Ground state energy and phase transitions of Long-range XXZ using VQE

TL;DR

This work uses a variational quantum eigensolver (VQE) to study the ground-state energy and phase transitions of the Long-Range XXZ (LRXXZ) model. By constructing a spin-conserving ansatz and evaluating the energy-difference error $E_d = E_{exact} - E_{VQE}$, the authors identify phase boundaries, including an infinite-order PM–AFM transition, through directional coherence in the gradient of $E_d$ across parameters $\Delta$ and $\alpha$. Comparisons with exact diagonalization validate the approach, and depth-2 VQE achieves accurate ground-state energies for $J=\pm 1$, demonstrating that quantum algorithms can reveal subtle phase structure with compact circuits. The results offer a practical route to probe global properties and phase transitions in long-range quantum systems using near-term quantum hardware.

Abstract

The variational quantum eigen solver (VQE), has been widely used to find the ground state energy of different Hamiltonians with no analytical solutions and are classically difficult to compute. In our work, we have used VQE to identify the phase transition boundary for an infinite order phase transition. We use long-range XXZ (LRXXZ) chain for our study. In order to probe infinite order phase transition, we propose to utilise the ground state energy obtained from VQE. The idea rests on the argument that VQE requires an ansatz circuit; therefore, the accuracy of the VQE will rely on this ansatz circuit. We have designed this 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 optimisation process. Consequently, the ansatz works in a certain phase where it gives relatively small random error, as it should, when compared to the error in ground state energy calculations of the other phases, where the ansatz fails. By identifying these changes in the behaviour of the error in ground state energy using VQE, we were able to determine the phase boundaries. Using exact diagonalisation, we also compare the behaviour of the energy gradient and energy gap across both the phase transition boundaries for this model. Further, by increasing the depth of the optimisation circuit, we also accurately evaluate the ground energy of the LRXXZ chain for the value of coupling constant, J equal to -1

Figures (6)

• Figure 1: Circuit representation of initialisation using three-qubits with alternate application of the X-gate.
• Figure 2: Three-qubit ansatz circuit for a single layer.
• Figure 3: Phase diagram of (a) directional coherence, (b) ground state energy gradient, and (c) energy gap. In (a) a dotted line is plotted at the point where the direction of arrow changes, directly coinciding with $\Delta=1$, which marks the phase transition from PM phase to the FM phase.
• Figure 4: Phase diagram of (a) directional coherence, (b) energy gradient, and (c) energy gap, as a function of $\Delta$ and $\alpha$ shows the clear depiction of AFM-PM phase transition in (a) as compared to (b) and (c). The dotted line in (a) separates the region of randomly oriented arrows (AFM phase depicted by blue region) with respect to the organised region (PM phase shown in red).
• Figure 5: The plot of relative error of VQE energy with respect to the exact diagonalization in the PM phase of LRXXZ with $J=-1$.