σ-VQE: Excited-state preparation of quantum many-body scars with shallow circuits

We present and benchmark a type of variational quantum eigensolver (VQE), which we denote the σ-VQE. It is designed to target mid-spectrum eigenstates and prepare quantum many-body scar states. The approach leverages the fact that noisy intermediate-scale quantum devices are limited in their ability to generate generic highly-entangled states. This modified VQE pairs a low-depth circuit with an energy-selective objective that explicitly penalizes energy variance around a chosen target energy. The cost function exploits the limited expressibility of the shallow circuit as atypical low-entanglement eigenstates such as scar states are preferentially selected. We validate this mechanism across two complementary families of models that contain many-body scar states: the Shiraishi-Mori embedding approach, and the matrix-product state parent Hamiltonian construction. We define an unbiased estimation scheme for the nonlinear cost function that is compatible with qubit-wise commuting grouping and bitstring reuse. A proof-of-principle demonstration using a small-system instance was carried out on IBM Fez (Heron r2 QPU). These results motivate its use both as a practical "scar detector" and as a state-preparation primitive for initializing nonthermal eigenstate-supported dynamics.