Domain-Wall / Unary Encoding in QUBO for Permutation Problems

QUBO is the input language of quantum computers based on quantum annealing such as the D-Wave systems and of the "quantum-inspired" dedicated hardware such as Fujitsu’s Digital Annealing Unit. We propose a new way to represent integer variables in QUBO together with constraints on those variables by using unary/domain-wall encoding in order to solve constrained optimization problem by quantum annealing. We detail in particular how to encode permutation constraints on integers within the unary/domain-wall encoding scheme, as these constraints are omnipresent in complex combinatorial problems. This new encoding in QUBO is interesting in the context of quantum annealing because it minimizes interactions between variables, and therefore connections needed between qubits in hardware. When compared to the usual one-hot encoding of integers in QUBO, we show than the new encoding scheme divides by nearly a factor 2 the number of connections between qubits for complex problems involving permutation constraints and makes it possible to cope with larger problem instances on current quantum and quantum-inspired hardware.