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Liquid crystals (LCs) can host robust topological defect structures that essentially determine their optical and elastic properties. Although recent experimental progress enables precise control over localization and dynamics of nematic LC defects, their practical potential for information storage and processing has yet to be explored. Here, we introduce the concept of nematic bits (nbits) by exploiting a quaternionic mapping from LC defects to the Poincaré-Bloch sphere. Through theory and simulations, we demonstrate how single-nbit operations can be implemented using electric fields, in close analogy with Pauli, Hadamard and other common quantum gates. Ensembles of two-nbit states can exhibit strong statistical correlations arising from nematoelastic interactions, which can be used as a computational resource. Utilizing nematoelastic interactions, we show how suitably arranged 4-nbit configurations can realize universal classical NOR and NAND gates. Finally, we demonstrate the implementation of generalized logical functions that take values on the Poincaré-Bloch sphere. These results open a new route towards the implementation of classical and non-classical computation strategies in topological soft matter systems.