比較不同古典優化器在量子近似優化演算法解決投資組合最佳化問題

Abstract

投資組合最佳化是金融領域中一個關鍵的問題，投資者需要依靠獨特的策略來管理投資組合，在達到最大化投資報酬的同時最小化風險，屬於一種最佳化問題，但隨著資產種類的增加該最佳化問題會越來越複雜，計算該最佳化問題的時間成本也快速上升，在合理時效內找出最佳解，變的困難許多，與此同時，量子演算法在解決此類最佳化問題上具備節省時間成本的潛力，所以本研究中，我們利用量子近似優化演算法解決投資組合最佳化問題，並想更進一步優化整個過程，我們利用不同古典優化器的收斂特性，比較不同古典優化器對於量子近似優化演算法解決投資組合最佳化問題上的表現。 實驗中我們運用量子近似優化演算法解決投資組合最佳化問題，並設置了相應的限制式，與此同時，我們使用了不同的古典優化器，試圖比較不同古典優化器的收斂結果，我們使用IBM開發的Qiskit（量子電腦開源框架）做模擬分析，該實驗區分成兩大部分。在第一部分，我們將投資組合最佳化問題轉換成二次不受限二進位最佳化問題並利用量子近似優化演算法解決該最佳化問題，第二部分，我們依照不同的量子近似優化演算法層數去設定古典優化器的評估次數上限，並分別探討4種古典優化器（2種無梯度相關古典優化器、2種梯度相關古典優化器）的收斂速度、收斂最終目標值及其對應之測試狀態找到最佳解的機率。 這兩部分的實驗皆用高階量子電腦模擬器Qiskit模擬。而實驗的第二部分，則是在不同層數的量子近似優化演算法下模擬無噪音環境及噪音環境。 實驗結果表明，在無噪音環境下，無梯度相關古典優化器具備計算效率的優勢，在平均收斂結果差異不大的情況下，能夠藉由更少的評估次數達成收斂。其中NELDER MEAD古典優化器收斂到的最終目標值全距數也相對較小，代表其收斂最終目標值更為一致，從而確保了最終目標值的品質。我們認為在上述的四種的古典優化器之中，無噪音環境下，NELDER MEAD 古典優化器在量子近似優化演算法中解決投資組合最佳化問題是最為合適的選擇之一。而在有噪音環境下，四種古典優化器的性能都受到影響。因為噪音環境影響，收斂最終目標值變大，且收斂最終目標值的全距數也有增加的趨勢，找到最佳解的機率相對於無噪音環境也顯著降低。然而，相對而言，ADAM古典優化器則是在噪音環境下提供了相對較好的結果。另外實驗中也發現隨著量子近似優化演算法的層數增加，參數空間也變得更加複雜。因此，除了選擇適合的古典優化器外，初始參數，也有機會影響量子近似優化演算法的計算效率和結果品質。

Portfolio optimization is a critical issue in the financial field, where investors need to rely on unique strategies to manage their portfolios, aiming to maximize returns while minimizing risk. This is an optimization problem, but as the number of asset increases, the complexity of this optimization problem also grows, and the computational cost rises rapidly. Finding the optimal solution within a reasonable time frame becomes much more challenging. Meanwhile, quantum algorithms have the potential to save time costs in solving such optimization problems. Therefore, in this study, we use the Quantum Approximate Optimization Algorithm (QAOA) to address the portfolio optimization problem and aim to further enhance the entire process. We leverage the convergence characteristics of different classical optimizers to compare their performance in solving the portfolio optimization problem using the QAOA. In the experiment, we applied the QAOA to solve the portfolio optimization problem and set up the corresponding constraints. Simultaneously, we used different classical optimizers to compare their convergence results. We conducted simulation analysis using Qiskit, an open-source quantum computing framework developed by IBM. The experiment was divided into two main parts. In the first part, we transformed the portfolio optimization problem into a Quadratic Unconstrained Binary Optimization (QUBO) problem and solved it using the QAOA. In the second part, we set the evaluation constraint of classical optimizers based on different layers of the QAOA. We explored the convergence speed, final convergence objective value, and the probability of finding the optimal solution under the test conditions of four classical optimizers (two gradient-free and two gradient-based classical optimizers). Both parts of the experiment were simulated using the advanced quantum computer simulator Qiskit. In the second part of the experiment, we simulated both noiseless and noisy environments under different layers of the QAOA. The experimental results show that in a noiseless environment, gradient-free classical optimizers have an advantage in computational efficiency, achieving convergence with fewer evaluations while having similar average convergence results. And the NELDER MEAD classical optimizer had a relatively smaller range of final objective values, indicating more consistent final objective values, thereby ensuring the quality of the final results. We believe that among the four classical optimizers, using the NELDER MEAD classical optimizer in the QAOA to solve the portfolio optimization problem is most suitable in a noiseless environment. In a noisy environment, the performance of all four classical optimizers is affected. Due to the influence of noise, the final objective values become larger, and the range of these values also tends to increase, with the probability of finding the optimal solution significantly lower than in a noiseless environment. However , the ADAM classical optimizer provided comparatively better results in a noisy environment. The experiment also found that as the layers of the QAOA increase, the parameter space becomes more complex. Therefore, in addition to choosing a suitable classical optimizer, the initial parameters can also impact the computational efficiency and quality of the results of the QAOA.