Develop symbol‑respecting, low‑depth mixers for binary‑coded colored‑permutation registers

Develop mixer operators for the binary-coded global-position colored-permutation encoding used for capacitated vehicle routing that are low depth, preserve symbol structure, are robust to padding leakage, and enable sufficiently rich exploration of the valid code space so that the compressed-register formulation inherits the sampling and concentration behavior of the Constraint‑Enhanced QAOA kernel.

Background

The paper proposes a global-position colored-permutation encoding for CVRP that is naturally aligned with the CE–QAOA kernel, but its one‑hot realization requires n^2K qubits. To reach near‑term industrially relevant sizes, the authors advocate compressing each position’s local alphabet from one‑hot to a binary-coded symbol (i,k), reducing qubit count to Θ(n log(nK)).

However, this compression removes the existing one‑hot manifold structure that CE–QAOA’s block‑XY mixer preserves. A binary-coded version needs new mixer primitives that operate directly on the compressed registers while maintaining low depth and symbol‑respecting dynamics, remain robust to padding leakage, and still explore the valid code space effectively. Establishing such mixers is identified as a key missing technical ingredient.