# Spectral quantum algorithm for passive scalar transport in shear flows

## TL;DR

This paper introduces a quantum algorithm to simulate how substances mix in fluid flows, using quantum computing techniques to solve complex fluid dynamics equations.

## Contribution

The novel contribution is a spectral quantum algorithm for simulating scalar transport in shear flows using quantum computational fluid dynamics.

## Key findings

- Quantum circuits were constructed to simulate arbitrary polynomial velocity profiles in multiple dimensions.
- The algorithm was tested with different flow types and boundary conditions using statevector simulations.
- The number of two-qubit gates required scales with the logarithm of grid points and the polynomial order of the velocity profile.

## Abstract

The mixing of scalar substances in fluid flows by stirring and diffusion is ubiquitous in natural flows, chemical engineering, and microfluidic drug delivery. Here, we present a spectral quantum algorithm for scalar mixing by solving the advection–diffusion equation in a quantum computational fluid dynamics framework. The exact gate decompositions of the advection and diffusion operators in spectral space are derived. For all but the simplest one-dimensional flows, these operators do not commute. Therefore, we use operator splitting to construct quantum circuits capable of simulating arbitrary polynomial velocity profiles in multiple dimensions, such as the Blasius profile of a laminar boundary layer. Periodic, Neumann, and Dirichlet boundary conditions can be imposed with the appropriate quantum spectral transform. We evaluate the approach in statevector simulations of a Couette flow, plane Poiseuille flow, and a polynomial Blasius profile approximation. For an advection–diffusion problem in one dimension, we compare the time evolution of an ideal quantum simulation with those of real quantum computers with superconducting and trapped-ion qubits. The required number of two-qubit gates grows with the logarithm of the number of grid points raised to one higher power than the order of the polynomial velocity profile.