  A central challenge in analog quantum simulation is to characterize desirable
physical properties of quantum states produced in experiments. However, in
conventional approaches, the extraction of arbitrary information requires
performing measurements in many different bases, which necessitates a high
level of control that present-day quantum devices may not have. Here, we
propose and analyze a scalable protocol that leverages the ergodic nature of
generic quantum dynamics, enabling the efficient extraction of many physical
properties. The protocol does not require sophisticated controls and can be
generically implemented in analog quantum simulation platforms today. Our
protocol involves introducing ancillary degrees of freedom in a predetermined
state to a system of interest, quenching the joint system under Hamiltonian
dynamics native to the particular experimental platform, and then measuring
globally in a single, fixed basis. We show that arbitrary information of the
original quantum state is contained within such measurement data, and can be
extracted using a classical data-processing procedure. We numerically
demonstrate our approach with a number of examples, including the measurements
of entanglement entropy, many-body Chern number, and various superconducting
orders in systems of neutral atom arrays, bosonic and fermionic particles on
optical lattices, respectively, only assuming existing technological
capabilities. Our protocol excitingly promises to overcome limited
controllability and, thus, enhance the versatility and utility of near-term
quantum technologies.
