微分方程的有效哈密顿量量子求解器：替代构造与函数编码

中文摘要 AI 辅助

微分方程可以通过构造一个正半定有效哈密顿量来编码为基态问题，该哈密顿量的最小能量态代表方程的解。我们在多个方向上扩展了这一框架。首先，我们展示了如何利用一个已知的非零参考条件，将多个非零初始条件、边界条件和数据条件纳入齐次形式 $A | f\rangle=0$ 中。然后，我们分析了基于 $A |f\rangle = | b\rangle$ 的替代公式，其哈密顿量为 $H_b=A^\dagger(I-|b\rangle\langle b|)A$，该公式通过增广系统纳入源项和非零约束。对于非线性微分方程，我们考察了张量积表示引入的非物理基态简并性，并考虑哈密顿量约束和拟设层面的限制，以针对物理上有效的乘积态。最后，我们开发了两种哈密顿量构造的网格值振幅编码版本，并将其与谱系数编码进行比较。通过线性和非线性示例，我们评估了哈密顿量公式和函数表示的选择如何影响解的恢复、谱间隙、简并性和读出。这些结果拓宽了基于基态的量子微分方程求解器的适用性，同时阐明了它们的主要实际权衡。

英文摘要

Differential equations can be encoded as ground-state problems by constructing a positive-semidefinite effective Hamiltonian whose minimum-energy state represents the solution. We extend this framework in several directions. First, we show how multiple nonzero initial, boundary, and data conditions can be incorporated into the homogeneous formulation $A | f\rangle=0$ using a known nonzero reference condition. We then analyse an alternative formulation based on $A |f\rangle = | b\rangle$, with Hamiltonian $H_b=A^\dagger(I-|b\rangle\langle b|)A$, which incorporates source terms and nonzero constraints through an augmented system. For nonlinear differential equations, we examine the unphysical ground-state degeneracy introduced by tensor-product representations and consider both Hamiltonian constraints and ansatz-level restrictions for targeting physically valid product states. Finally, we develop grid-value amplitude-encoded versions of both Hamiltonian constructions and compare them with spectral coefficient encoding. Through linear and nonlinear examples, we assess how the choice of Hamiltonian formulation and function representation affects solution recovery, spectral gap, degeneracy, and readout. These results broaden the applicability of ground-state-based quantum differential-equation solvers while clarifying their principal practical trade-offs.

发表机构

• Fujitsu Research of Europe Ltd.（富士通欧洲研究院有限公司）

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