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The rapid development of quantum computing technologies already made itpossible to manipulate a collective state of several dozen of qubits. Thissuccess poses a strong demand on efficient and reliable methods forcharacterization and verification of large-scale many-body quantum states.Traditional methods, such as quantum tomography, which require storing andoperating wave functions on classical computers, become problematic to use inthe regime of large number of degrees of freedom. In this paper, we propose anumerically cheap procedure to describe and distinguish quantum states which isbased on a limited number of simple projective measurements in at least twodifferent bases and computing inter-scale dissimilarities of the resultingbit-string patterns via coarse-graining. The information one obtains throughthis procedure can be viewed as a "hash function" of quantum state -- a simpleset of numbers which is specific for a concrete many-body wave function and canbe used for certification. By studying a number of archetypal examples, we showthat it is enough to characterize quantum states with different structure ofentanglement, including the chaotic quantum states. The connection of thedissimilarity to standard measures of quantum correlations such as von Neumannentropy is discussed. We also demonstrate that our approach can be employed todetect phase transitions of different nature in many-body quantum magneticsystems.

Certification of quantum states with hidden structure of their bitstrings