Application of the Quantum Equation of Motion to the Lipkin-Meshkov-Glick model.

Classical numerical methods for solving the nuclear many-body problem face the impediment of exponential growth of the dimension of the Hilbert space. Quantum algorithms have become an attractive alternative for practitioners. However, due to the limitations of current quantum hardware, a class of hybrid classical-quantum algorithms has been developed to achieve quantum advantage. The Quantum Equation of Motion (qEOM), which is an extension of the Variational Quantum Eigensolver (VQE), is one of the hybrid algorithms recently proposed to calculate excitation energies of molecular Hamiltonians. In this work, we apply the qEOM to find the excitation energies of the Lipkin-Meshkov-Glick (LMG) Hamiltonian and benchmark its accuracy compared with the Hartree-Fock (HF), Random Phase Approximation (RPA), second RPA, and exact analytical solution. We found that the qEOM accurately produced the energy spectrum of the LMG Hamiltonian and has a reasonable scaling of less than O(N^1.9) with particle number N. We discuss possible ways to improve the qEOM method by using the qubit-ADAPT-VQE to reduce the circuit depth, and the Finite Amplitude Method to avoid diagonalizing the large classical matrix equation.