TY - JOUR

T1 - Two quantum algorithms for solving the one-dimensional advection–diffusion equation

AU - Ingelmann, Julia

AU - Bharadwaj, Sachin S.

AU - Pfeffer, Philipp

AU - Sreenivasan, Katepalli R.

AU - Schumacher, Jörg

N1 - Publisher Copyright:
© 2024 The Author(s)

PY - 2024/8/30

Y1 - 2024/8/30

N2 - Two quantum algorithms are presented for the numerical solution of a linear one-dimensional advection–diffusion equation with periodic boundary conditions. Their accuracy and performance with increasing qubit number are compared point-by-point with each other. Specifically, we solve the linear partial differential equation with a Quantum Linear Systems Algorithm (QLSA) based on the Harrow–Hassidim–Lloyd method and a Variational Quantum Algorithm (VQA), for resolutions that can be encoded using up to 6 qubits, which corresponds to N=64 grid points on the unit interval. Both algorithms are hybrid in nature, i.e., they involve a combination of classical and quantum computing building blocks. The QLSA and VQA are solved as ideal statevector simulations using the in-house solver QFlowS and open-access Qiskit software, respectively. We discuss several aspects of both algorithms which are crucial for a successful performance in both cases. These are the accurate eigenvalue estimation with the quantum phase estimation for the QLSA and the choice of the algorithm of the minimization of the cost function for the VQA. The latter algorithm is also implemented in the noisy Qiskit framework including measurement noise. We reflect on the current limitations and suggest some possible routes of future research for the numerical simulation of classical fluid flows on a quantum computer.

AB - Two quantum algorithms are presented for the numerical solution of a linear one-dimensional advection–diffusion equation with periodic boundary conditions. Their accuracy and performance with increasing qubit number are compared point-by-point with each other. Specifically, we solve the linear partial differential equation with a Quantum Linear Systems Algorithm (QLSA) based on the Harrow–Hassidim–Lloyd method and a Variational Quantum Algorithm (VQA), for resolutions that can be encoded using up to 6 qubits, which corresponds to N=64 grid points on the unit interval. Both algorithms are hybrid in nature, i.e., they involve a combination of classical and quantum computing building blocks. The QLSA and VQA are solved as ideal statevector simulations using the in-house solver QFlowS and open-access Qiskit software, respectively. We discuss several aspects of both algorithms which are crucial for a successful performance in both cases. These are the accurate eigenvalue estimation with the quantum phase estimation for the QLSA and the choice of the algorithm of the minimization of the cost function for the VQA. The latter algorithm is also implemented in the noisy Qiskit framework including measurement noise. We reflect on the current limitations and suggest some possible routes of future research for the numerical simulation of classical fluid flows on a quantum computer.

KW - Quantum computing

KW - Quantum linear systems algorithm

KW - Variational quantum algorithm

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U2 - 10.1016/j.compfluid.2024.106369

DO - 10.1016/j.compfluid.2024.106369

M3 - Article

AN - SCOPUS:85199497071

SN - 0045-7930

VL - 281

JO - Computers and Fluids

JF - Computers and Fluids

M1 - 106369

ER -