$\sigma$-VQE: Excited-state preparation of quantum many-body scars with shallow circuits

TL;DR

The paper introduces $\sigma$-VQE, a shallow-circuit variational algorithm designed to efficiently prepare and detect quantum many-body scar states on noisy quantum devices.

Contribution

It proposes a novel energy-selective VQE approach that targets low-entanglement scar states using limited circuit depth and validates it on models and a real quantum device.

Findings

Successfully prepared scar states on IBM Fez quantum processor.

Demonstrated the method's ability to target mid-spectrum eigenstates.

Validated the approach across different models with many-body scars.

Abstract

We present and benchmark a type of variational quantum eigensolver (VQE), which we denote $\sigma$-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 a 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 performed on IBM Fez (Heron r2 QPU). These results motivate its use as a practical algorithm for detecting quantum many-body scars and variationally generating states with appreciable scar state overlap.