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 noncommuting 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.

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• Quantum algorithms
• Quantum measurements
• Quantum spin liquid
• Quantum tomography
• Topological order
• Quantum many-body systems
• Machine learning