Generalizations of Various Results in Quantum Computation Theory to Mixed-Dimensional Quantum Systems

Abstract

The majority of the research in quantum computation theory is based on quantum mechanical systems consisting of quantum subsystems that are all of the same dimension. Following and expanding on the research conducted by Randall R. Holmes and Frederic Texier in their paper “A Generalization of the Deutsch-Jozsa Quantum Algorithm”, we generalize several results in quantum computation theory to quantum systems consisting of quantum subsystems of varying dimensions, which amounts to a generalization to arbitrary finite abelian groups. These results include the Bernstein-Vazirani algorithm, Simon’s algorithm, the Pauli group and algebra, the Clifford group, and stabilizer codes. We also further generalize the Deutsch-Jozsa algorithm to arbitrary finite groups. Additionally, we expand on the topic of orthogonal complements of subgroups of finite abelian groups as introduced by Holmes and Texier, and we define the symplectic complement of a subgroup of a group G ⊕ G, where G is a finite abelian group. Lastly, we define pseudo-unitarity and find results relating this definition to quantum computation theory, where the prefix ‘pseudo’ comes from the Moore-Penrose pseudo-inverse.