A QSVT-Based Quantum Jacobi Algorithm for Linear Systems with Application to the Poisson Equation

Computational fluid dynamics (CFD) relies heavily on solving large, sparse systems of linear equations derived from discretized partial differential equations. While classical solvers frequently use iterative methods like the Jacobi method, existing quantum linear system solvers (QLSS) often focus on direct matrix inversion. This paper addresses the need for quantum iterative methods that align more closely with established CFD solution strategies.

The authors develop a quantum Jacobi algorithm using the Quantum Singular Value Transformation (QSVT) framework. By reformulating the classical Jacobi iteration as a polynomial transformation of a block-encoded operator, the authors replace the costly multiplication of block encodings used in previous quantum implementations with efficient QSVT subroutines. This construction allows the algorithm to maintain a constant number of ancilla qubits relative to the number of iterations, while the circuit depth scales only linearly with the iteration count.

The proposed algorithm is shown to be more resource-efficient than previous gate-based quantum Jacobi implementations, which required ancilla registers that grew linearly with the number of iterations. The authors demonstrate the algorithm's effectiveness by solving one- and two-dimensional Poisson problems, including the pressure Poisson equation required for lid-driven cavity flow simulations. Numerical simulations confirm that the quantum circuit reproduces classical Jacobi results with high precision, and the approximation introduced to satisfy the hermiticity requirement has a negligible impact on the final pressure field.

This work bridges the gap between quantum computing and practical CFD workflows. By providing a building block that mirrors classical iterative solvers, the algorithm offers a path toward integrating quantum acceleration into larger simulation frameworks, such as multigrid methods or preconditioning techniques. It demonstrates that polynomial-based quantum implementations are well-suited for early fault-tolerant quantum computers, provided that efficient block encodings for physical operators are available.