Polynomial optimization with linear combination of unitaries

In optimization, a pivotal challenge involves minimizing or maximizing polynomial functions, which is often tackled through iterative methods. Although quantum algorithms hold promise to enhance these techniques, a practical quantum protocol is still elusive. In this paper, first, we utilize the linear combination of unitaries framework for gradient -based optimization of an arbitrary polynomial, where the entire optimization process can be represented as cascaded circuits. One standout feature of this protocol is its explicit implementation circuit, which has the ability to circumvent the necessity for a quantum random access memory structure. Then the feasibility of this protocol is initially assessed through numerical simulation, exploring its robustness in several noise scenarios. Finally, the paper explores the associated computational cost, sample, architecture, and time costs, and compares them with those of existing methods. Notably, the proposed protocol shows significant efficiency gains in terms of architecture cost. If certain conditions are met, both sample and time costs can be significantly reduced, addressing a critical weakness in previous works.