Advances in data analysis, machine learning, and combinatorial optimization have accelerated the discovery of new materials with desired properties; this paradigm is commonly referred to as Materials 4.0. Quantum computing represents a new computing framework that uses the principles of quantum mechanics such as entanglement and superposition. Quantum algorithms such as Grover search has been proven to have quadratic speedup over the classical search methods. This paper considers the application of Grover search-based combinatorial optimization for material discovery. In particular, this paper considers the identification of the Nickel-Titanium (Ni-Ti) based shape memory alloy of the composition, Ti50 Ni{mathrm {{50-x-y}}} Cux Fey that has the minimum thermal hysteresis associated with the austenite-martensite transformations. The paper details the construction of quantum circuits to carry out the optimization analysis, and demonstrates the analysis on a simulation platform using the Python Qiskit package.