Finding Optimal Qubit Permutations for IBM's Quantum Computer Architectures

IBM offers quantum processors for Clifford+T circuits. The only restriction is that not all CNOT gates are implemented and must be substituted with alternate sequences of gates. Each CNOT has its own mapping with a respective cost. However, by permuting the qubits, the number of CNOT that need mappings can be reduced. The problem is to find a good permutation without an exhaustive search. In this paper we propose a solution for this problem. The permutation problem is formulated as an Integer Linear Programming (ILP) problem. Solving the ILP problem, the lowest cost permutation for the CNOT mappings is guaranteed. To test and validated the proposed formulation, quantum architectures with 5 and 16 qubits were used. The ILP formulation along with mapping techniques found circuits with up to 64% fewer gates than other approaches.