The Travelling Salesman Problem (TSP) is a classic combinatorial optimisation problem which is NP-hard in optimisation form and NP-complete as a decision problem. The standard question asks for the shortest tour that visits every city exactly once and returns to the origin. This paper studies a threshold-fraction variant, which we call the robust TSP, formulated as a promise problem: given a distance threshold $K$ and a promised fraction $p$, decide whether at least a fraction $p$ of all tours have total length at most $K$, or whether no such tour exists. We show that Grover's quantum search algorithm decides this problem using $O(1/\sqrt{p})$ oracle queries, independent of the number of cities $N$, with each query implemented by a quantum circuit of polynomial size in $N$. For any fixed robustness level $p$, the total runtime is therefore polynomial in $N$. Measured in oracle queries, the quantum algorithm offers a quadratic improvement in the dependence on $1/p$ over classical random sampling, which needs $Θ(1/p)$ samples; because each quantum query is costly, however, it is faster in total running time only when $p$ is very small (roughly below $N^{-4}$), where the number of queries is no longer constant. We describe the problem formulation, detail the oracle construction (edge-length loading, sequential quantum addition, and threshold comparison), and analyse the query complexity to achieve the required result.

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