Optimized QAOA ansatz circuit design for two-body Hamiltonian problems.

Quantum Approximate Optimization Algorithm (QAOA) is a prospective candidate for providing quantum advantage in finding approximate solutions to optimization problems using near-term quantum devices. In this paper, the goal is to reduce the number of CNOT gates and the depth of quantum ansatz circuits for QAOA. First, we present a generalized QAOA formulation for any Hamiltonian that involves upto two-body interactions, as a graph. The circuit realization of the depth-p QAOA requires $2mp$ CNOT gates where the graph for the Hamiltonian has n vertices and m edges. Presently, a CNOT gate is one of the primary sources of error. We propose a graph-theoretic algorithm which can eliminate $n-1$ CNOTs from the QAOA ansatz while retaining functional equivalence with the original circuit. This improves upon previously proposed Depth First Search (DFS) method by restricting the increase in depth of the circuit, which was a drawback of the method, by $\simeq 84.8\%$ for Erdős-Renyi graphs. Finally, if the underlying connectivity (hardware coupling map) and the initial qubit placement for a hardware are known, then we present how to extend the greedy heuristic method in order to obtain a functionally equivalent circuit with $\simeq 5\%$; fewer SWAP gates for Erdős-Renyi graphs.