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

Abstract

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 Schr\"odinger-type dynamics on an enlarged Hilbert space, \[ i\frac{d}{dt}\Psi(t)=P\widehat H P\Psi(t), \] where $\widehat H$ 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. For Stokes, the Zeno-reduced generator has the projected square factorization \[ S_h=-\Pi_h\Delta_h\Pi_h=(G_h\Pi_h)^\dagger(G_h\Pi_h), \] which can be represented through a Gaussian moment dilation and implemented as a Gaussian superposition of unitary Zeno evolutions generated by a first-order square-root Hamiltonian. In the generic sparse-access model, this gives a simulation-stage cost $\widetilde O(h^{-2}\sqrt t)$, up to the usual postselection factor for preparing the normalized dissipative state. The results provide a first step toward understanding the intersection of quantum algorithms, DAEs, constrained PDE dynamics, and square-root Gaussian dilations.