Quantum Iterative Methods for Solving Differential Equations with Application to Computational Fluid Dynamics

TL;DR

This work develops quantum iterative solvers for differential equations pertinent to computational fluid dynamics by embedding Jacobi and Gauss-Seidel iterations into gate-based quantum circuits using block encodings and the linear combination of unitaries (LCU). The Jacobi method is first realized, with $A=D+R$ and the classical update $x_k = D^{-1}(b - R x_{k-1})$ mapped to a sequence of block-encoded unitaries, enabling the $k^ ext{th}$ iterate to be assembled as an LCU of $k+1$ terms; Gauss-Seidel is extended through a Woodbury-based summation $\Omega= sum_{l=0}^L (-D^{-1}B)^l$. The implementation uses Givens-rotation block encodings and probabilistic statevector subtraction, and is validated on CFD-relevant problems such as the viscous Burgers equation and the 2D linearized Euler acoustics, demonstrating rapid convergence with modest quantum resources. The study analyzes circuit width and depth, highlighting substantial memory advantages (scaling as $\mathcal{O}(\log N)$) and discusses the role of these quantum iterative solvers as preconditioners within multigrid frameworks for industrial CFD, with potential improvements via QSP-based encodings and readout techniques for practical deployment.

Abstract

We propose quantum methods for solving differential equations that are based on a gradual improvement of the solution via an iterative process, and are targeted at applications in fluid dynamics. First, we implement the Jacobi iteration on a quantum register that utilizes a linear combination of unitaries (LCU) approach to store the trajectory information. Second, we extend quantum methods to Gauss-Seidel iterative methods. Additionally, we propose a quantum-suitable resolvent decomposition based on the Woodbury identity. From a technical perspective, we develop and utilize tools for the block encoding of specific matrices as well as their multiplication. We benchmark the approach on paradigmatic fluid dynamics problems. Our results stress that instead of inverting large matrices, one can program quantum computers to perform multigrid-type computations and leverage corresponding advances in scientific computing.

Figures (7)

• Figure 1: The algorithmic pipeline used to apply the quantum iterative methods via a state preparation protocol that is based on block encodings. The input is the differential equation and the output is the $k^\text{th}$ iterate solution. The steps are summarized in the boxes above, and details of each step are described in the text.
• Figure 2: The quantum circuit used to implement the Jacobi iterative method. (a) The multiplication circuit used to obtain the block encoding $U_j$ of each expansion term. The block encoding $U_\alpha$ represents the vector encodings $U_b$ or $U_0$. The block encodings $U_\beta$ and $U_\gamma$ represent the matrix encodings $U_{D^{-1}}$ and $U_R$. After multiplying the block encodings, the top $\log_2(N)$ qubits encodes the $j^\text{th}$ expansion term. (b) The LCU circuit used to solve for the $k^\text{th}$ iterate $\ket{x_k}$ using $a=\lceil\log_2(k+1)\rceil$ ancillary qubits. The unitaries $U_j$ represent the block encodings of the $j^\text{th}\in[1,2^a]$ expansion term. The unitary $V$ encodes the normalization constants associated with each block encoded expansion term. The final result is obtained after taking a projective measurement on the ancillary qubits. (c) The block encoding circuit used to encode matrices $D^{-1}$ and $R$ and vectors $\vb*{x}_0$, $\vb*{b}$ and $\vb*{v}$. The circuit is constructed from a product of $g$ Givens rotations over $n$ qubits. Each Givens rotation $G(\theta,i,j)$ consists of permutation gates $P(j,2^n)$, $P(i,2^n-1)$ and a $y$-axis rotation by angle $\theta$ that is controlled by $n-1$ qubits denoted $R_y(\theta)$.
• Figure 3: A demonstration of how the output from the quantum implementation of the Jacobi method iteratively converges towards the true solution. (a) The problem is defined by the viscous Burgers equation, whose true solution is a shock wave as displayed by the solid curve. The dashed curves represent the $k^\text{th}\in[0,1,4,10]$ iterate solution obtained from the quantum Jacobi solver. (b) The logarithmic decrease in the error of the quantum Jacobi solution over $30$ iterations.
• Figure 4: The solution to the Burgers equation using the quantum circuit implementation of the Jacobi method after $80$ iterations. The scalar field surface $f(x,t)$ represents a travelling sinusoidal wave in a dissipative system.
• Figure 5: The solution to the Euler equations using the quantum circuit implementation of the Jacobi method after $12$ iterations. The pressure field solution $p(x,y)$ represents the transmission of a Gaussian-modulated noise, emitted from a static point source, through quiescent air.