Quantum

Quantum Heuristic Separation of Comb Inequalities for the Travelling Salesman Problem

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Briefings

A new quantum heuristic algorithm was proposed to separate comb inequalities for the travelling salesman problem. The method uses polynomially many gates and qubits to prepare a uniform superposition of all combs with a fixed number of teeth. It encodes Hamming weight in one-hot form and searches for violations using the Grover-Mixer Quantum Alternating Operator Ansatz. While the running time is polynomial under specified budgets, the evidence does not confirm if these budgets are sufficient to find a violated comb with constant probability.

Key points

• The algorithm uses polynomially many gates and qubits to prepare a uniform superposition of all combs with a fixed number of teeth.
• The running time is polynomial under prescribed polynomial budgets for the number of QAOA rounds, parameter optimization steps and samples gathered, although no evidence exists that such budgets suffice to find a violated comb with constant probability when one exists.

Context

Comb inequalities are essential cutting planes for the travelling salesman problem, yet their separation remains computationally challenging. This research proposes a quantum heuristic separation algorithm to address this challenge.

Limitations

The evidence does not confirm if the specified polynomial budgets are sufficient to find a violated comb with constant probability when one exists. The resource analysis provides an explicit circuit construction but lacks probabilistic guarantees on success rates.

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