Revisiting Majumdar-Ghosh spin chain model and Max-cut problem using variational quantum algorithms

TL;DR

This work benchmarks variational quantum algorithms (QAOA and VQE) on the Majumdar-Ghosh spin-chain and a 17-node Max-cut graph to understand how circuit depth, ansatz choice, and optimizers affect performance in both noiseless and noisy scenarios. By comparing ground- and first-excited-state energies (via VQD) against classical solvers, and by analyzing optimizer behavior (SPSA, SLSQP, QNSPSA), the authors reveal that VQE often offers more robust convergence on NISQ-like hardware, while QAOA can approach exact results with sufficiently deep circuits in ideal conditions. The Max-cut benchmark corroborates the trend that VQE can outperform QAOA for certain COPs, and the results support the Lieb-Schultz-Mattis expectations for MGM. Collectively, the paper highlights practical trade-offs between circuit depth, parameter count, and noise, informing the selection of ansätze and optimizers for near-term quantum applications in condensed matter and combinatorial optimization.

Abstract

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.

Figures (17)

• Figure 1: Schematic flow chart of VQA framework.
• Figure 2: A spin ladder representation of MGM with $J$ and $\alpha J$
• Figure 3: (Color online) Efficient SU2 ansatz (repetition = 1, also ' reps' in qiskit to denote the repetitions of the ansatz) for 3 spins. Here, $R_x, R_y, R_z$ are the single qubit rotation gates that rotate the state of the qubit with $\theta$ degrees about $x,y,z$ axis respectively.
• Figure 4: (Color online) QAOA ansatz (reps = 1) for 3 spins. The quantum circuit shown above is a single circuit where the output of the top (middle) panel is to be viewed as the input of the middle (lower) panel. Here, U is a single-qubit rotation gate with 3 Euler angles named as $\theta,\phi,\lambda$.
• Figure 5: (Color online) Ground state energies comparison from 4 to 15 spins chain.