The complexity of quantum circuit mapping with fixed parameters

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