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 $\tilde{\mathcal{O}}(\sqrt{\|A\| T \log(1/\varepsilon)})$, 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 $A\mathbf{x}=\mathbf{b}$ with $A=L^\dagger L$, achieving $\tilde{\mathcal{O}}(κ^{3/2}\log^2(1/\varepsilon))$ queries and improving the condition-number dependence over standard quantum linear-system algorithms in this factorized setting.

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