AI 中文总结
Koopman-von Neumann框架结合玻色子编码，扩展了高维PDE的埃尔米特-伽辽金离散化方法，可应用于带高斯先验与观测噪声的贝叶斯逆问题，实现后验态制备简化。
AI 中文摘要
Koopman-von Neumann框架已被提出用于设计非线性动力学的量子算法，它将非线性常微分方程映射为控制概率幅的线性偏微分方程（PDE）。此前的研究将该概率幅表示为埃尔米特函数基，等价于一个玻色子态，并截断总埃尔米特阶数，得到在$\boldsymbol{\theta(m\boldsymbol{\text{log}}N)}$个量子比特上的表示，其中$N$为变量数，$m$为截断阶数。我们将该方法扩展到更广泛的一类线性PDE，其微分算子具有结构化多项式形式。我们在明确的正则性与稳定性假设下，证明了该截断对时间依赖动力学和带隙基态问题的收敛性。随后，我们引入一种量子比特编码，支持对截断算子的高效块编码。最后，我们将该框架应用于具有高斯先验和观测噪声的贝叶斯逆问题，将后验态制备简化为结构化哈密顿量的基态制备。
英文摘要
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 $Θ(m\log N)$ 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.
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