  In an ideal quantum measurement, the wave function of a quantum system
collapses to an eigenstate of the measured observable, and the corresponding
eigenvalue determines the measurement outcome. If the observable commutes with
the system Hamiltonian, repeated measurements yield the same result and thus
minimally disturb the system. Seminal quantum optics experiments have achieved
such quantum non-demolition (QND) measurements of systems with few degrees of
freedom. In contrast, here we describe how the QND measurement of a complex
many-body observable, the Hamiltonian of an interacting many-body system, can
be implemented in a trapped-ion analog quantum simulator. Through a single-shot
measurement, the many-body system is prepared in a narrow band of (highly
excited) energy eigenstates, and potentially even a single eigenstate. Our QND
scheme, which can be carried over to other platforms of quantum simulation,
provides a framework to investigate experimentally fundamental aspects of
equilibrium and non-equilibrium statistical physics including the eigenstate
thermalization hypothesis and quantum fluctuation relations.
