  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.
