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

TL;DR

This paper introduces a compact quantum encoding for the Traveling Salesperson Problem that represents tours as quantum states with embedded feasibility and cost information, enabling quantum algorithms to process TSP solutions.

Contribution

It presents a novel quantum encoding of TSP tours using route generation, validity, and cost oracles, with efficient qubit usage and circuit depth, compatible with amplitude amplification techniques.

Findings

Encoding uses O(n log n) qubits.

Circuit depth scales quadratically with n.

Exponential complexity persists due to small valid tour fraction.

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.