  The weighted MAX k-CUT problem consists of finding a k-partition of a given
weighted undirected graph G(V,E) such that the sum of the weights of the
crossing edges is maximized. The problem is of particular interest as it has a
multitude of practical applications. We present a formulation of the weighted
MAX k-CUT suitable for running the quantum approximate optimization algorithm
(QAOA) on noisy intermediate scale quantum (NISQ)-devices to get approximate
solutions. The new formulation uses a binary encoding that requires only
|V|log_2(k) qubits. The contributions of this paper are as follows: i) A novel
decomposition of the phase separation operator based on the binary encoding
into basis gates is provided for the MAX k-CUT problem for k >2. ii) Numerical
simulations on a suite of test cases comparing different encodings are
performed. iii) An analysis of the resources (number of qubits, CX gates) of
the different encodings is presented. iv) Formulations and simulations are
extended to the case of weighted graphs. For small k and with further
improvements when k is not a power of two, our algorithm is a possible
candidate to show quantum advantage on NISQ devices.
