From Nonlinear to Unitary: Selection-Correct Linear Embeddings for Viscosity Solutions

Hamilton-Jacobi equations and scalar conservation laws develop shocks and caustics in finite time, beyond which physically relevant solutions are selected by viscosity and entropy conditions. Since quantum computation natively implements linear unitary evolution, post-shock quantum algorithms require linear representations that respect these selection principles. This paper presents a self-contained account of two such selection-correct linear lifts. We first examine the quadratic Hamilton-Jacobi and viscous Burgers equations, where the Cole-Hopf transform maps the viscous regularization to the heat equation, yielding the correct vanishing-viscosity limit. We then address general convex Hamiltonians using entropy penalization and soft-min Hopf-Lax formulas to convert dynamic-programming minimizations into linear positive integral operators. We derive the discrete-time linear dynamics, prove soft-min error bounds, and demonstrate that discretization yields positive matrices with controllable operator norms. These linear maps are then explicitly embedded into unitaries via Halmos dilation, providing a direct route to quantum simulation compatible with Schrödingerization-based pipelines.