A Quantum Encoding of Traveling Salesperson Tours via Route Generation, Cost Phases, and a Valid-Permutation Oracle

Abstract

We present a compact quantum encoding of the Traveling Salesperson Problem (TSP) based on a time-register representation of tours. A candidate route is represented as a sequence of $n$ city labels over discrete time steps, with one fixed start city and the remaining cities encoded in binary registers. We describe three ingredients of the construction: uniform route generation over the route register, a reversible oracle for marking valid tours, and a phase oracle that encodes the total tour cost. The validity oracle distinguishes permutations of the non-start cities from invalid assignments, while the cost oracle accumulates the contribution of the start edge, intermediate transitions, and return edge into a tour-dependent phase. This yields a coherent superposition of candidate routes with feasibility and tour-length information embedded directly in the quantum state. The number of qubits required is $\Order{n\log_2(n)}$ and the circuit depth scales quadratically in $n$. The encoding is compatible with amplitude amplification or spectral filtering techniques such as the quantum singular value transform (QSVT) or Grover's algorithm. However, due to the exponentially small fraction of valid tours, the overall complexity remains exponential even when combined with amplitude amplification.

Figures (1)

• Figure 1: Illustrative TSP instance with 5 cities. (a) Directed weighted graph defining the cost matrix. Optimal solution in red. (b) All valid tours with cost, invalid tours have already been excluded as one would do using a Grover style operationbrassard2000quantumhoyer2000arbitrary. In total, there are $n^n = 5^5$ total tour candidates of which $n! = 5!$ are valid.

Theorems & Definitions (2)

• Definition 1: Valid tour
• Definition 2: Tour cost