Low-Depth Fixed-Layer Variational Quantum Eigensolver for Small Transverse-Field Ising Ground States

Abstract

Exact diagonalization provides exact spectra and ground-state references for small quantum many-body systems, but its cost grows rapidly with Hilbert-space dimension. This paper studies a fixed low-depth unitary circuit as a variational ansatz for approximating ground states of small transverse-field Ising-model (TFIM) systems. Each of the N qubits contains three rotation gates in each of the L layers and each rotation gate contains a trainable parameter, yielding a total of 3NL tunable parameters. And a PyTorch-based Variational Quantum Eigensolver (PyTorch-VQE), optimized with Adam and automatic differentiation, is evaluated in CPU simulations for open chains with different system sizes N, nearest-neighbor coupling strengths J, transverse-field strengths h, circuit depths L, and five fixed initializations per setting. Exact diagonalization supplies the ground-state energy and vector used to evaluate the variational approximation. Across the 27 tested settings, the three-depth circuit gives a mean relative ground-state energy error of \(0.598\%\) and a mean fidelity of \(90.466\%\). These results demonstrate that the selected fixed-depth ansatz can well approximates ground states for the tested small TFIM systems.