Transmutation based Quantum Simulation for Non-unitary Dynamics

We present a quantum algorithm for simulating dissipative diffusion dynamics generated by positive semidefinite operators of the form A=L^dagger L, a structure that arises naturally in standard discretizations of elliptic operators. Our main tool is the Kannai transform, which represents the diffusion semigroup e^-TA as a Gaussian-weighted superposition of unitary wave propagators. This representation leads to a linear-combination-of-unitaries implementation with a Gaussian tail and yields query complexity {mathcal{O}}\(sqrt{\|A\| T log(1/varepsilon\)}), up to standard dependence on state-preparation and output norms, improving the scaling in \|A\|, T and varepsilon compared with generic Hamiltonian-simulation-based methods. We instantiate the method for the heat equation and biharmonic diffusion under non-periodic physical boundary conditions, and we further use it as a subroutine for constant-coefficient linear parabolic surrogates arising in entropy-penalization schemes for viscous Hamilton--Jacobi equations. In the long-time regime, the same framework yields a structured quantum linear solver for Amathbf{x}=mathbf{b} with A=L^dagger L, achieving {mathcal{O}}\(κ^3/2log²(1/varepsilon\)) queries and improving the condition-number dependence over standard quantum linear-system algorithms in this factorized setting.