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 $e^{-AT}$, where $T$ is the final simulation time, as a Gaussian-weighted superposition of unitary wave propagators. For target accuracy $\varepsilon$, this representation leads to a linear-combination-of-unitaries implementation with a Gaussian tail and yields query complexity $\widetilde{O}(\sqrt{\|A\|\,T\,\log(1/\varepsilon)})$, up to the standard dependence on state-preparation and output-norm factors, 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 further use it as a subroutine for constant-coefficient linear parabolic surrogates arising in entropy-penalization schemes for the viscous Hamilton--Jacobi equations. In the long-time regime, under a spectral-gap assumption, the same framework gives a structured quantum linear solver by exploiting convergence to the steady state. For normalized positive definite systems $σ(A)\subset[1,κ]$, the solver outputs an $\varepsilon$-approximation to the state proportional to $\mathbf{x}=A^{-1}\mathbf{b}$ with query complexity $\widetilde{O}\left(\frac{\|\mathbf{b}\|}{\|\mathbf{x}\|}\sqrtκ\log^2\frac{\|\mathbf{b}\|}{\varepsilon\|\mathbf{x}\|}\right)$.