An edge-based and subspace reduction encoding scheme to solve the traveling salesman problem in quantum computers

TL;DR

This work introduces two quantum encoding strategies for the Traveling Salesman Problem to reduce qubit requirements: an edge-based encoding and a Subspace Reduction Encoding (SRE). Using QAOA as the optimization engine, the authors compare these schemes against conventional node-based encoding on 4–6 city TSPs, showing edge-based encoding achieves better performance while using fewer qubits in simulation. The SRE further minimizes resource use by restricting the search to feasible tours, enabling small-instance tests on real quantum hardware, including a 4-city optimal result on IQM Garnet. The results highlight the potential of diagonal-cost Hamiltonians built via tensor products and conditioned subspaces to improve resource efficiency for quantum combinatorial optimization.

Abstract

This paper introduces a novel edge-based encoding technique for solving the Traveling Salesman Problem (TSP) on a quantum computer, reducing the required number of qubits. For implementation in real quantum devices, we applied the subspace reduction encoding to further reduce the dimension of the TSP solution space. We attack the TSP for 4-, 5-, and 6-city instances in both simulators and real quantum computers across different encoding frameworks. Optimal solutions of the 4-city TSP instance are obtained on state-of-the art IQM quantum computer. Our study presents a comparative analysis between edge-based encoding scheme and the node-based encoding methodology in the literature. Our findings indicate that the proposed encoding scheme outperforms conventional methods in terms of statistical measures, quantum resource utilization, and computational efficiency when applied to smaller TSP instances.

Figures (6)

• Figure 1: Alignment of cities for $n=4$. $\circ$ denotes a city and $i$ is the number of a city. $a$ denotes the step. For example, $A_{2,1}$ denotes the 2nd city in the first step of the travel. The lines denote the flights the salesman takes. This example corresponds to the travel $1 \to 2 \to 3 \to 4 \to 1$.
• Figure 2: Four cities of the 4-city TSP and distances between pairs of cities. $\circ$ denotes a city and the number attached to each line denotes the distance between two cities.
• Figure 3: Comparison of QAOA performance on 4- and 5-city TSP instances using node-based and edge-based encoding schemes. The figure presents the Success Rate (SR) representing the percentage of times the optimal route was found, and the Feasibility percentage (Feas. %) indicating the proportion of valid solutions obtained. All simulations were executed on the simulator.
• Figure 4: Cost of travel as a function of the number of QAOA ansatz layers for the 4-city TSP, comparing two encoding schemes: edge-based and node-based.
• Figure 5: Figure: Runtimes of the QAOA algorithm as a function of the number of QAOA ansatz layers for the 4-city Traveling Salesman Problem (TSP), comparing two encoding schemes: edge-based and node-based.