Variationally Learning Grover's Quantum Search Algorithm

TL;DR

This paper explores how a variational quantum algorithm can learn and recover Grover's search algorithm parameters, demonstrating potential improvements and insights into hybrid quantum-classical optimization for quantum algorithms.

Contribution

It shows that variational algorithms can effectively learn Grover's algorithm parameters across different qubit sizes, highlighting the efficiency of classical optimization in quantum settings.

Findings

Achieved up to 5.77% improvement in success probability for three-qubits.

Demonstrated near-unity success probability oscillations for five-qubits.

Proved that optimal parameters can be found without varying over a family of circuits.

Abstract

Given a parameterized 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 illustrate the promise of using hybrid quantum classical approaches to improve quantum algorithms. This paper further demonstrates that to find optimal parameters one doesn't 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 arbitrary number of qubits.