Two-step approach to scheduling quantum circuits

As the effort to scale up existing quantum hardware proceeds, it becomes necessary to schedule quantum gates in a way that minimizes the number of operations. There are three constraints that have to be satisfied: the logical dependency of the quantum gates in the 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 connectivity. Here we present a two-step approach in which logical gates are initially scheduled neglecting connectivity considerations and routing operations are added at a later step in ways that minimize their overhead. We rephrase the subtasks of gate scheduling in terms of graph problems like edge-coloring and maximum subgraph isomorphism. While this approach is general, we specialize to a one-dimensional array of qubits to propose a routing scheme that is minimal in the number of exchange operations. 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 that results from different scheduling strategies.