S6 - Computational Fluid Dynamics III

This submission is about matrix inversion with quantum computers, a foundational subroutine in quantum algorithms.
A large variety of real-world problems can be described by linear algebra. Even non-linear systems can be described by linear algebra by a process called linearisation. Whenever a system of equations, or an operator, must be inverted quantum matrix inversion could be a subroutine of choice. This spans application areas from computational fluid dynamics, power grids, structural mechanics, machine learning, finance, and many more. 
While algorithms for quantum matrix inversion are well known, classical preprocessing required can be significant. Here, we focus on the classical preprocessing for the QSVT (quantum singular value transformation) matrix inversion algorithm. Our findings reduce classical preprocessing runtime and/or quantum circuit length.

Many real-world scientific and engineering problems-particularly in computational fluid dynamics (CFD), heat transfer, and nonlinear dynamics-reduce to solving large, sparse systems of linear equations or their time-evolution counterparts. Examples include discretized partial differential equations such as the heat equation and nonlinear systems like the Burgers' equation. These problems are central to applications ranging from climate modeling and aerodynamics to energy systems and materials science.
Quantum algorithms such as quantum linear solvers and Quantum Singular Value Transformation (QSVT) promise asymptotic speedups for solving such systems. However, a major bottleneck preventing their practical use is the lack of efficient, hardware-compatible implementations of block encoding, which is required to represent sparse matrices in quantum circuits. Existing constructions often incur significant overhead due to multi-controlled operations, poor qubit connectivity, and inefficient amplitude manipulation, making them impractical on near-term quantum hardware.
This work addresses a concrete and critical gap: how to translate theoretically efficient quantum linear system algorithms into gate-level circuits that respect real hardware constraints. By focusing on structured sparse matrices arising from discretized differential equations, we target a class of problems with clear real-world relevance and known classical baselines. The goal is not only to demonstrate quantum feasibility but also to identify regimes where quantum advantage could realistically emerge, given current and near-term hardware limitations.

Thermodynamic and fluid dynamic phenomena occur in a wide range of applications in both industry and academia [2, 10, 13, 14]. However, detailed analysis requires resolution on smaller length scales [12], for which Quantum Computers (QCs) offer significantly greater computational resources compared to their classical counterparts [8, 9].

To this end, time-marching computational fluid dynamics methods can be adapted to run on fault-tolerant QCs [1, 11]. This increases the demands on the probabilistic design of quantum algorithms, as high success probabilities are essential. In particular, the modeling of dissipation, due to its irreversible nature, breaks the unitariness of quantum operations and thereby reduces the success probabilities. Consequently, any sequential application of non-unitary operations leads to an exponential decay in the cumulative success probability [3], making a direct implementation on a QC impractical without additional techniques, which in turn introduce a quadratic scaling with simulation time [6, 7].