Variational Quantum Eigensolver Identifies Phase Transition Boundary in Long-Range XXZ Chain

Understanding the behaviour of complex quantum systems presents a significant challenge in modern physics, and determining the boundaries between different phases of matter is crucial for materials discovery. Mrinal Dev and Shraddha Sharma, both from the Department of Physics and Astronomy at the National Institute of Technology Rourkela, investigate these phase transitions within the long-range XXZ model, a system known for its complex quantum properties. They employ the variational quantum eigen solver, a powerful computational technique, to pinpoint the precise boundaries between phases, even those exhibiting subtle, infinite-order transitions that are notoriously difficult to characterise using traditional methods. This research demonstrates a novel approach to leveraging quantum computation for materials science, offering a new pathway to understand and predict the behaviour of complex quantum materials by carefully analysing the accuracy of ground state energy calculations.

Finite order phase transitions typically reveal themselves through changes in finite-order derivatives of the ground state energy. However, infinite order phase transitions present a challenge, as they require a globally ranged order parameter for detection and are conventionally difficult to evaluate using ground state energy alone. This work investigates the long-range XXZ (LRXXZ) chain, a system known to exhibit two distinct types of phase transition. One is a first-order phase transition, which can be readily evaluated by examining the gradient of the ground energy. The second is an infinite order phase transition, which poses significant challenges for conventional evaluation methods based on ground state energy analysis.

Phase Transitions Mapped with Quantum Eigen Solver

Scientists have achieved a breakthrough in identifying phase transitions in complex quantum systems using a variational quantum eigen solver (VQE). Their work focuses on the long-range XXZ (LRXXZ) chain, a model system exhibiting subtle phase changes that are classically difficult to compute. The team designed a novel approach leveraging the ground state energy obtained from the VQE to pinpoint the boundaries between different phases of matter. This method relies on a specifically crafted quantum circuit, or ansatz, whose accuracy is sensitive to the phase of the system being evaluated. The core of the technique involves designing an ansatz circuit that maintains a constant net spin throughout the optimization process.

This constraint ensures the estimated ground state energy exhibits a distinct response depending on the phase, producing smaller random errors in certain phases compared to others. By carefully analyzing changes in the error of the ground state energy calculated by the VQE, the researchers successfully determined the phase boundaries. Experiments revealed that by increasing the depth of the optimization circuit, they could accurately evaluate the ground energy of the LRXXZ chain for a coupling constant, J, equal to -1. Further analysis involved comparing the behaviour of the energy gradient and energy gap across phase transition boundaries using exact diagonalisation.

The team introduced directional coherence as a powerful probe for detecting these transitions, calculating it by analysing the gradient vectors of the energy difference between exact diagonalisation and VQE results. Measurements confirm that the directional coherence effectively differentiates between phases, exhibiting a clear change in behaviour at the phase transition boundaries. For example, the team observed a transition from a paramagnetic (PM) to a ferromagnetic (FM) phase at ∆ = -1, evidenced by a distinct change in the orientation of the gradient vectors and a corresponding strip in the directional coherence phase diagram. The team also demonstrated that the energy gap and energy gradient serve as excellent indicators of phase transitions, aligning with existing mean-field theory results. Specifically, the directional coherence, energy gradient, and energy gap all converged on the same phase transition point, providing strong validation of the technique. These results demonstrate a significant advancement in the ability to characterize complex quantum systems and open new avenues for exploring exotic phases of matter.

Quantumly Detecting Material Phase Transitions

Researchers have demonstrated that the variational quantum eigensolver can accurately identify phase transitions in materials. Their work focuses on the long-range XXZ model, a system exhibiting both conventional and more subtle, infinite-order phase transitions, which are notoriously difficult to detect using traditional methods. The team designed a specific computational circuit that allows the accuracy of the energy calculations to be sensitive to changes in the material’s phase, enabling the identification of these boundaries. By comparing the energy calculated using the quantum method with the exact energy of the system, the researchers successfully pinpointed both first-order and infinite-order phase transitions.

Importantly, they observed that the behaviour of the energy gradient clearly distinguishes between different phases, aligning with predictions from mean-field theory. This achievement demonstrates that identifying phase transitions does not necessarily require detailed knowledge of the system’s behaviour, but can be accomplished by focusing solely on the ground state energy. Future work could explore the application of this approach to other materials and transitions where the ground state symmetries change, offering a new avenue for materials discovery and characterisation. The team suggests that by adapting the computational circuit to suit the problem, this method holds promise as a versatile tool for identifying phase transitions in a wide range of physical systems.