Imaginary Time Spectral Transforms for Excited State Preparation

Excited states of many-body quantum systems play a key role in a wide range of physical and chemical phenomena. Despite this, there is a notable lack of scalable algorithms capable of preparing highly excited eigenstates. To address this challenge, we introduce a general approach that directly targets eigenstates near a chosen energy, applicable to both classical and quantum simulation frameworks. Our approach combines the shift-invert mechanism with imaginary time evolution, enabling the construction of excited states of large many-body quantum systems. We demonstrate the technique classically by computing mid-spectrum eigenstates of disordered spin chains with up to $L=128$ sites. Based on this, we propose a hybrid scheme compatible with near-term quantum hardware.