Characterizing QUBO Reformulations of the Max-k-Cut Problem for Quantum Computing

Quantum computing offers significant potential for solving NP-hard combinatorial (optimization) problems that are beyond the reach of classical computers. One way to tap into this potential is by reformulating combinatorial problems as a quadratic unconstrained binary optimization (QUBO) problem. The solution of the QUBO reformulation can then be addressed using adiabatic quantum computing devices or appropriate quantum computing algorithms on gate-based quantum computing devices. In general, QUBO reformulations of combinatorial problems can be readily obtained by properly penalizing the violation of the problem's constraints in the original problem's objective. However, characterizing tight (i.e., minimal but sufficient) penalty coefficients for this purpose is critical for enabling the solution of the resulting QUBO in current and near-term quantum computing devices. Along these lines, we here focus on the (weighted) max $k$-cut problem, a fundamental combinatorial problem with wide-ranging applications that generalizes the well-known max cut problem. We present closed-form characterizations of tight penalty coefficients for two distinct QUBO reformulations of the max $k$-cut problem whose values depend on the (weighted) degree of the vertices of the graph defining the problem. These findings contribute to the ongoing effort to make quantum computing a viable tool for solving combinatorial problems at scale. We support our theoretical results with illustrative examples. Further, we benchmark the proposed QUBO reformulations to solve the max $k$-cut problem on a quantum computer simulator.