Quantum Algorithm for Solving the Advection Equation using Hamiltonian Simulation

Abstract

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, multi-dimensional 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. Statevector 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.

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Cited by in corpus (12)

• Hamiltonian simulation for hyperbolic partial differential equations by scalable quantum circuits
• Compact quantum algorithms for time-dependent differential equations
• Boundary Treatment for Variational Quantum Simulations of Partial Differential Equations on Quantum Computers
• Towards Variational Quantum Algorithms for generalized linear and nonlinear transport phenomena
• Quantum Algorithm for the Advection-Diffusion Equation by Direct Block Encoding of the Time-Marching Operator
• Quantum Dynamics Simulation of the Advection-Diffusion Equation
• Tensor-Programmable Quantum Circuits for Solving Differential Equations
• Quantum algorithms for solving a drift-diffusion equation: A complexity analysis
• Quantum circuits for partial differential equations in Fourier space
• A Quantum-Inspired Algorithm for Wave Simulation Using Tensor Networks
• Performance evaluation of variational quantum eigensolver and quantum dynamics algorithms on the advection-diffusion equation
• Spectral quantum algorithm for passive scalar transport in shear flows