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arXiv:2601.03616

Transmutation based Quantum Simulation for Non-unitary Dynamics

Shi Jin, Chuwen Ma, Enrique Zuazua

quant-phSubmitted Jan 7, 2026• Announced Jan 8, 2026• 0 citations• Revised Jun 27, 2026 (v2)
v1 · Jan 7, 2026v2 · Jun 27, 2026v3 · Sep 21, 2026
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Abstract

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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We present a quantum algorithm for simulating dissipative diffusion dynamics generated by positive semidefinite operators of the form A=L†LA=L^\dagger LA=L†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−TAe^{-TA}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 O~(∥A∥Tlog⁡(1/ε))\tilde{\mathcal{O}}(\sqrt{\|A\| T \log(1/\varepsilon)})O~(∥A∥Tlog(1/ε)​), up to standard dependence on state-preparation and output norms, improving the scaling in ∥A∥,T\|A\|, T∥A∥,T and ε\varepsilonε 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 Ax=bA\mathbf{x}=\mathbf{b}Ax=b with A=L†LA=L^\dagger LA=L†L, achieving O~(κ3/2log⁡2(1/ε))\tilde{\mathcal{O}}(κ^{3/2}\log^2(1/\varepsilon))O~(κ3/2log2(1/ε)) queries and improving the condition-number dependence over standard quantum linear-system algorithms in this factorized setting.
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