  Protein design is a technique to engineer proteins by modifying their
sequence to obtain novel functionalities. In this method, amino acids in the
sequence are permutated to find the low energy states satisfying the
configuration. However, exploring all possible combinations of amino acids is
generally impossible to achieve on conventional computers due to the
exponential growth of possibilities with the number of designable sites. Thus,
sampling methods are currently used as a conventional approach to address the
protein design problems. Recently, quantum computation methods have shown the
potential to solve similar types of problems. In the present work, we use the
general idea of Grover's algorithm, a pure quantum computation method, to
design circuits at the gate-based level and address the protein design problem.
In our quantum algorithms, we use custom pair-wise energy tables consisting of
eight different amino acids. Also, the distance reciprocals between designable
sites are included in calculating energies in the circuits. Due to the noisy
state of current quantum computers, we mainly use quantum computer simulators
for this study. However, a very simple version of our circuits is implemented
on real quantum devices to examine their capabilities to run these algorithms.
Our results show that using $\mathcal{O}(\sqrt N)$ iterations, the circuits
find the correct results among all $N$ possibilities, providing the expected
quadratic speed up of Grover's algorithm over classical methods.
