  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
Poincar\'e-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 Poincar\'e-Bloch sphere. These results open a new route towards
the implementation of classical and non-classical computation strategies in
topological soft matter systems.
