The weighted MAX $$k$$ 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 $$k$$ 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 $$|V|\log _2k$$ | 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 $$k$$ k -CUT problem for $$k>2$$ 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.

No abstract

The weighted MAX k-CUT problem aims at finding a k-partition of a given weighted undirected graph G(V, E) so as to maximize the sum of the weights of the crossing edges. 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. This is an exponential improvement with respect to k on the number of qubits when compared to previously known encodings. We provide a decomposition of the phase separation operator into basis gates and we show that the circuit depth is very shallow if k is a power of two. The resulting circuits are implemented using Qiskit [2], which is used to benchmark our algorithm on a suite of test cases. The simulations on random graphs with up to 10 vertices show that we are able to achieve competitive approximation ratios for all cases. We provide an analysis of the circuit depth, which shows that the circuit depth is very shallow if k is a power of two. 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.

No abstract