The Structure of the Majorana Clifford Group

TL;DR

This paper explores the structure of Majorana Clifford groups in quantum information, revealing their algebraic properties, generation methods, and relation to stabilizer codes, with implications for fermionic quantum systems.

Contribution

It characterizes the Majorana Clifford group as an orthogonal group over a binary field and demonstrates how it can be generated and used for stabilizer codes.

Findings

Majorana Clifford subgroup is isomorphic to an orthogonal group over .

The group can be generated by braiding operators.

Frame potential analysis shows equivalence to ordinary Clifford group.

Abstract

In quantum information science, Clifford operators and stabilizer codes play a central role for systems of qubits (or qudits). In this paper, we study their analogues for systems composed of Majorana fermions. In this case, a crucial role is played by fermion parity symmetry, which is an unbreakable symmetry present in any system with fundamentally fermionic degrees of freedom. We prove that the subgroup of parity-preserving Majorana Cliffords can be represented by the orthogonal group over the binary field $\F$, and we show how it can be generated by braiding operators and used to construct any (even-parity) Majorana stabilizer code. We also analyze the frame potential for this so-called p-Clifford group when acting on a fixed-parity sector of the Hilbert space, proving that it is equivalent to the frame potential of the ordinary Clifford group acting on the same sector.