Construct a fault-tolerant universal gate set for exchange-only encodings

Construct a fault-tolerant CNOT gate between two exchange-only or singlet-only exchange-only qubits and determine whether additional single-qubit gates can be made fault tolerant using auxiliary spins, with the aim of obtaining a fault-tolerant universal gate set.

Background

The paper establishes fault-tolerant implementations for three single-qubit rotations and SWAP operations in the SOEO encoding, but emphasizes that these operations do not constitute a fault-tolerant universal gate set. In particular, the proposed architecture does not yet provide a fault-tolerant CNOT between encoded exchange-only qubits, and it is unresolved whether auxiliary spins can make further single-qubit operations fault tolerant.