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A quantum circuit must be preprocessed before implementing on NISQ devices due to the connectivity constraint. Quantum circuit mapping (QCM) transforms the circuit into an equivalent one that is compliant with the NISQ device's architecture constraint by adding SWAP gates. The QCM problem asks the minimal number of auxiliary SWAP gates, and is NP-complete. The complexity of QCM with fixed parameters is studied in the paper. We give an exact algorithm for QCM, and show that the algorithm runs in polynomial time if the NISQ device's architecture is fixed. If the number of qubits of the quantum circuit is fixed, we show that the QCM problem is NL-complete by a reduction from the undirected shortest path problem. Moreover, the fixed-parameter complexity of QCM is W[1]-hard when parameterized by the number of qubits of the quantum circuit. We prove the result by a reduction from the clique problem. If taking the depth of the quantum circuits and the coupling graphs as parameters, we show that the QCM problem is still NP-complete over shallow quantum circuits, and planar, bipartite and degree bounded coupling graphs.

The quantum circuit mapping approach is an indispensable part of the software stack for the noisy intermediatescale quantum (NISQ) device. It has a significant impact on the reliability of computational tasks on NISQ devices. To improve the overall fidelity of physical circuits, we propose a quantum circuit mapping method based on multi-agent cooperation. This approach considers the Spatio-temporal variation of quantum operation quality on the NISQ device when inserting ancillary operation. It consists of two core components: the qubit placement algorithm and the qubit routing method. The qubit placement algorithm exploits the iterated local search framework to find a desirable initial mapping for the reduced symmetric form of the original circuit. The qubit routing method generates the physical circuit through multi-agent communication and collaboration. Each agent inserts the ancillary gates independently according to its environment state. The quality of the physical circuit evolves according to an information-exchanging mechanism between agents, which combines the local search and global search. To experiment on the benchmark circuits (with hundreds of quantum gates) beyond the capacity of current NISQ devices, we build a noisy simulator with gate error 10x lower than that of the latest NISQ device of IBM. The experimental results confirm the performance of our approach in improving circuit fidelity. Compared with the stateof-the-art method, our method can improve the success rate by 25.86% on average and 95.42% at maximum., Comment: 13 pages, 10 figures