Quantum Simulation of Differential-Algebraic Equations with Applications to Unsteady Stokes Flow

Differential-algebraic equations (DAEs) arise naturally in constrained dynamical systems, but their algebraic constraints and hidden compatibility conditions make them more subtle than standard ordinary differential equations. This paper initiates a quantum-algorithmic study of constrained linear DAEs. We introduce a dilation framework that embeds the generally non-Hermitian constrained evolution into a projected Schrödinger-type dynamics on an enlarged Hilbert space, \[ i\frac{d}{dt}Ψ(t)=P\tH PΨ(t), \] where tH is Hermitian and P is the orthogonal projector onto the lifted constraint subspace. This identifies the DAE evolution with a quantum Zeno-type dynamics and enables the use of block encodings, QSVT-based projector construction, and Hamiltonian simulation. We apply the framework to structure-preserving discretizations of the unsteady Stokes equations, where the pressure enforces the discrete incompressibility constraint. We derive the corresponding projected Hamiltonian formulation, identify low-energy spectral cutoffs motivated by solution smoothness, and discuss the resulting quantum simulation cost in comparison with classical projection-type methods. The results provide a first step toward understanding the potential intersection of quantum algorithms, DAEs, and constrained PDE dynamics.