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

Fumio Uchida, Koichi Miyamoto, Soichiro Yamazaki, Kotaro Fujisawa, Naoki Yoshida

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 $ψ$, we apply a sequence of quantum gates to solve the resulting linear equation and obtain the quantum state $\vertψ\rangle$ that encodes the solution $ψ$. We also propose an efficient way to extract stochastic properties of $u$, namely the multi-point functions of $u$, from the quantum state of $\vertψ\rangle$. 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.

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

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 to a new field , we apply a sequence of quantum gates to solve the resulting linear equation and obtain the quantum state that encodes the solution . We also propose an efficient way to extract stochastic properties of , namely the multi-point functions of , from the quantum state of . 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.

Paper Structure

This paper contains 21 sections, 70 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Burgers flow
  3. Burgers equation and Cole--Hopf transformation
  4. Statistical quantities of Burgers turbulence
  5. Quantum algorithm
  6. Implementing the initial condition
  7. Integration of the heat equation
  8. Retrieving the desired information
  9. Two-point function
  10. Three-point function
  11. Higher-order multi-point function
  12. Taking ensemble averages
  13. Discussion
  14. Total complexity
  15. Validity of the assumptions and generalization
  16. ...and 6 more sections

Figures (4)

  • Figure 1: An example of the evolution of $\psi$ with a random initial condition given in Eq. \ref{['eq:randomini']} (Red-solid line at $t=0$, green-solid at $t=0.01$, and yellow-solid at $t=0.02$). We also show the spatial average $\bar{\psi}$, which is constant in time,
  • Figure 2: Comparison between the values of flatness $\beta$ obtained by the zero-th order approximation, Eq. \ref{['eq:dpsi_to_u']} with $\tilde{\psi}_j=\bar{\psi}$
  • Figure 3: The evolution of the Reynolds number (red-solid) and of its square (blue-solid), and the relative errors of the flatness, $\varepsilon_0:=\vert\beta/\beta_0-1\vert$ (green-dashed) and $\varepsilon_1:=\vert\beta/\beta_1-1\vert$ (yellow dash-dot). Our numerical code used to make this plot is public at mycode.
  • Figure 4: Shock-like structure (Top) and rarefaction wave (Bottom). In each panel, solid lines correspond to solutions at $t=0$ (red), $t=1$ (green), and $t=2$ (yellow). Our numerical code used to make this plot is public at mycode.