  We rigorously investigate the quantum non-Markovian dissipative dynamics of a
system coupled to a harmonic oscillator bath by deriving hierarchical
Schrodinger equations of motion (HSEOM) and studying their dynamics. The HSEOM
are the equations for wave functions derived on the basis of the Feynman-Vernon
influence functional formalism for the density operator, $\langle q|\rho(t)|q'
\rangle$, where $\langle q|$ and $|q' \rangle$ are the left- and right-hand
elements. The time evolution of $\langle q|$ is computed from time $0$ to $t$,
and subsequently, the time evolution of $|q' \rangle$ is computed from time $t$
to $0$ along a contour in the complex time plane. By appropriately choosing
functions for the bath correlation function and the spectral density, we can
take advantage of an HSEOM method to carry out simulations without the need for
a great amount of computational memory. As a demonstration, quantum annealing
simulation for a ferromagnetic $p$-spin model is studied.
