Test case optimization (TCO) reduces software testing cost while preserving its effectiveness, but solving TCO problems for large-scale and complex systems requires substantial computational resources. Quantum approximate optimization algorithms (QAOAs) are promising combinatorial optimization algorithms that rely on quantum computational resources, with the potential efficiency advantages over classical approaches. Several proof-of-concept applications of QAOAs for solving combinatorial problems, such as portfolio optimization, energy systems, and job scheduling, have been proposed. Given the lack of investigation into QAOA's application to TCO problems, and motivated by the computational challenges of TCO problems and the potential of QAOAs, we present IGDec-QAOA to formulate a TCO problem as a QAOA problem and solve it on both ideal and noisy quantum computer simulators, as well as on a real quantum computer. To solve bigger TCO problems that require many qubits, which are unavailable currently, we integrate a problem decomposition strategy with the QAOA. We performed an empirical evaluation with five TCO problems and four publicly available industrial datasets from ABB, Google, and Orona to compare various configurations of IGDec-QAOA, assess its decomposition strategy of handling large datasets, and compare its performance with classical algorithms (i.e., GA and Random Search). Based on the evaluation results achieved on an ideal simulator, we recommend the best configuration of our approach for TCO problems. We also demonstrate that it can reach the same effectiveness as GA and outperform GA in two out of five test case optimization problems. In addition, we observe that, on a noisy simulator, IGDec-QAOA achieved similar performance to that from an ideal simulator. Finally, we demonstrate the feasibility of IGDec-QAOA on a real quantum computer in the presence of noise.