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

IEEE International Conference on Quantum Computing and Engineering (QCE)

Hamiltonian Simulation for Solving the Advection Equation With Arbitrary Velocity Field

Niladri Gomes, Gautam Sharma, Jay Pathak

Published Aug 30, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1109/qce65121.2025.00202

Related papers

Quantum Circuits for partial differential equations via Schrödingerisation

Junpeng Hu, Shi Jin, Nana Liu +1 more

Quantum computing has emerged as a promising avenue for achieving significant speedup, particularly in large-scale PDE simulations, compared to classical computing. One of the main quantum approaches involves utilizing Hamiltonian simulation, which is directly applicable only to Schrödinger-type equations. To address this limitation, Schrödingerisation techniques have been developed, employing the warped transformation to convert general linear PDEs into Schrödinger-type equations. However, despite the development of Schrödingerisation techniques, the explicit implementation of the corresponding quantum circuit for solving general PDEs remains to be designed. In this paper, we present detailed implementation of a quantum algorithm for general PDEs using Schrödingerisation techniques. We provide examples of the heat equation, and the advection equation approximated by the upwind scheme, to demonstrate the effectiveness of our approach. Complexity analysis is also carried out to demonstrate the quantum advantages of these algorithms in high dimensions over their classical counterparts.

arXiv:2403.10032•Quantum•Dec 12, 2024•27 citations

Quantum algorithm and its implementation for solving partial differential equations

Yuki Sato, Hiroyuki Tezuka, Ruho Kondo +1 more

Partial differential equations (PDEs) play an important role in engineering since various physical phenomena can be well described by PDEs. The key problems in solving PDEs are memory usage and computational time when the size of the system of interest grows. Given the potential of quantum computing, we propose a quantum algorithm for solving PDEs based on the Hamiltonian simulation, which is well known as the possible application of quantum computing for physical simulation. First, we review the method of linear combination of Hamiltonian simulation and propose its implementation via a tensor network technique. Second, we discuss how to deal with the spatially varying physical constants via a logic minimizer technique. Finally, we provide numerical experiments to demonstrate our proposed method.

Keisan Rikigaku Koenkai koen ronbunshu/Keisan Rikigaku Kouenkai kouen rombunshuu•Jan 1, 2024•0 citations

A quantum algorithm for the lattice-Boltzmann method advection-diffusion equation

David Wawrzyniak, Josef M. Winter, Steffen J. Schmidt +4 more

We present a versatile and efficient quantum algorithm based on the Lattice Boltzmann method (LBM) approximate solution of the linear advection-diffusion equation (ADE). We emphasize that the LBM approximation modifies the diffusion term of the underlying exact ADE and leads to a modified equation (mADE). Due to its versatility in terms of operator splitting, the proposed quantum LBM algorithm for the mADE provides a building block for future quantum algorithms to solve the linearized Navier-Stokes equation on quantum computers. We split the algorithm into four operations: initialization, collision, streaming, and calculation of the macroscopic quantities. We propose general quantum building blocks for each operator, which adapt intrinsically from the general three-dimensional case to smaller dimensions and apply to arbitrary lattice-velocity sets. Based on (sub-linear) amplitude data encoding, we propose improved initialization and collision operations with reduced complexity and efficient sampling-based simulation. Quantum streaming algorithms are based on previous developments. The proposed quantum algorithm allows for the computation of successive time steps, requiring full state measurement and reinitialization after every time step. It is validated by comparison with a digital implementation and based on analytical solutions in one and two dimensions. Furthermore, we demonstrate the versatility of the quantum algorithm for two cases with non-uniform advection velocities in two and three dimensions. Various velocity sets are considered to further highlight the flexibility of the algorithm. We benchmark our optimized quantum algorithm against previous methods employed in sampling-based quantum simulators. We demonstrate sampling efficiency, with sampling accelerated convergence requiring fewer shots. • Novel LBM quantum algorithm for solving the linear advection-diffusion equation with variable velocity as a building block. • Modular quantum algorithm for 1 through 3 spatial dimensions and all common lattice-velocity sets. • Significant reduction in quantum-algorithm complexity achieved through efficient encoding strategy leveraging symmetries. • Improved collision quantum-algorithm building block leading to higher success rates for the collision operator. • Benchmarking shows faster convergence and reduced shot requirements compared to previous algorithms.

Computer Physics Communications•Sep 10, 2024•33 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

Quantum dynamics simulation of the advection-diffusion equation

Hirad Alipanah, Feng Zhang, Yong-Xin Yao +6 more

The advection-diffusion equation is simulated via several quantum algorithms. Three formulations are considered: (1) Trotterization, (2) variational quantum time evolution (VarQTE), and (3) adaptive variational quantum dynamics simulation (AVQDS). These schemes were originally developed for the Hamiltonian simulation of many-body quantum systems. The finite-difference discretized operator of the transport equation is formulated as a Hamiltonian and solved without the need for ancillary qubits. Computations are conducted on a quantum simulator (IBM Qiskit Aer) and a superconducting quantum hardware (IBM Fez). The former emulates the latter without the noise. The actual hardware implementation experiences significant noise. The results of the quantum simulator are compared with data from direct numerical simulation (DNS) with infidelities of the order 10 − 5 . In the quantum simulator, Trotterization is observed to have the lowest infidelity and is suitable for fault-tolerant computation. The AVQDS algorithm requires the lowest gate count and circuit depth. The VarQTE algorithm is the next best in terms of gate counts, but the number of its optimization variables is directly proportional to the number of qubits. Due to current hardware limitations, Trotterization cannot be implemented, as it has an overwhelmingly large number of operations. Meanwhile, AVQDS and VarQTE can be executed at the hardware level. These algorithms present a new paradigm for computational transport phenomena on quantum computers.

arXiv:2503.13729•Physical Review Research•Oct 19, 2025•6 citations

Resource-efficient quantum simulation of transport phenomena via Hamiltonian embedding

Joseph Li, Gengzhi Yang, Jiaqi Leng +1 more

Transport phenomena play a key role in a variety of application domains, and efficient simulation of these dynamics remains an outstanding challenge. While quantum computers offer potential for significant speedups, existing algorithms either lack rigorous theoretical guarantees or demand substantial quantum resources, preventing scalable and efficient validation on realistic quantum hardware. To address this gap, we develop a comprehensive framework for simulating classes of transport equations, offering both rigorous theoretical guarantees -- including exponential speedups in specific cases -- and a systematic, hardware-efficient implementation. Central to our approach is the Hamiltonian embedding technique, a white-box approach for end-to-end simulation of sparse Hamiltonians that avoids abstract query models and retains near-optimal asymptotic complexity. Empirical resource estimates indicate that our approach can yield an order-of-magnitude (e.g., $42\times$) reduction in circuit depth given favorable problem structures. We then apply our framework to solve linear and nonlinear transport PDEs, including the first experimental demonstration of a 2D advection equation on a trapped-ion quantum computer.

quant-pharXiv:2602.03099•Quantum Science and Technology•Feb 4, 2026•0 citations
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.
Quantum computing has emerged as a promising avenue for achieving significant speedup, particularly in large-scale PDE simulations, compared to classical computing. One of the main quantum approaches involves utilizing Hamiltonian simulation, which is directly applicable only to Schrödinger-type equations. To address this limitation, Schrödingerisation techniques have been developed, employing the warped transformation to convert general linear PDEs into Schrödinger-type equations. However, despite the development of Schrödingerisation techniques, the explicit implementation of the corresponding quantum circuit for solving general PDEs remains to be designed. In this paper, we present detailed implementation of a quantum algorithm for general PDEs using Schrödingerisation techniques. We provide examples of the heat equation, and the advection equation approximated by the upwind scheme, to demonstrate the effectiveness of our approach. Complexity analysis is also carried out to demonstrate the quantum advantages of these algorithms in high dimensions over their classical counterparts.
Partial differential equations (PDEs) play an important role in engineering since various physical phenomena can be well described by PDEs. The key problems in solving PDEs are memory usage and computational time when the size of the system of interest grows. Given the potential of quantum computing, we propose a quantum algorithm for solving PDEs based on the Hamiltonian simulation, which is well known as the possible application of quantum computing for physical simulation. First, we review the method of linear combination of Hamiltonian simulation and propose its implementation via a tensor network technique. Second, we discuss how to deal with the spatially varying physical constants via a logic minimizer technique. Finally, we provide numerical experiments to demonstrate our proposed method.
We present a versatile and efficient quantum algorithm based on the Lattice Boltzmann method (LBM) approximate solution of the linear advection-diffusion equation (ADE). We emphasize that the LBM approximation modifies the diffusion term of the underlying exact ADE and leads to a modified equation (mADE). Due to its versatility in terms of operator splitting, the proposed quantum LBM algorithm for the mADE provides a building block for future quantum algorithms to solve the linearized Navier-Stokes equation on quantum computers. We split the algorithm into four operations: initialization, collision, streaming, and calculation of the macroscopic quantities. We propose general quantum building blocks for each operator, which adapt intrinsically from the general three-dimensional case to smaller dimensions and apply to arbitrary lattice-velocity sets. Based on (sub-linear) amplitude data encoding, we propose improved initialization and collision operations with reduced complexity and efficient sampling-based simulation. Quantum streaming algorithms are based on previous developments. The proposed quantum algorithm allows for the computation of successive time steps, requiring full state measurement and reinitialization after every time step. It is validated by comparison with a digital implementation and based on analytical solutions in one and two dimensions. Furthermore, we demonstrate the versatility of the quantum algorithm for two cases with non-uniform advection velocities in two and three dimensions. Various velocity sets are considered to further highlight the flexibility of the algorithm. We benchmark our optimized quantum algorithm against previous methods employed in sampling-based quantum simulators. We demonstrate sampling efficiency, with sampling accelerated convergence requiring fewer shots. • Novel LBM quantum algorithm for solving the linear advection-diffusion equation with variable velocity as a building block. • Modular quantum algorithm for 1 through 3 spatial dimensions and all common lattice-velocity sets. • Significant reduction in quantum-algorithm complexity achieved through efficient encoding strategy leveraging symmetries. • Improved collision quantum-algorithm building block leading to higher success rates for the collision operator. • Benchmarking shows faster convergence and reduced shot requirements compared to previous algorithms.
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.
The advection-diffusion equation is simulated via several quantum algorithms. Three formulations are considered: (1) Trotterization, (2) variational quantum time evolution (VarQTE), and (3) adaptive variational quantum dynamics simulation (AVQDS). These schemes were originally developed for the Hamiltonian simulation of many-body quantum systems. The finite-difference discretized operator of the transport equation is formulated as a Hamiltonian and solved without the need for ancillary qubits. Computations are conducted on a quantum simulator (IBM Qiskit Aer) and a superconducting quantum hardware (IBM Fez). The former emulates the latter without the noise. The actual hardware implementation experiences significant noise. The results of the quantum simulator are compared with data from direct numerical simulation (DNS) with infidelities of the order 10 − 5 . In the quantum simulator, Trotterization is observed to have the lowest infidelity and is suitable for fault-tolerant computation. The AVQDS algorithm requires the lowest gate count and circuit depth. The VarQTE algorithm is the next best in terms of gate counts, but the number of its optimization variables is directly proportional to the number of qubits. Due to current hardware limitations, Trotterization cannot be implemented, as it has an overwhelmingly large number of operations. Meanwhile, AVQDS and VarQTE can be executed at the hardware level. These algorithms present a new paradigm for computational transport phenomena on quantum computers.
Transport phenomena play a key role in a variety of application domains, and efficient simulation of these dynamics remains an outstanding challenge. While quantum computers offer potential for significant speedups, existing algorithms either lack rigorous theoretical guarantees or demand substantial quantum resources, preventing scalable and efficient validation on realistic quantum hardware. To address this gap, we develop a comprehensive framework for simulating classes of transport equations, offering both rigorous theoretical guarantees -- including exponential speedups in specific cases -- and a systematic, hardware-efficient implementation. Central to our approach is the Hamiltonian embedding technique, a white-box approach for end-to-end simulation of sparse Hamiltonians that avoids abstract query models and retains near-optimal asymptotic complexity. Empirical resource estimates indicate that our approach can yield an order-of-magnitude (e.g., 42×42\times42×) reduction in circuit depth given favorable problem structures. We then apply our framework to solve linear and nonlinear transport PDEs, including the first experimental demonstration of a 2D advection equation on a trapped-ion quantum computer.