Single-shot quantum measurements sketch quantum many-body states

Quantum measurements allow us to observe quantum many-body systems consisting of a multitude of microscopic degrees of freedom. However, the quantum uncertainty and the exponentially large Hilbert space pose natural barriers to simple interpretations of the quantum measurement outcomes. We propose a nonlinear "measurement energy" based upon the measurement outcomes and a general approach akin to quantum machine learning to extract the most probable states (maximum likelihood estimates), naturally reconciling noncom -muting observables and getting more out of the quantum measurements. Compatible with established quantum many-body ansatzes and efficient optimization, our strategy offers state-of-art capacity with control and full information. We showcase the versatility and accuracy of our strategy on random long-range fermion and Kitaev quantum spin-liquid models, where smoking-gun signatures were lacking.