Systematic construction of topological-nontopological hybrid universal quantum gates based on many-body Majorana fermion interactions

Topological quantum computation by way of braiding of Majorana fermions is not universal quantum computation. There are several attempts to make universal quantum computation by introducing some additional quantum gates or quantum states. However, there is a serious problem in embedding an $M$-qubit quantum gate in the $N$-qubit system with $N>M$. This is an inherent problem to the Majorana system, where quantum gates for logical qubits become nonlocal in terms of physical qubits in general because braiding operations preserve the fermion parity. For instance, the CZ gate could not be embedded in the three-qubit system. We overcome this embedding problem by introducing $2(N+1)$-body interactions of Majorana fermions in the $N$-qubit system. A universal set of quantum gates is constructed for $N$ logical qubits, leading to topological-nontopological hybrid universal quantum computation. It would be more robust than conventional universal quantum computation because the quantum gates generated by braiding are topologically protected.