This paper explores the explicit design of quantum circuits for quantum simulation of partial differential equations (PDEs) with physical boundary conditions. These equations and/or their discretized forms usually do not evolve via unitary dynamics, thus are not suitable for quantum simulation. Boundary conditions (either time-dependent or independent) make the problem more difficult. To tackle this challenge, the Schrodingerisation method can be employed, which converts linear partial and ordinary differential equations with non-unitary dynamics into systems of Schrodinger-type equations, via the so-called warped phase transformation that maps the equation into one higher dimension. Despite advancements in Schrodingerisation techniques, the explicit implementation of quantum circuits for solving general PDEs, especially with physical boundary conditions, remains underdeveloped. We present two methods for handling the inhomogeneous terms arising from time-dependent physical boundary conditions. One approach utilizes Duhamel's principle to express the solution in integral form and employs linear combination of unitaries (LCU) for coherent state preparation. Another method applies an augmentation to transform the inhomogeneous problem into a homogeneous one. We then apply the quantum simulation technique from [CJL23] to transform the resulting non-autonomous system to an autonomous system in one higher dimension. We provide detailed implementations of these two methods and conduct a comprehensive complexity analysis in terms of queries to the time evolution input oracle.

翻译：本文探讨了针对具有物理边界条件的偏微分方程量子模拟的量子电路显式设计。这些方程和/或其离散形式通常不遵循幺正动力学演化，因此不适合直接进行量子模拟。边界条件（无论是时变还是时不变）使问题更具挑战性。为应对这一难题，可采用薛定谔化方法，该方法通过所谓的扭曲相变换将方程映射到高一维空间，从而将具有非幺正动力学的线性和常微分方程转换为薛定谔型方程组。尽管薛定谔化技术已取得进展，但用于求解一般偏微分方程（尤其是具有物理边界条件）的量子电路显式实现仍不完善。我们提出了两种处理时变物理边界条件所产生的非齐次项的方法。一种方法利用杜阿梅尔原理将解表示为积分形式，并采用幺正线性组合技术实现相干态制备。另一种方法通过增广变换将非齐次问题转化为齐次问题。随后，我们应用[CJL23]中的量子模拟技术，将所得非自治系统转化为高一维空间中的自治系统。我们详细阐述了这两种方法的实现过程，并针对时间演化输入预言机的查询次数进行了全面的复杂度分析。