In this work, energy levels of the Majumdar-Ghosh model (MGM) are analyzed up to 15 spins chain in the noisy intermediate-scale quantum framework using noisy simulations. This is a useful model whose exact solution is known for a particular choice of interaction coefficients. We have solved this model for interaction coefficients other than that required for the exactly solvable conditions as this solution can be of help in understanding the quantum phase transitions in complex spin chain models. The solutions are obtained using quantum approximate optimization algorithms (QAOA), and variational quantum eigensolver (VQE). To obtain the solutions, the one-dimensional lattice network is mapped to a Hamiltonian that corresponds to the required interaction coefficients among spins. Then, the ground states energy eigenvalue of this Hamiltonian is found using QAOA and VQE. Further, the validity of the Lieb-Schultz-Mattis theorem in the context of MGM is established by employing variational quantum deflation to find the first excited energy of MGM. Solution for an unweighted Max-cut graph for 17 nodes is also obtained using QAOA and VQE to know which one of these two techniques performs better in a combinatorial optimization problem. Since the variational quantum algorithms used here to revisit the Max-cut problem and MGM are hybrid algorithms, they require classical optimization. Consequently, the results obtained using different types of classical optimizers are compared to reveal that the QNSPSA optimizer improves the convergence of QAOA in comparison to the SPSA optimizer. However, VQE with EfficientSU2 ansatz using the SPSA optimizer yields the best results.

翻译：本文在含噪中等规模量子计算框架下，利用含噪模拟分析了最多15个自旋链的 Majumdar-Ghosh 模型（MGM）的能级。该模型在特定相互作用系数选择下存在精确解，是一个具有研究价值的模型。我们针对非精确可解条件的相互作用系数求解该模型，该解有助于理解复杂自旋链模型中的量子相变。通过量子近似优化算法（QAOA）和变分量子本征求解器（VQE）获得解。为求解该模型，将一维晶格网络映射为对应自旋间所需相互作用系数的哈密顿量，随后利用QAOA和VQE求得该哈密顿量的基态能量本征值。进一步，通过运用变分量子退平方法寻找MGM的第一激发态能量，验证了 Lieb-Schultz-Mattis 定理在MGM中的有效性。同时，利用QAOA和VQE求解17节点无权重最大割图问题，以评估这两种技术在组合优化问题中的性能优劣。由于本文用于重访最大割问题和MGM的变分量子算法属于混合算法，需要经典优化步骤。因此，通过比较不同类型经典优化器所得结果，发现与SPSA优化器相比，QNSPSA优化器能提升QAOA的收敛性。然而，采用EfficientSU2拟设的VQE配合SPSA优化器取得了最佳结果。