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The Quantum Approximate Optimization Algorithm (QAOA) is an algorithmicframework for finding approximate solutions to combinatorial optimizationproblems, derived from an approximation to the Quantum Adiabatic Algorithm(QAA). In solving combinatorial optimization problems with constraints in thecontext of QAOA or QAA, one needs to find a way to encode problem constraintsinto the scheme. In this paper, we formalize different constraint types tolinear equalities, linear inequalities, and arbitrary form. Based on this, wepropose constraint-encoding schemes well-fitting into the QAOA framework forsolving NP combinatorial optimization problems. The implemented algorithmsdemonstrate the effectiveness and efficiency of the proposed scheme by thetesting results of varied instances of some well-known NP optimizationproblems. We argue that our work leads to a generalized framework for finding,in the context of QAOA, high-quality approximate solutions to combinatorialproblems with various types of constraints.

Quantum approximate algorithm for NP optimization problems with constraints