Variational quantum algorithms for permutation-based combinatorial problems: Optimal ansatz generation with applications to quadratic assignment problems and beyond

We present a quantum variational algorithm based on a novel circuit that generates all permutations that can be spanned by one- and two-qubits permutation gates. The construction of the circuits follows from group-theoretical results, most importantly the Bruhat decomposition of the group generated by the cx gates. These circuits require a number of qubits that scale logarithmically with the permutation dimension, and are therefore employable in near-term applications. We further augment the circuits with ancilla qubits to enlarge their span, and with these we build ansatze to tackle permutation-based optimization problems such as quadratic assignment problems, and graph isomorphisms. The resulting quantum algorithm, QuPer, is competitive with respect to classical heuristics and we could simulate its behavior up to a problem with 256 variables, requiring 20 qubits.