  We study the propagation of a specific class of instrumental systematics to
the reconstruction of the B-mode power spectrum of the cosmic microwave
background (CMB). We focus on the non-idealities of the half-wave plate (HWP),
a polarization modulator that is to be deployed by future CMB experiments, such
as the phase-A satellite mission LiteBIRD. We study the effects of non-ideal
HWP properties, such as transmittance, phase shift, and cross-polarization. To
this end, we developed a simple, yet stand-alone end-to-end simulation pipeline
adapted to LiteBIRD. We analyzed the effects of a possible mismatch between the
measured frequency profiles of HWP properties (used in the mapmaking stage of
the pipeline) and the actual profiles (used in the sky-scanning step). We
simulated single-frequency, CMB-only observations to emphasize the effects of
non-idealities on the BB power spectrum. We also considered multi-frequency
observations to account for the frequency dependence of HWP properties and the
contribution of foreground emission. We quantified the systematic effects in
terms of a bias $\Delta r$ on the tensor-to-scalar ratio, $r$, with respect to
the ideal case without systematic effects. We derived the accuracy requirements
on the measurements of HWP properties by requiring $\Delta r < 10^{-5}$ (1% of
the expected LiteBIRD sensitivity on $r$). Our analysis is introduced by a
detailed presentation of the mathematical formalism employed in this work,
including the use of the Jones and Mueller matrix representations.
