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

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.

💡 Research Summary

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This paper presents two novel quantum encoding schemes for the Traveling Salesman Problem (TSP) that dramatically reduce qubit requirements and improve algorithmic performance compared with the conventional node‑based approach. The first scheme, called edge‑based encoding, represents a tour by the sequence of edges rather than by a binary matrix of city‑step assignments. By encoding each city index in binary, the number of qubits scales as (n‑1)·⌈log₂(n‑1)⌉ instead of n² for the node‑based formulation. The authors construct the cost Hamiltonian H_edge entirely from diagonal matrices using tensor products: D₁ and D₂ encode the start‑and‑end edges, while a diagonal matrix C encodes all intermediate edges. A large penalty term γ·P is added to raise the energy of illegal configurations (repeated cities or inclusion of “fake” cities). Because H_edge is diagonal, the variational quantum algorithm does not need to implement complex non‑diagonal interactions, simplifying circuit depth and connectivity requirements.

The second scheme, Subspace Reduction Encoding (SRE), further compresses the search space by pre‑selecting only the (n‑1)! legal tours and embedding them in a subspace of dimension K = ⌈log₂((n‑1)!)⌉. This is achieved by constructing a diagonal matrix that contains only the legal tour indices and restricting the variational optimizer to this subspace. The result is a faster convergence of the QAOA optimizer and reduced exposure to hardware noise, since the algorithm never explores illegal states.

Experimental validation is performed on 4‑, 5‑, and 6‑city instances. The authors employ the Quantum Approximate Optimization Algorithm (QAOA) with depths p = 1–3, running both on classical simulators and on a state‑of‑the‑art IQM superconducting quantum processor. For the 4‑city case, the optimal tour is obtained on the real device, confirming feasibility. For the 5‑ and 6‑city cases, high‑quality solutions are achieved in simulation. Performance metrics include success probability, expected tour cost, standard deviation, circuit depth, and qubit count. Compared with the node‑based encoding, the edge‑based + SRE combination reduces qubit usage by roughly 60–70 % (e.g., 4‑city: 16 → 6 qubits, further to 5 with SRE), shortens circuit depth, and improves statistical measures by 20–35 % on average.

Key contributions are:

1. Logarithmic qubit scaling through binary encoding of city indices.
2. Diagonal Hamiltonian construction via tensor products of simple diagonal matrices, avoiding costly multi‑qubit gates.
3. Subspace reduction that limits the optimizer to the legal tour manifold, accelerating convergence and mitigating noise.
4. Experimental demonstration on a real quantum processor, establishing practical viability for small‑scale TSP.

Limitations include the focus on very small instances (≤ 6 cities) and the sensitivity of the penalty parameter γ, which must be tuned to exceed the maximum edge cost. Future work should explore higher QAOA depths, error‑mitigation techniques, and dynamic subspace expansion to handle larger TSP instances, potentially integrating problem‑specific heuristics to guide the variational search.