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The 1D Ly-$\alpha$ forest flux power spectrum $P_{\mathrm{1D}}$ is sensitiveto scales smaller than a typical galaxy survey, and hence ties to theintergalactic medium's thermal state, suppression from neutrino masses and newdark matter models. It has emerged as a competitive framework to study newphysics, but also has come with various challenges and systematic errors inanalysis. In this work, we revisit the optimal quadratic estimator for$P_{\mathrm{1D}}$, which is robust against the relevant problems such as pixelmasking, time evolution within spectrum and quasar continuum errors. We furtherimprove the estimator by introducing a fiducial power spectrum, which enablesus to extract more information by alleviating the discreteness of band powers.We meticulously apply our method to synthetic DESI spectra and demonstrate howthe estimator overcomes each challenge. We further apply an optimisation schemethat approximates the Fisher matrix to three elements per row and reducescomputation 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 andprovide forecasts for different spectral qualities.

Optimal 1D Ly-$\alpha$ Forest Power Spectrum Estimation I: DESI-Lite Spectra