Abstract. This paper illustrates fundamental principles about small traveling salesman problems (TSPs) which are a current application for quantum annealing computers. The 2048 qubit, quantum annealing computer manufactured by D-Wave Systems is estimated to be able to solve all TSPs on 8 cities, which advances a recent 4-city result on a quantum simulator. Additionally, the D-Wave quantum computer is expected to find all optimal tours for each TSP. To prepare for this, we show the expected quantum output for 5,000 randomly generated TSPs on 6, 8 and 10 cities. The examples in the TSPLIB have 14 or more cities. These are too large for the current quantum annealing processors. We have included an annotated bibliography about solving the TSP on a quantum computer.
Keywords:Traveling salesman, optimal tour, combinatorial analysis, discrete optimization, quantum annealing AMS Classification Codes: 68R05, 81P68, 90C27
IntroductionRecently first steps have been taken to solve the traveling salesman problem (TSP) using the computational advantages promised by quantum algorithms [13]. As with any emerging effort, the initial steps have been limited and focused on a small number of cities that fit on the hardware. The major contributions of this paper are for 6, 8 and 10 city TSPs. We present empirical evidence based on 15,000 random samples, 5,000 for each number of cities. The majority of evidence indicates that the number of optimal tours and the length of an optimal tour increase together. All of the data shows that as the number of cities increases, the number of optimal tours increases. In all cases, the distances between cities are random integers in the interval [1,21].Outside the quantum world, the traveling salesman problem continues to be a hot topic that attracts wide interest. The problem is easy to explain and has several diverse applications [9,18], as well as theoretical studies [2,20]. There are many fascinating ways to solve it, usually approximately. Given a set of cities and the directed distances between each pair of cities, the TSP asks for a shortest route that visits each city exactly once and returns to the starting city. There are excellent, non-quantum TSP surveys [5,7,10].Quantum computing, particularly quantum annealing [3], is a new paradigm for discrete optimization that could use TSP benchmarks for the hardware. Quantum computing opens the possibility of large speedup with high probability of optimal answers, but requires new techniques for solving the TSP [11,19].The paper is organized as follows: Section 2 has references for other reports that use quantum computing algorithms for solving the TSP. Section 3 contains the results from a study of the expected output for TSPs from current quantum annealing computers. The output is envisioned to be all optimal tours, since these are obtained on a quantum annealing simulator. We have classified this output for 6, 8 and 10-city TSPs which is the estimated size that can be solved currently. The last section points to future quantum work for the TS...
