Portfolio optimization, rooted in the principles of Mod ern Portfolio Theory (MPT) as introduced by Harry Markowitz in 1952, seeks to maximize expected returns while minimizing risk through the strategic allocation of assets. This process typically involves the use of a covari ance matrix to analyze the relationships between various assets, alongside metrics such as the Sharpe ratio to eval uate performance. However, the complexity of these op timization problems often categorizes them as NP-hard, presenting significant challenges in computational feasi bility. Traditional methods, including Monte Carlo simu lations and maximum diversification techniques, struggle to efficiently navigate the vast solution space inherent in large portfolios. In recent years, advancements in quantum computing have opened new avenues for tackling these complex op timization problems. Notable contributions from orga nizations such as IBM and D-Wave have demonstrated the potential of quantum algorithms to outperform clas sical approaches. Quantum algorithms, such as the Vari ational Quantum Eigensolver (VQE), offer unique advan tages in optimizing asset allocation and risk management by leveraging quantum superposition and entanglement. These techniques can explore multiple solutions si multaneously, significantly reducing the time required to identify optimal strategies in dynamic market conditions. As the financial industry continues to embrace these in novations, we can expect a paradigm shift in how invest ment strategies are developed and executed, ultimately leading to more robust and resilient portfolios. Thus, it makes sense that the motivation behind this challenge lies in understanding the potential of quantum comput ing to revolutionize traditional financial models and en hance decision-making processes through an analysis of time and space complexity as well as scalability
!pip install qiskit
!pip install qiskit_finance
!pip install qiskit_optimization
!pip install qiskit_aer
!pip install qiskit_ibm_runtime
!pip install qiskit_algorithms
!pip install scienceplots