An Adaptive Hybrid Quantum Algorithm for the Metric Traveling Salesman Problem

In this paper, we design, analyze, and evaluate a hybrid quantum algorithm for the metric traveling salesman problem (TSP). TSP is a well-studied NP-complete problem that many algorithmic techniques have been developed for, on both classic computers and quantum computers. The existing literature of algorithms for TSP are neither adaptive to input data nor suitable for processing medium-size data on the modern classic and quantum machines. In this work, we leverage the classic computers' power (large memory) and the quantum computers' power (quantum parallelism), based on the input data, to fasten the hybrid algorithm's overall running time. Our algorithmic ideas include trimming the input data efficiently using a classic algorithm, finding an optimal solution for the postprocessed data using a quantum-only algorithm, and constructing an optimal solution for the untrimmed data input efficiently using a classic algorithm. We conduct experiments to compare our hybrid algorithm against the state-of-the-art classic and quantum algorithms on real data sets. The experimental results show that our solution truly outperforms the others and thus confirm our theoretical analysis. This work provides insightful quantitative tools for people and compilers to choose appropriate quantum or classical or hybrid algorithms, especially in the NISQ (noisy intermediate-scale quantum) era, for NP-complete problems such as TSP.