Bosonic Encodings for Hermite-Galerkin Discretizations of High-Dimensional PDEs and Bayesian Inverse Problems

Abstract:The Koopman-von Neumann framework has been proposed to design quantum algorithms for non-linear dynamics. It maps a non-linear ordinary differential equation to a linear partial differential equation (PDE) governing a probability amplitude. Previous works represents this amplitude in the Hermite-function basis, equivalently as a bosonic state, and truncates the total Hermite degree to obtain a representation over $\Theta(m\log N)$ qubits, where $N$ is the number of variables and $m$ the truncation order. We extend this approach to a broader class of linear PDEs whose differential operators have a structured polynomial form. We prove convergence of the truncation for both time-dependent dynamics and gapped ground-state problems under explicit regularity and stability assumptions. We then introduce a qubit encoding that supports efficient block encodings of the truncated operators. Finally, we apply the framework to Bayesian inverse problems with Gaussian priors and observation noise, reducing posterior-state preparation to the preparation of a structured Hamiltonian's ground state.

arXiv:2608.26268 [quant-ph]

(or arXiv:2608.26268v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2608.26268 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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