From Linear Differential Equations to Unitaries: A Moment-Matching Dilation Framework with Near-Optimal Quantum Algorithms

Quantum speed-ups for dynamical simulation usually demand unitary time-evolution, whereas the large ODE/PDE systems encountered in realistic physical models are generically non-unitary. We present a universal moment-fulfilling dilation that embeds any linear, non-Hermitian flow $\dot{x}=Lx$ with $L=-iH+K$ into a strictly unitary evolution on an enlarged Hilbert space: $$((l\mathrm{\mid }\otimes I)\mathcal{T}{e}^{-i\int (I_A\otimes H+iF\otimes K)dt}(\mathrm{\mid }r)\otimes I)=\mathcal{T}{e}^{\int Ldt},$$ provided the triple $(F,(l\mathrm{\mid },\mathrm{\mid }r))$ satisfies the compact moment identities $(l\mathrm{\mid }{F}^k\mathrm{\mid }r)=1$ for all $k\ge 0$ in the ancilla space. This algebraic criterion recovers both \emph{Schrödingerization} [Phys. Rev. Lett. 133, 230602 (2024)] and the linear-combination-of-Hamiltonians (LCHS) scheme [Phys. Rev. Lett. 131, 150603 (2023)], while also unveiling whole families of new dilations built from differential, integral, pseudo-differential, and difference generators. Each family comes with a continuous tuning parameter \emph{and} is closed under similarity transformations that leave the moments invariant, giving rise to an overwhelming landscape of design space that allows quantum dilations to be co-optimized for specific applications, algorithms, and hardware. As concrete demonstrations, we prove that a simple finite-difference dilation in a finite interval attains near-optimal oracle complexity. Numerical experiments on Maxwell viscoelastic wave propagation confirm the accuracy and robustness of the approach.