Ground state energy and phase transitions of the long-range XXZ chain using variational quantum eigensolver

The variational quantum eigensolver (VQE) has been widely used to find the ground state energy of different Hamiltonians with no analytical solutions that are classically difficult to compute. In our work, we use the VQE to identify the phase transition boundary for an infinite-order phase transition. Typically, in a finite-order phase transition, finite-order derivatives of the ground state energy would signal the phase transition. However, that is not the case for infinite-order phase transitions, for which a global-range order parameter is required. In this work, we use a long-range XXZ (LRXXZ) chain for our study. It has been observed that two types of phase transitions for this model exist. One is a first-order phase transition, straightforwardly evaluated using the gradient of the ground energy. The second is an infinite-order phase transition, which is conventionally unfeasible to evaluate using the ground state energy. Therefore, it has been observed that the ground state energy is not sufficient to probe both of these transition boundaries. However, we propose a simple technique to utilize the ground state energy from the VQE to identify both phase transitions. The idea stems from the argument that the VQE requires an ansatz circuit; therefore, the accuracy of the VQE relies on this ansatz circuit. We have designed the ansatz circuit such that the estimated ground state energy is sensitive to the phase it is evaluated in. It is achieved by applying the constraint that the net spin remains constant throughout the optimization process. Consequently, the ansatz works in a certain phase in which it gives a relatively small random error, as it should, when compared to the error in the ground state energy calculations of the other phases, in which the ansatz fails. Identifying these changes in the behavior of the error in the ground state energy evaluation using the VQE, we are able to identify the phase boundaries. Using Exact Diagonalization (ED), we also compare the behavior of the energy gradient and energy gap across both phase transition boundaries for this model. Further, by increasing the depth of the optimization circuit, we also accurately evaluate the ground energy of the LRXXZ chain for $J>0$ and $J<0$ in the paramagnetic region.