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This paper concerns quantum heuristics based on Mixer Hamiltonians that allowto restrict investigation on a specific subspace. Mixer Hamiltonian basedapproaches can be included in QAOA algorithm and we can state that MixerHamiltonians are mapping functions from the set of qubit-strings to the set ofsolutions. Mixer Hamiltonian offers an approach very similar to indirectrepresentations commonly used in routing or in scheduling community fordecades. After the initial publication of Cheng et al. in 1996 (Cheng et al.,1996), numerous propositions in OR lies on 1-to-n mapping functions, includingthe split algorithm that transform one TSP solution into a VRP solution. Theobjective is at first to give a compact and readable presentation of theseMixer Hamiltonians considering the functional analogies that exist between theOR community practices and the quantum field. Our experiments encompassnumerical evaluations of circuit using the Qiskit library of IBM meeting thetheoretical considerations.

Mixer Hamiltonian with QAOA for Max k-coloring : numerical evaluations