  CNOT circuits are a common building block of general quantum circuits. The
problem of synthesizing and optimizing such circuits has received a lot of
attention in the quantum computing literature. This problem is especially
challenging for quantum devices with restricted connectivity, where two-qubit
gates can only be placed between adjacent qubits. The state-of-the-art
algorithms for optimizing the number of CNOT gates are heuristic algorithms
that are based on Gaussian elimination and that use Steiner trees to connect
between different subsets of qubits. In this article, we suggest considering
weighted Steiner trees, and we present a simple low-cost heuristic to compute
weights. The simulated evaluation shows that the suggested heuristic is almost
always beneficial and reduces the number of CNOT gates by up to 10%.
