Excitation Gaps of Ground and Excited State Energy of the Fermi-Hubbard Model Using Variational Quantum Eigensolver

The Hubbard model is a challenging quantum many-body problem and serves as a benchmark for quantum computing research. Accurate computation of its ground and excited state energies is essential for understanding correlated electron systems. In this study, the ground, first, and second excited state energies of 4$\times$1 and 2$\times$2 Hubbard lattices are obtained using a newly designed ansatz circuit. The ansatz is constructed by combining concepts from the Hamiltonian Variational Ansatz (HVA) and the Number-Preserving Ansatz (NPA). A hybrid optimization strategy is applied, where COBYLA is used for coarse convergence and L-BFGS for fine-tuning. The resulting energies are evaluated, and the corresponding physical properties of the systems are analyzed through phase diagrams of the energy excitation gaps for different charge and spin configurations.