Transversal Construction of Topological Gates on Multiqubit Quantum Codes

We study the possibility of constructing quantum gates using topological phases, which originate from local SU(2) evolution of entangled multiqubit systems. For this purpose, logical codewords using two-, three- and nine-qubit entangled states are defined and possible implementations of topological gates on these codes, are examined. For two-qubit systems, it is shown that for only two of the Pauli gates, a topological implementation is possible, the third must be non-topological. Furthermore, it is shown that a topological implementation of Hadamard gate is also possible on the two-qubit code. For the three-qubit code, the logical Pauli gates are found to be topologically implementable and a topological implementation of the logical S gate seems to be possible as well. Lastly, for the nine-qubit code, the logical Pauli gates, the logical S gate and the logical T gate are shown to be implementable topologically on the code. It remains an open question whether topological implementation of logical Hadamard gate by invertible local operators is possible on the nine-qubit code.