Algorithmic and architectural options and trade-offs between performance and complexity/power dissipation are necessary to consider in the transfer of coherent technology from long-haul transmission to short-reach application. An adaptive equalization algorithm cascading polarization demultiplexing and frequency domain equalization is proposed to target passive optical network applications that feature extremely low computational complexity. Three coordinate rotation digital computer-based vector rotators are introduced to realize polarization demultiplexing without performing multiplications. A simplified gradient descent algorithm is proposed to search and trace the state of polarization. A nonbutterfly adaptive equalizer in the frequency domain for each polarization is designed to handle link chromatic dispersion and the imperfection of the transceiver frequency response. The performance of the proposed equalization scheme is experimentally verified by a 32-GBaud dual-polarization Nyquist quadrature phase-shift keying system over 20-km standard single-mode fiber transmission. With the proposed method, the number of hardened multipliers for field-programmable gate array realization is reduced by ∼85  %   compared with a conventional 2  ×  2 multiple-input multiple-output finite impulse response filter with the same receiver sensitivity.