Nonlinear dynamics as a ground-state solution on quantum computers

Abstract

For the solution of time-dependent nonlinear differential equations, we present variational quantum algorithms (VQAs) that encode both space and time in qubit registers. The spacetime encoding enables us to obtain the entire time evolution from a single ground-state computation. We describe a general procedure to construct efficient quantum circuits for the cost function evaluation required by VQAs. To mitigate the barren plateau problem during the optimization, we propose an adaptive multigrid strategy. The approach is illustrated for the nonlinear Burgers equation. We classically optimize quantum circuits to represent the desired ground-state solutions, run them on IBM Q System One and Quantinuum System Model H1, and demonstrate that current quantum computers are capable of accurately reproducing the exact results.

20 pages, 19 figures

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• Numerical solution of nonlinear Schrödinger equation by a hybrid pseudospectral-variational quantum algorithm
• Quantum Dynamics Simulation of the Advection-Diffusion Equation
• Observation of Braid-Protected Unpaired Exceptional Points
• Tensor-Programmable Quantum Circuits for Solving Differential Equations
• Quantum circuits for partial differential equations in Fourier space
• Simulating Quantum Turbulence with Matrix Product States
• A time-marching quantum algorithm for simulation of the nonlinear Lorenz dynamics
• An Efficient Decomposition of the Carleman Linearized Burgers' Equation
• A Quantum-Inspired Algorithm for Wave Simulation Using Tensor Networks
• Resource-efficient quantum algorithm for linear systems of equations
• Performance evaluation of variational quantum eigensolver and quantum dynamics algorithms on the advection-diffusion equation
• Quantum-assisted tracer dispersion in turbulent shear flow
• Error and Resource Estimates of Variational Quantum Algorithms for Solving Differential Equations Based on Runge-Kutta Methods
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