A Study of Entanglement and Ansatz Expressivity for the Transverse-Field Ising Model using Variational Quantum Eigensolver

The Variational Quantum Eigensolver (VQE) is a leading hybrid quantum-classical algorithm for simulating many-body systems in the NISQ era. Its effectiveness, however, depends on the faithful preparation of eigenstates, which becomes challenging in degenerate and strongly entangled regimes. We study this problem using the transverse-field Ising model (TFIM) with periodic boundary conditions in one, two, and three dimensions, considering systems of up to 27 qubits. We employ different ansatzes: the hardware-efficient EfficientSU2 from Qiskit, the physics-inspired Hamiltonian Variational Ansatz (HVA) and HVA with symmetry breaking, and benchmark their performance using energy variance, entanglement entropy, spin correlations, and magnetization.

💡 Research Summary

The paper presents a systematic benchmark of three variational quantum circuit families—Hardware‑Efficient Ansatz (HEA, specifically Qiskit’s EfficientSU2), Hamiltonian Variational Ansatz (HVA), and a symmetry‑breaking extension of HVA (HVA‑SB)—applied to the transverse‑field Ising model (TFIM) in one, two, and three dimensions with periodic boundary conditions. Using exact state‑vector simulations on NVIDIA CUDA‑Q for up to 27 qubits, the authors evaluate each ansatz in terms of expressivity (via frame potential), optimization landscape (L‑BFGS for the smooth HEA, COBYLA for the more rugged HVA/HVA‑SB), circuit depth (4, 8, 10, 15 layers), and physical observables: ground‑state energy, energy variance, magnetization, spin‑spin correlation, and von Neumann entanglement entropy.

Key findings include: (i) HEA exhibits the lowest frame‑potential values, confirming the highest expressivity, but its energy error improves only gradually with depth and it systematically underestimates entanglement, especially in the low‑field (high‑entanglement) regime. (ii) HVA, built from Trotterized TFIM terms, restricts the variational space to the Hamiltonian’s natural subspace; as a result, its performance shows a sharp transition from poor to accurate energy estimates once a sufficient depth is reached. (iii) Adding a single‑qubit Rz layer (HVA‑SB) breaks the Z₂ parity symmetry, allowing the ansatz to access parity‑violating states that dominate the low‑field sector in finite systems. This leads to lower energy variance and a more faithful reproduction of the entanglement entropy peak near the quantum critical point.

In 1‑D with ten spins, HVA‑SB captures the expected entanglement entropy of ≈ ln 2, while HEA and plain HVA give smaller values. Spin‑correlation functions display a clear change near the critical transverse field, and all ansätze reproduce this qualitative feature, though quantitative accuracy varies. In 2‑D (4 × 4 lattice) optimization becomes more unstable; HVA performs poorly in the low‑entanglement regime, whereas HEA maintains reasonable energy accuracy but continues to underestimate entanglement. In 3‑D lattices, the authors mitigate optimization difficulty by removing Rz rotations (real‑amplitude ansatz) and initializing parameters from nearby field values, achieving size‑independent ground‑state energies and clear entanglement signatures of the phase transition.

Overall, the study highlights a trade‑off: high expressivity (HEA) versus a smoother optimization landscape (HVA) versus the need for symmetry breaking (HVA‑SB) to capture degenerate ground‑state manifolds. The authors suggest future work on adaptive ansatz construction and machine‑learning‑driven optimizers to overcome these limitations and scale VQE to larger, more strongly correlated systems.