Spectral quantum algorithm for passive scalar transport in shear flows

TL;DR

The paper tackles non-unitary advection–diffusion of a passive scalar in fluids using a spectral quantum algorithm. It leverages the quantum Fourier transform to diagonalize advection and diffusion operators, and employs operator splitting (Lie–Trotter and Strang) to manage non-commuting terms in 2D laminar shear flows, while implementing boundary conditions via quantum cosine/sine transforms. Explicit quantum circuits are derived for advection, diffusion, and Neumann/Dirichlet boundaries, with statevector simulations validating accuracy and near-term hardware experiments on IBM and IonQ platforms demonstrating practical feasibility. The work suggests a viable path for quantum-accelerated CFD on regular meshes, with gate counts that scale polylogarithmically in grid size and clear guidance for future hardware improvements.

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.

Figures (8)

• Figure 1: (a) Quantum circuit to implement the advection evolution operator on $n$ qubits for a constant velocity $u$ in a periodic domain by diagonalization under the QFT with $\alpha = 2\pi u t/L$. (b) Illustration of the action of the phase gates by the circuit with a color coding that corresponds to the gates in panel (a).
• Figure 2: Quantum circuits using $n=3$ qubits per spatial dimension for simulating the advection equation (a) in a Couette flow, (b) a Poiseuille (channel) flow, and (c) a boundary layer described by a third-order approximation to the Blasius solution. The top three qubits correspond to the $x$ direction and the bottom three qubits correspond to the $y$ direction. The phase coefficients for the Couette, Poiseuille and Blasius profiles are calculated from Eqs. \ref{['eq:couette']}, \ref{['eq:poiseuille']}, and \ref{['eq:blasius']}, respectively. Here, $\alpha=2\pi U t / L$ as the variable velocity profile $u(y)$ is encoded in the prefactors of $\alpha$ in each gate.
• Figure 3: Quantum circuit for solving the one-dimensional heat equation with spectral accuracy on $N=16$ grid points with $O(\log^2N)$ gates. Periodic boundary conditions are used. The circuit implements the products in Eqs. \ref{['eq:exp_jsquared']} and \ref{['eq:exp_jp1_squared']}. The unitary $U$ is defined in Eq. \ref{['eq:U']}, and the ancilla is measured to be $\lvert0\rangle$ after each application. The outer CNOT gates flip the second half of the spectrum to exploit the mirroring of the squared wavenumbers, as discussed in the text.
• Figure 4: Quantum circuit for simulating a Trotter step of the advection-diffusion equation in a two-dimensional laminar shear flow using $n=3$ qubits per spatial dimension, where the top three qubits correspond to the $x$ direction and the next three qubits correspond to the $y$ direction. Ancilla qubit measurements of $\lvert0\rangle$ are required for successful execution. The boundaries are periodic in $x$ and homogeneous Neumann in $y$, leading to $\beta_1=Dt(2\pi/L)^2$ and $\beta_2=Dt(\pi/L)^2$. Example advection block implementations are shown in Fig. \ref{['fig:advection_shear_circuits']}. While the QCT is a unitary transformation, existing efficient circuits require one additional ancilla.
• Figure 5: Statevector simulations of the one-dimensional advection-diffusion equation, showing (a) the evolution of the quantum amplitudes (and thus the solution) for $N=128$ grid points, and (b) the error norm versus grid size $N$. The error norm is obtained with respect to the normalized solution of Eq. \ref{['eq:analytical_solution']}. The time evolution is done by a single time step.