An Exact Branch and Bound Algorithm for the generalized Qubit Mapping Problem

Abstract: Quantum circuits are typically represented by a (ordered) sequence of gates over a set of virtual qubits. During compilation, the virtual qubits of the gates are assigned to the physical qubits of the underlying quantum hardware, a step often referred to as the qubit assignment problem. To ensure that the resulting circuit respects hardware connectivity constraints, additional SWAP gates are inserted as needed, which is known as the qubit routing problem. Together, they are called the Qubit Mapping Problem (QMP), which is known to be NP-hard. A very common way to deal with the complexity of the QMP is to partition the sequence of gates into a sequence of gate groups (or layers). However, this imposes a couple of important restrictions: (1) SWAP gates can only be added between pairs of consecutive groups, and (2) all the gates belonging to a certain group have to be executed (in parallel) in the same time slot. The first one prevents gates to be re-arranged optimally, while the second one imposes a time discretization that practically ignores gate execution time. While this clearly reduces the size of the feasible space, little is still known about how much is actually lost by imposing a fixed layering when looking at the minimization of either the number of SWAPs or the makespan of the compiled circuit. In this paper, we present a flexible branch and bound algorithm for a generalized version of the QMP that either considers or ignores the gate layering and the gate execution time. The algorithm can find find proven optimal solutions for all variations of the QMP, but also offers a great platform for different heuristic algorithms. We present results on several benchmark sets of small quantum circuits, and we show how ignoring the layering can significantly improve some key performance indicators of the compiled circuit.