Quantum Error Correction for Complex and Majorana Fermion Qubits

TL;DR

This paper develops error-correcting codes specifically designed for fermionic quantum systems, including Majorana fermions, to address parity-violating and conserving errors, with potential physical implementations.

Contribution

It introduces a generic method to construct fermion error-correcting codes from classical binary codes, including the shortest code for quasiparticle poisoning correction.

Findings

Constructed the shortest fermion code for quasiparticle poisoning

Developed translationally-invariant fermion codes

Proposed physical realization in mesoscopic superconductors

Abstract

We introduce error-correcting codes that can correct for fermion parity-violating (quasiparticle poisoning) and parity-conserving errors in systems of complex fermions and of Majorana fermions. After establishing properties of fermion codes, we introduce a generic construction of fermion codes from weakly self-dual classical, binary error-correcting codes. We use this method to construct (i) the shortest fermion code to correct for quasiparticle poisoning errors, (ii) translationally-invariant fermion codes and (iii) other codes that correct higher-weight errors. We conclude by discussing a possible physical realizations of one such code in mesoscopic superconductors hosting Majorana zero modes.