Quantum Computing with Majorana Fermion Codes

TL;DR

This paper introduces a unified framework for fault-tolerant quantum computation using Majorana fermion codes, enabling efficient Clifford gates and new code families with reduced overhead.

Contribution

It presents a novel protocol for Pauli measurements with Majorana modes, adapts twist-based lattice surgery for fermionic codes, and proposes Majorana color codes for improved efficiency.

Findings

Zero time overhead for logical Clifford gates.

Implementation of local parity measurements with tetrons and hexons.

Introduction of Majorana color codes with reduced overhead.

Abstract

We establish a unified framework for Majorana-based fault-tolerant quantum computation with Majorana surface codes and Majorana color codes. All logical Clifford gates are implemented with zero time overhead. This is done by introducing a protocol for Pauli product measurements with tetrons and hexons which only requires local 4-Majorana parity measurements. An analogous protocol is used in the fault-tolerant setting, where tetrons and hexons are replaced by Majorana surface code patches, and parity measurements are replaced by lattice surgery, still only requiring local few-Majorana parity measurements. To this end, we discuss twist defects in Majorana fermion surface codes and adapt the technique of twist-based lattice surgery to fermionic codes. Moreover, we propose a new family of codes that we refer to as Majorana color codes, which are obtained by concatenating Majorana surface codes with small Majorana fermion codes. Majorana surface and color codes can be used to decrease the space overhead and stabilizer weight compared to their bosonic counterparts.