Quantum Error-Correction for Fermionic Qubits

Recently, there has been a growing effort to store quantum information in fermion states, so that a qubit is encoded in the fermion occupation number or the fermion number parity. Using fermions as the carriers of quantum information necessitates a new computational model and new error-correcting codes, as Fermi statistics forbids a mapping of local quantum gates acting on fermions to local gates acting on bosons. Motivated by on-going experiments, we introduce a variety of fermionic quantum codes that are able to correct for fermion parity-preserving and parity-violating ("quasiparticle poisoning") errors. First, we discuss a surface code of Majorana fermions, along with its implementation, and demonstrate that the threshold error-rates for this code are superior to that of bosonic surface codes. We then introduce a generic construction of fermion codes from weakly self-dual classical, binary error-correcting codes, and use this method to find the shortest fermion code to correct for quasiparticle poisoning errors and other codes that correct higher-weight errors. We conclude by discussing physical implementations of codes with shorter code distance.