Quantum EigenGame for excited state calculation

Quantum computing offers an alternative approach to solving complex computational tasks, potentially reducing the time and space complexity compared to classical methods. Quantum algorithms -like Quantum Phase Estimation [1], the Deutsch-Jozsa algorithm [2], and Grover's algorithm [3]- demonstrate superior performance in ideal, noiseless conditions. However, in the Noisy Intermediate-Scale Quantum (NISQ) era [4], noise remains a significant challenge, influencing the stability and reliability of quantum computations [5-8]. Performing optimization tasks under noisy settings is a common scenario in the algorithmic literature. In optimization and machine learning, errors that propagate throughout iterations critically influence performance metrics and outcomes [9-12]. Understanding and mitigating error propagation is crucial for enhancing the practical utility of algorithms in real-world applications. Particularly relevant to the present work, consider the case of derivative-free optimization (DFO) [13-18]: DFO is employed effectively in scenarios where traditional gradient-based methods falter [16]. However, the efficiency of DFO methods often lags, particularly for high-dimensional problems, due to their reliance on sampling routines that may require many function evaluations to approximate gradients [15]. Further, DFO may struggle with precision near minima [17].