The Structure of the Majorana 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 the analogous objects for systems of Majorana fermions. A crucial role is played by fermion parity symmetry, which is an unbreakable symmetry present in any system in which the fundamental degrees of freedom are fermionic. We prove that the subgroup of parity-preserving fermionic Cliffords can be represented by the orthogonal group over the binary field $\mathbb{F}_2$, 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, proving that it is equivalent to the frame potential of the ordinary Clifford group acting on a fixed-parity sector of the Hilbert space.