate non-trivial, spatially varying velocity fields and is applicable to both 2D and 3D advection problems. To ensure numerical stability, the algorithm incorporates an upwinding discretization scheme, which we adapt for quantum implementation through a combination of approximation and optimization techniques. We demonstrate the algorithm's effectiveness on benchmark cases involving rotational and shear flows in two and three dimensions. Moreover, we successfully implement the 2D advection equation using 16 qubits on IBM Quantum hardware, validating the approach and showcasing its practical applicability and robustness.

IEEE International Conference on Quantum Computing and Engineering (QCE)•Aug 30, 2025•0 citations

Finding flows of a Navier–Stokes fluid through quantum computing

Frank Gaitan

Abstract There is great interest in using quantum computers to efficiently simulate a quantum system’s dynamics as existing classical computers cannot do this. Little attention, however, has been given to quantum simulation of a classical nonlinear continuum system such as a viscous fluid even though this too is hard for classical computers. Such fluids obey the Navier–Stokes nonlinear partial differential equations, whose solution is essential to the aerospace industry, weather forecasting, plasma magneto-hydrodynamics, and astrophysics. Here we present a quantum algorithm for solving the Navier–Stokes equations. We test the algorithm by using it to find the steady-state inviscid, compressible flow through a convergent-divergent nozzle when a shockwave is (is not) present. We find excellent agreement between numerical simulation results and the exact solution, including shockwave capture when present. Finally, we compare the algorithm’s computational cost to deterministic and random classical algorithms and show that a significant speed-up is possible. Our work points to a large new application area for quantum computing with substantial economic impact, including the trillion-dollar aerospace industry, weather-forecasting, and engineered-plasma technologies.

npj Quantum Information•Jul 16, 2020•175 citations

Shock-Capturing Quantum Algorithm for the Linear Advection Operator

Samuel Hagele, William Gregory, Yuan Shi

The linear advection operator is an ubiquitous building block in fluid and plasma problems. We develop a quantum algorithm for enacting the operator. When the advection velocity is constant in space, our algorithm is exponentially more efficient per time step than classical and avoids spurious oscillations near steep gradients. The algorithm is most cleanly illustrated using the one-dimensional advection equation on a uniform spatial grid with periodic boundary conditions, which can be extended to higher dimensions. The algorithm uses a first-order upwind scheme, which captures discontinuities in the wave envelope but is not unitary. We embed the non-unitary upwind scheme using Linear Combinations of Unitaries (LCUs), and develop an efficient quantum gate decomposition of the upwind unitary, which performs one step of advection using $O(n^2)$ two-qubit gates, where $N=2^n$ is the number of spatial grid points, as opposed to a classical computer which costs $O(N)$. Although LCUs introduces a small bounded probability of failure per time step, we show that the accumulation of failures does not lead to exponential-in-time complexity as one would naively expect. Moreover, when LCUs fails, we develop a probabilistic scheme to recover from the failure state, which avoids a full restart of the simulation. The failure recovery scheme uses quantum Fourier transform (QFT) and effectively achieves quantum indefinite integration of an unknown quantum state. The recovery, which can itself fail, is more efficient than a full restart if the wave envelop is well-resolved to include only low Fourier modes. We emulate our scheme classically and demonstrate small problems on Quantinuum's trapped-ion qubits. Our quantum algorithm provides a subroutine for physics simulations that involve linear advection.

quant-pharXiv:2609.21153•Sep 21, 2026•0 citations

Solving Burgers’ equation with quantum computing

Furkan Oz, Rohit K. S. S. Vuppala, Kursat Kara +1 more

Abstract Computational fluid dynamics (CFD) simulations are a vital part of the design process in the aerospace industry. Although reliable CFD results can be obtained with turbulence models, direct numerical simulation of complex bodies in three spatial dimensions (3D) is impracticable due to the massive amount of computational elements. For instance, a 3D direct numerical simulation of a turbulent boundary-layer over the wing of a commercial jetliner that resolves all relevant length scales using a serial CFD solver on a modern digital computer would take approximately 750 million years or roughly 20% of the earth’s age. Over the past 25 years, quantum computers have become the object of great interest worldwide as powerful quantum algorithms have been constructed for several important, computationally challenging problems that provide enormous speed-up over the best-known classical algorithms. In this paper, we adapt a recently introduced quantum algorithm for partial differential equations to Burgers’ equation and develop a quantum CFD solver that determines its solutions. We used our quantum CFD solver to verify the quantum Burgers’ equation algorithm to find the flow solution when a shockwave is and is not present. The quantum simulation results were compared to: (i) an exact analytical solution for a flow without a shockwave; and (ii) the results of a classical CFD solver for flows with and without a shockwave. Excellent agreement was found in both cases, and the error of the quantum CFD solver was comparable to that of the classical CFD solver.

Quantum Information Processing•Dec 31, 2021•55 citations
In this paper, we solve for shocks and travelling waves in advection , inviscid Burgers’ and Burgers’ equations by implementing a recently established quantum algorithm in the literature. The quantum algorithm has been successful in solving Navier–Stokes, flow generated by Burgers’ and submarine tephra flow equations under certain initial and boundary conditions . Here, we further study the efficacy of the quantum algorithm by extending the application to advection , inviscid Burgers’ and Burgers’ equations under different kinds of initial and boundary conditions. In addition to central differencing and upwinding, Lax–Wendroff discretization scheme has also been introduced in the quantum algorithm to observe how numerical dissipation and dispersion are affected. We recover known travelling waves, and shocks with rarefaction and expansion.
We present quantum numerical methods for the typical initial boundary value problems (IBVPs) of convection-diffusion equations in fluid dynamics. The IBVP is discretized into a series of linear systems via finite difference methods and explicit time marching schemes. To solve these discrete systems in quantum computers, we design a series of quantum circuits, including four stages of encoding, amplification, adding source terms, and incorporating boundary conditions. In the encoding stage, the initial condition is encoded in the amplitudes of quantum registers as a state vector to take advantage of quantum algorithms in space complexity. In the following three stages, the discrete differential operators in classical computing are converted into unitary evolutions to satisfy the postulate in quantum systems. The related arithmetic calculations in quantum amplitudes are also realized to sum up the increments from these stages. The proposed quantum algorithm is implemented within the open-source quantum computing framework Qiskit [2]. By simulating one-dimensional transient problems, including the Helmholtz equation, the Burgers’ equation, and Navier-Stokes equations, we demonstrate the capability of quantum computers in fluid dynamics.
A quantum algorithm for solving the advection equation by embedding the discrete time-marching operator into Hamiltonian simulations is presented. One-dimensional advection can be simulated directly since the central finite-difference operator for first-order derivatives is anti-Hermitian. Here this is extended to industrially relevant multidimensional flows with realistic boundary conditions and arbitrary finite-difference stencils. A single copy of the initial quantum state is required and the circuit depth grows linearly with the required number of time steps, the sparsity of the time-marching operator, and the inverse of the allowable error. State-vector simulations of a scalar transported in a two-dimensional channel flow and lid-driven cavity configuration are presented as a proof of concept of the proposed approach. Published by the American Physical Society 2024
We present a novel approach to solving the advection equation under arbitrary velocity fields using a quantuminspired ‘Schrödingerisation’ technique for Hamiltonian simulation. Although numerous methods exist for solving partial differential equations (PDEs), Hamiltonian simulation remains a relatively underexplored yet promising direction-particularly in the context of long-term, fault-tolerant quantum computing. Building on this potential, our quantum algorithm is designed to accommodate non-trivial, spatially varying velocity fields and is applicable to both 2D and 3D advection problems. To ensure numerical stability, the algorithm incorporates an upwinding discretization scheme, which we adapt for quantum implementation through a combination of approximation and optimization techniques. We demonstrate the algorithm's effectiveness on benchmark cases involving rotational and shear flows in two and three dimensions. Moreover, we successfully implement the 2D advection equation using 16 qubits on IBM Quantum hardware, validating the approach and showcasing its practical applicability and robustness.
Abstract There is great interest in using quantum computers to efficiently simulate a quantum system’s dynamics as existing classical computers cannot do this. Little attention, however, has been given to quantum simulation of a classical nonlinear continuum system such as a viscous fluid even though this too is hard for classical computers. Such fluids obey the Navier–Stokes nonlinear partial differential equations, whose solution is essential to the aerospace industry, weather forecasting, plasma magneto-hydrodynamics, and astrophysics. Here we present a quantum algorithm for solving the Navier–Stokes equations. We test the algorithm by using it to find the steady-state inviscid, compressible flow through a convergent-divergent nozzle when a shockwave is (is not) present. We find excellent agreement between numerical simulation results and the exact solution, including shockwave capture when present. Finally, we compare the algorithm’s computational cost to deterministic and random classical algorithms and show that a significant speed-up is possible. Our work points to a large new application area for quantum computing with substantial economic impact, including the trillion-dollar aerospace industry, weather-forecasting, and engineered-plasma technologies.
The linear advection operator is an ubiquitous building block in fluid and plasma problems. We develop a quantum algorithm for enacting the operator. When the advection velocity is constant in space, our algorithm is exponentially more efficient per time step than classical and avoids spurious oscillations near steep gradients. The algorithm is most cleanly illustrated using the one-dimensional advection equation on a uniform spatial grid with periodic boundary conditions, which can be extended to higher dimensions. The algorithm uses a first-order upwind scheme, which captures discontinuities in the wave envelope but is not unitary. We embed the non-unitary upwind scheme using Linear Combinations of Unitaries (LCUs), and develop an efficient quantum gate decomposition of the upwind unitary, which performs one step of advection using O(n2)O(n^2)O(n2) two-qubit gates, where N=2nN=2^nN=2n is the number of spatial grid points, as opposed to a classical computer which costs O(N)O(N)O(N). Although LCUs introduces a small bounded probability of failure per time step, we show that the accumulation of failures does not lead to exponential-in-time complexity as one would naively expect. Moreover, when LCUs fails, we develop a probabilistic scheme to recover from the failure state, which avoids a full restart of the simulation. The failure recovery scheme uses quantum Fourier transform (QFT) and effectively achieves quantum indefinite integration of an unknown quantum state. The recovery, which can itself fail, is more efficient than a full restart if the wave envelop is well-resolved to include only low Fourier modes. We emulate our scheme classically and demonstrate small problems on Quantinuum's trapped-ion qubits. Our quantum algorithm provides a subroutine for physics simulations that involve linear advection.
Abstract Computational fluid dynamics (CFD) simulations are a vital part of the design process in the aerospace industry. Although reliable CFD results can be obtained with turbulence models, direct numerical simulation of complex bodies in three spatial dimensions (3D) is impracticable due to the massive amount of computational elements. For instance, a 3D direct numerical simulation of a turbulent boundary-layer over the wing of a commercial jetliner that resolves all relevant length scales using a serial CFD solver on a modern digital computer would take approximately 750 million years or roughly 20% of the earth’s age. Over the past 25 years, quantum computers have become the object of great interest worldwide as powerful quantum algorithms have been constructed for several important, computationally challenging problems that provide enormous speed-up over the best-known classical algorithms. In this paper, we adapt a recently introduced quantum algorithm for partial differential equations to Burgers’ equation and develop a quantum CFD solver that determines its solutions. We used our quantum CFD solver to verify the quantum Burgers’ equation algorithm to find the flow solution when a shockwave is and is not present. The quantum simulation results were compared to: (i) an exact analytical solution for a flow without a shockwave; and (ii) the results of a classical CFD solver for flows with and without a shockwave. Excellent agreement was found in both cases, and the error of the quantum CFD solver was comparable to that of the classical CFD solver.