  This thesis explores hybrid algorithms that combine classical and quantum
computing to enhance the performance of classical algorithms. Two approaches
are studied: a hybrid search and sample optimization algorithm and a classical
algorithm that assesses the cost and performance of quantum algorithms in
chemistry. Hybrid algorithms are vital due to limitations in both classical and
quantum computing, offering a solution by leveraging the strengths of both. The
first algorithm, quantum Metropolis Solver (QMS), adapts a quantum walk to a
Metropolis-Hastings algorithm for industrial applications, demonstrating
advantages over classical counterparts in various sectors. The second
algorithm, TFermion, is a classical tool for evaluating the cost of T-type
gates in quantum chemistry algorithms, aiding in the comparison and execution
of these algorithms on real quantum hardware, and applied to the design of more
efficient electric batteries.
