N. G. Karaçaylı, P. Martini, J. Aguilar, S. Ahlen, E. Armengaud, S. Bailey, A. Bault, D. Bianchi, A. Brodzeller, D. Brooks, J. Chaves-Montero, T. Claybaugh, A. Cuceu, A. de la Macorra, A. Dey, B. Dey, P. Doel, S. Ferraro, A. Font-Ribera, J. E. Forero-Romero, E. Gaztañaga, S. Gontcho A Gontcho, G. Gutierrez, J. Guy, C. Hahn, H. K. Herrera-Alcantar, K. Honscheid, M. Ishak, R. Kehoe, D. Kirkby, A. Kremin, M. Landriau, J. M. Le Goff, L. Le Guillou, M. E. Levi, M. Manera, A. Meisner, R. Miquel, P. Montero-Camacho, S. Nadathur, G. Niz, N. Palanque-Delabrouille, Z. Pan, W. J. Percival, Matthew M. Pieri, F. Prada, I. Pérez-Ràfols, C. Ravoux, G. Rossi, E. Sanchez, C. Saulder, D. Schlegel, M. Schubnell, H. Seo, M. Siudek, D. Sprayberry, T. Tan, Ji-Jia Tang, G. Tarlé, M. Walther, B. A. Weaver, J. Yu, R. Zhou, H. Zou

The one-dimensional power spectrum $P_{\mathrm{1D}}$ of Ly$\alpha$ forest offers rich insights into cosmological and astrophysical parameters, including constraints on the sum of neutrino masses, warm dark matter models, and the thermal state of the intergalactic medium. We present the measurement of $P_{\mathrm{1D}}$ using the optimal quadratic maximum likelihood estimator applied to over 300,000 Ly$\alpha$ quasars from Data Release 1 (DR1) of the Dark Energy Spectroscopic Instrument (DESI) survey. This sample represents the largest to date for $P_{\mathrm{1D}}$ measurements and is larger than the Extended Baryon Oscillation Spectroscopic Survey (eBOSS) by a factor of 1.7. We conduct a meticulous investigation of instrumental and analysis systematics and quantify their impact on $P_{\mathrm{1D}}$. This includes the development of a cross-exposure estimator that eliminates the need to model the pipeline noise and has strong potential for future $P_{\mathrm{1D}}$ measurements. We also present new insights into metal contamination through the 1D correlation function. Using a fitting function we measure the evolution of the Ly$\alpha$ forest bias with high precision: $b_F(z) = (-0.218\pm0.002)\times((1 + z) / 4)^{2.96\pm0.06}$. In a companion validation paper, we substantially extend our previous suite of CCD image simulations to quantify the pipeline's exquisite performance accurately. In another companion paper, we present DR1 $P_{\mathrm{1D}}$ measurements using the Fast Fourier Transform (FFT) approach to power spectrum estimation. These two measurements are consistent with each other and constitute the most precise $P_{\mathrm{1D}}$ measurement to date, while being in good agreement with results from the DESI early data release.