A quantum advection-diffusion solver using the quantum singular value transform

TL;DR

This work develops a quantum algorithm for the linear advection-diffusion equation by encoding high-order finite-difference operators as block-encodings and processing them with the quantum singular value transform (QSVT). It provides a detailed end-to-end framework, including construction of block encodings for finite-difference operators, polynomial approximations via Chebyshev expansions, and a rigorous end-to-end complexity analysis, together with numerical simulations in 1D and 2D that demonstrate the efficiency gains of higher-order methods. The authors also discuss the extension to higher dimensions, the associated post-selection costs, and potential pathways to non-linear models and real-world applications such as weather prediction. The accompanying code repository enables reproducibility on simulators and hardware, laying the groundwork for scalable quantum fluid dynamics simulations. Overall, the paper offers a concrete, implementable quantum approach that can outperform low-order methods under realistic resource budgets and sets the stage for future hardware demonstrations and methodological extensions.

Abstract

We present a quantum algorithm for the simulation of the linear advection-diffusion equation based on block encodings of high order finite-difference operators and the quantum singular value transform. Our complexity analysis shows that the higher order methods significantly reduce the number of gates and qubits required to reach a given accuracy. The theoretical results are supported by numerical simulations of one- and two-dimensional benchmarks.

Figures (8)

• Figure 1: The quantum circuit $U_\Phi$ for an angle sequence $\Phi = (\phi_1,\cdots,\phi_d)$ with $d$ even.
• Figure 2: The quantum circuit $U_{(\Phi^{(1)},\Phi^{(2)})}$ for a pair of angle sequences $\Phi^{(1)} = (\phi_1^{(1)},\ldots,\phi^{(1)}_d)$ and $(\Phi^{(2)} = (\phi^{(2)}_1,\ldots,\phi^{(2)}_{d+1})$ with $d$ odd.
• Figure 3: Comparison of methods of order $2$ and $6$ and exact solution (bottom panel) for the QSVT-based solution of the 1D advection equation with speed $c=1$ and Gaussian initial conditions $u_0(x) = \exp(-10(x-5/3)^2)$ (top panel). The number of spatial qubits used is denoted by spq.
• Figure 4: Comparison of QSVT-based methods of order $2$, $4$ and $6$ and exact solution (bottom panel) for the pure diffusion equation with $\nu = 0.2$ and initial condition a sum of sine waves (top panel, Eq. \ref{['eq:sum_sine']}). The number of spatial qubits used is denoted by spq.
• Figure 5: Comparison of QSVT-based methods of order $6$ and $14$ and exact solution (bottom panel) for the advection-diffusion equation with $c = 1$, $\nu = 10^{-3}$ and initial data a wave packet (top panel, Equation \ref{['eq:wavepack']})

Theorems & Definitions (60)

• Theorem 2.1
• Remark
• Remark
• Theorem 3.1
• Remark
• Corollary 3.1.1
• Remark
• Lemma 3.2
• proof
• Definition 4.1