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Despite rapid advances in quantum computing technologies, the qubitconnectivity limitation remains to be a critical challenge. Both near-term NISQquantum computers and relatively long-term scalable quantum architectures donot offer full connectivity. As a result, quantum circuits may not be directlyexecuted on quantum hardware, and a quantum compiler needs to perform qubitrouting to make the circuit compatible with the device layout. During the qubitrouting step, the compiler inserts SWAP gates and performs circuittransformations. Given the connectivity topology of the target hardware, thereare typically multiple qubit routing candidates. The state-of-the-art compilersuse a cost function to evaluate the number of SWAP gates for different routesand then select the one with the minimum number of SWAP gates. After qubitrouting, the quantum compiler performs gate optimizations upon the circuit withthe newly inserted SWAP gates.  In this paper, we observe that the aforementioned qubit routing is notoptimal, and qubit routing should \textit{not} be independent on subsequentgate 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 tothe development of our qubit routing algorithm, NASSC (Not All Swaps have theSame Cost). NASSC is the first algorithm that considers the subsequentoptimizations during the routing step. Our optimization-aware qubit routingleads to better routing decisions and benefits subsequent optimizations. Wealso propose a new optimization-aware decomposition for the inserted SWAPgates. Our experiments show that the routing overhead compiled with our routingalgorithm is reduced by up to $69.30\%$ ($21.30\%$ on average) in the number ofCNOT gates and up to $43.50\%$ ($7.61\%$ on average) in the circuit depthcompared with the state-of-the-art scheme, SABRE.

Not All SWAPs Have the Same Cost: A Case for Optimization-Aware Qubit Routing