Quantum simulation of Burgers turbulence: Nonlinear transformation and direct evaluation of statistical quantities

• URL: http://arxiv.org/abs/2412.17206v1
• Date: Mon, 23 Dec 2024 01:17:26 GMT
• Title: Quantum simulation of Burgers turbulence: Nonlinear transformation and direct evaluation of statistical quantities
• Authors: Fumio Uchida, Koichi Miyamoto, Soichiro Yamazaki, Kotaro Fujisawa, Naoki Yoshida,
• Abstract summary: It is still challenging to solve nonlinear equations in fluid dynamics, such as the Burgers equation, using quantum computers. We propose a novel quantum algorithm to solve the Burgers equation.
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• Abstract: Fault-tolerant quantum computing is a promising technology to solve linear partial differential equations that are classically demanding to integrate. It is still challenging to solve non-linear equations in fluid dynamics, such as the Burgers equation, using quantum computers. We propose a novel quantum algorithm to solve the Burgers equation. With the Cole-Hopf transformation that maps the fluid velocity field $u$ to a new field $\psi$, we apply a sequence of quantum gates to solve the resulting linear equation and obtain the quantum state $\vert\psi\rangle$ that encodes the solution $\psi$. We also propose an efficient way to extract stochastic properties of $u$, namely the multi-point functions of $u$, from the quantum state of $\vert\psi\rangle$. Our algorithm offers an exponential advantage over the classical finite difference method in terms of the number of spatial grids when a perturbativity condition in the information-extracting step is met.

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