Tower of Structured Excited States from Measurements

Preparing highly entangled quantum states is a key challenge in quantum metrology and quantum information science. Measurements, especially those of global observables, offer a simple and efficient way to generate entanglement between subsystems when they are measured as a whole. We introduce a log-depth protocol leveraging quantum phase estimation to measure a global observable, such as total magnetization and momentum. We demonstrate its capability to prepare towers of structured excited states that are useful in quantum metrology; examples include quantum many-body scars in various models, including the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, the constrained domain-wall model, and the spin-$\frac{1}{2}$ and spin-$1$ XX chains. The same method is also applicable to preparing the Dicke states of high weight. In addition, we propose a protocol for momentum measurement that avoids disturbing the system, facilitating the preparation of states beyond the above construction, such as the Arovas $A$ state of the AKLT Hamiltonian. Our results expand the utility of measurement-based approaches to accessing highly entangled states in quantum many-body systems.