Gate scheduling for quantum algorithms

As the effort to scale up existing quantum hardware proceeds, the necessity of effective methods to schedule quantum gates and minimize the number of operations becomes more compelling. Three are the constraints that have to be taken into account: The order or dependency of the quantum gates in the specific algorithm, the fact that any qubit may be involved in at most one gate at a time, and the restriction that two-qubit gates are implementable only between connected qubits. The last aspect implies that the compilation depends not only on the algorithm, but also on hardware properties like its connectivity. Here we suggest a two-step approach in which logical gates are initially scheduled neglecting the limited connectivity, while routing operations are added at a later step and their number is empirically minimized. In the context of the two-step approach, we propose a solution of the routing problem that is optimal for an one dimensional array of qubits. As a practical application, we schedule the Quantum Approximate Optimization Algorithm in a linear geometry and quantify the reduction in the number of gates and circuit depth by increasing the efficacy of the scheduling strategies.