Quasi-Clifford to qubit mappings

2025/08/02 by Felix Huber, Huber, Felix
Computer Science · Mathematics · #Advanced Operator Algebra Research #Algebraic and Geometric Analysis #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Computing Algorithms and Architecture #Quantum Physics (quant-ph)

paper · pdf · doi:10.48550/arxiv.2508.01470

openalex publication_date 2025/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 α₁, …, α_n is is given by the relations α_i² = k_i and α_j α_i = (-1)^χ_ij α_i α_j, where k_i ∈ ℂ and χ_ij ∈ \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.

Citations

• Frustration graph formalism for qudit observables
• Strongly interacting fermions are non-trivial yet non-glassy
• Limitations and Separations in the Quantum Sum-of-squares, and the Quantum Knapsack Problem
• Minimal qubit representations of Hamiltonians via conserved charges
• Optimizing Strongly Interacting Fermionic Hamiltonians
• Gastineau-Hills' quasi-Clifford algebras and plug-in constructions for Hadamard matrices

Related

• A complete theory of the Clifford commutant
• Basic Calculations on Clifford Algebras
• Clifford Transformations for Fermionic Quantum Systems: From Paulis to Majoranas to Fermions
• The State-Operator Clifford Compatibility: A Real Algebraic Framework for Quantum Information
• Commuting Clifford actions