Quantum simulation of Liouville equation in geometrical optics with partial transmission and reflection via Schrödingerization

This paper investigates quantum simulation algorithms for the Liouville equation in geometrical optics with partial transmission and reflection at sharp interfaces, based on the Schrödingerization method. By means of a warped phase transformation in one higher dimension, the Schrödingerization method converts linear partial differential equations into a system of Schrödinger-type equations with unitary evolution, thereby rendering them suitable for quantum simulation. In this work, the Schrödingerization method is combined with a Hamiltonian-preserving scheme that incorporates partial transmission and reflection into the numerical flux. A main difficulty is that the interface treatment in the classical scheme relies on threshold-dependent "if/else" procedures, making it highly nontrivial to reformulate the method in a matrix form suitable for quantum simulation. To overcome this difficulty, we encode the interface conditions into a partial transmission and reflection matrix prepared a priori, rather than during the time evolution. We present detailed constructions of the resulting quantum algorithms and show through complexity analysis that the proposed methods achieve polynomial quantum advantage in the precision parameter ε over their classical counterparts.