Rigorous Establishment of Polynomial Quantum Advantage for the Single-Qubit TSP Algorithm

Establish a rigorous polynomial quantum advantage for the single-qubit Bloch-sphere algorithm solving Travelling Salesman Problem instances by formally defining and incorporating a quantum phase oracle and an inversion-about-the-average operation that underpin a quantum search formulation of optimal-path selection.

Background

The authors draw an analogy between their superposition-based single-qubit approach to TSP and Grover’s quantum search, noting prior results that such superposition-based algorithms can achieve polynomial speedups. While they expect a similar advantage, they have not yet provided a rigorous formulation.

They explicitly state that formal definitions of a "quantum phase oracle" and "inversion about the average" tailored to their algorithm are required, and identify this as future work. This marks an unresolved task needed to substantiate the claimed advantage.