Quasi-Clifford to qubit mappings

TL;DR

This paper introduces a mapping from quasi-Clifford algebras to Pauli algebras, enabling block-diagonalization and symmetry reduction in quantum information processing, with applications to semidefinite programs and anti-commuting subsets.

Contribution

It presents a novel mapping from quasi-Clifford algebras to Pauli algebras, facilitating algebraic decompositions and applications in quantum computation.

Findings

Provides a Wedderburn decomposition of matrix groups with quasi-Clifford structure.

Enables block-diagonalization of Pauli groups and recovers Jordan-Wigner transform for Majorana operators.

Discusses applications in symmetry reduction of semidefinite programs and constructing maximal anti-commuting sets.

Abstract

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.