Optimal 1D Ly-$\alpha$ Forest Power Spectrum Estimation I: DESI-Lite
  Spectra
  The 1D Ly-$\alpha$ forest flux power spectrum $P_{\mathrm{1D}}$ is sensitive
to scales smaller than a typical galaxy survey, and hence ties to the
intergalactic medium's thermal state, suppression from neutrino masses and new
dark matter models. It has emerged as a competitive framework to study new
physics, but also has come with various challenges and systematic errors in
analysis. In this work, we revisit the optimal quadratic estimator for
$P_{\mathrm{1D}}$, which is robust against the relevant problems such as pixel
masking, time evolution within spectrum and quasar continuum errors. We further
improve the estimator by introducing a fiducial power spectrum, which enables
us to extract more information by alleviating the discreteness of band powers.
We meticulously apply our method to synthetic DESI spectra and demonstrate how
the estimator overcomes each challenge. We further apply an optimisation scheme
that approximates the Fisher matrix to three elements per row and reduces
computation time by 60%. We show that we can achieve percent precision in
$P_{\mathrm{1D}}$ with 5-year DESI data in the absence of systematics and
provide forecasts for different spectral qualities.
