  Despite rapid advances in quantum computing technologies, the qubit
connectivity limitation remains to be a critical challenge. Both near-term NISQ
quantum computers and relatively long-term scalable quantum architectures do
not offer full connectivity. As a result, quantum circuits may not be directly
executed on quantum hardware, and a quantum compiler needs to perform qubit
routing to make the circuit compatible with the device layout. During the qubit
routing step, the compiler inserts SWAP gates and performs circuit
transformations. Given the connectivity topology of the target hardware, there
are typically multiple qubit routing candidates. The state-of-the-art compilers
use a cost function to evaluate the number of SWAP gates for different routes
and then select the one with the minimum number of SWAP gates. After qubit
routing, the quantum compiler performs gate optimizations upon the circuit with
the newly inserted SWAP gates.
  In this paper, we observe that the aforementioned qubit routing is not
optimal, and qubit routing should \textit{not} be independent on subsequent
gate optimizations. We find that with the consideration of gate optimizations,
not all of the SWAP gates have the same basis-gate cost. These insights lead to
the development of our qubit routing algorithm, NASSC (Not All Swaps have the
Same Cost). NASSC is the first algorithm that considers the subsequent
optimizations during the routing step. Our optimization-aware qubit routing
leads to better routing decisions and benefits subsequent optimizations. We
also propose a new optimization-aware decomposition for the inserted SWAP
gates. Our experiments show that the routing overhead compiled with our routing
algorithm is reduced by up to $69.30\%$ ($21.30\%$ on average) in the number of
CNOT gates and up to $43.50\%$ ($7.61\%$ on average) in the circuit depth
compared with the state-of-the-art scheme, SABRE.
