Quasi-Clifford to qubit mappings

Algebras with given (anti-)commutativity structure are widespread in quantum mechanics. This structure is captured by quasi-Clifford algebras (QCA): a QCA generated by ${\alpha }_1,\dots ,{\alpha }_n$ is is given by the relations ${\alpha }_i^2=k_i$ and ${\alpha }_j{\alpha }_i=(-1{)}^{{\chi }_{ij}}{\alpha }_i{\alpha }_j$, where $k_i\in \mathbb{C}$ and ${\chi }_{ij}\in \{0,1\}$. We present a mapping from QCA to Pauli algebras and discuss its use in quantum information and computation. The mapping also provides a Wedderburn decomposition of matrix groups with quasi-Clifford structure. This provides a block-diagonalization for e.g. Pauli groups, while for Majorana operators the Jordan-Wigner transform is recovered. Applications to the symmetry reduction of semidefinite programs and for constructing maximal anti-commuting subsets are discussed.