Variational learning of Grover's quantum search algorithm

Abstract

Given a parametrized quantum circuit such that a certain setting of these real-valued parameters corresponds to Grover's celebrated search algorithm, can a variational algorithm recover these settings and hence learn Grover's algorithm? We studied several constrained variations of this problem and answered this question in the affirmative, with some caveats. Grover's quantum search algorithm is optimal up to a constant. The success probability of Grover's algorithm goes from unity for two qubits, decreases for three and four qubits, and returns near unity for five qubits, then oscillates ever so close to unity, reaching unity in the infinite qubit limit. The variationally approach employed here found an experimentally discernible improvement of $5.77%$ and $3.95%$ for three and four qubits, respectively. Our findings are interesting as an extreme example of variational search, and they illustrate the promise of using hybrid quantum classical approaches to improve quantum algorithms. This paper further demonstrates that to find optimal parameters, one does not need to vary over a family of quantum circuits to find an optimal solution. This result looks promising and points out that there is a set of variational quantum problems with parameters that can be efficiently found on a classical computer for an arbitrary number of qubits.