Quantum dynamics, typically expressed in the form of a time-dependent Schr\"odinger equation with a Hermitian Hamiltonian, is a natural application for quantum computing. However, when simulating quantum dynamics that involves the emission of electrons, it is necessary to use artificial boundary conditions (ABC) to confine the computation within a fixed domain. The introduction of ABCs alters the Hamiltonian structure of the dynamics, and existing quantum algorithms can not be directly applied since the evolution is no longer unitary. The current paper utilizes a recently introduced Schr\"odingerisation method (Jin et al. arXiv:2212.13969 and arXiv:2212.14703) that converts non-Hermitian dynamics to a Schr\"odinger form, for the artificial boundary problems. We implement this method for three types of ABCs, including the complex absorbing potential technique, perfectly matched layer methods, and Dirichlet-to-Neumann approach. We analyze the query complexity of these algorithms, and perform numerical experiments to demonstrate the validity of this approach. This helps to bridge the gap between available quantum algorithms and computational models for quantum dynamics in unbounded domains.

翻译：量子动力学通常以含时薛定谔方程的形式描述，其哈密顿量为厄米型，是量子计算的自然应用场景。然而，在模拟涉及电子发射的量子动力学时，必须使用人工边界条件（ABC）将计算限制在固定域内。引入ABC会改变动力学的哈密顿结构，由于演化不再具有幺正性，现有量子算法无法直接应用。本文采用近期提出的薛定谔化方法（Jin等人，arXiv:2212.13969和arXiv:2212.14703），将非厄米动力学转换为薛定谔形式，以解决人工边界问题。我们将该方法应用于三类ABC：复吸收势技术、完美匹配层方法和Dirichlet-to-Neumann方法。我们分析了这些算法的查询复杂度，并通过数值实验验证了该方法的有效性。这有助于弥合现有量子算法与无界域中量子动力学计算模型之间的差距。