  Due to the limited connectivity of gate model quantum devices, logical
quantum circuits must be compiled to target hardware before they can be
executed. Often, this process involves the insertion of SWAP gates into the
logical circuit, usually increasing the depth of the circuit, achieved by
solving a so-called qubit assignment and routing problem. Recently, a number of
integer linear programming (ILP) models have been proposed for solving the
qubit assignment and routing problem to proven optimality. These models encode
the objective function and constraints of the problem, and leverage the use of
automated solver technology to find hardware-compliant quantum circuits. In
this work, we propose constraint programming (CP) models for this problem and
compare their performance against ILP for circuit depth minimization for both
linear and two-dimensional grid lattice device topologies on a set of randomly
generated instances. Our empirical analysis indicates that the proposed CP
approaches outperform the ILP models both in terms of solution quality and
runtime.
