Comparing QUBO models for quantum annealing: integer encodings for permutation problems

QUBO (quadratic unconstrained binary optimization) has become the modeling language for quantum annealing and quantum-inspired annealing solvers. We present different approaches in QUBO for the magic square problem and the quadratic assignment problem (QAP), which can be modeled by linear equations and a permutation constraint over integer variables. Different ways of encoding integers by Booleans in QUBO amount to models, the implementation of which could have very different performance. Experiments performed on the Fixstars Amplify Annealer Engine, a quantum-inspired annealing solver, show that, compared to the classical one-hot encoding, using unary encoding for integers performs slightly better for the QAP and much better for magic square.