  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.
