  Level spectroscopy stands as a powerful method for identifying the transition
point that delineates distinct quantum phases. Since each quantum phase
exhibits a characteristic sequence of excited states, the crossing of energy
levels between low-lying excited states offers a reliable mean to estimate the
phase transition point. While approaches like the Variational Quantum
Eigensolver are useful for approximating ground states of interacting systems
using quantum computing, capturing low-energy excitations remains challenging.
In our study, we introduce an equivariant quantum circuit that preserves the
total spin and the translational symmetry to accurately describe singlet and
triplet excited states in the $J_1$-$J_2$ Heisenberg model on a chain, which
are crucial for characterizing its transition point. Additionally, we assess
the impact of noise on the variational state, showing that conventional
mitigation techniques like Zero Noise Extrapolation reliably restore its
physical properties.
