Wave Function Network Description and Kolmogorov Complexity of Quantum Many-Body Systems

Programmable quantum devices are now able to probe wave functions atunprecedented levels. This is based on the ability to project the many-bodystate of atom and qubit arrays onto a measurement basis which producessnapshots of the system wave function. Extracting and processing informationfrom such observations remains, however, an open quest. One often resorts toanalyzing low-order correlation functions - i.e., discarding most of theavailable information content. Here, we introduce wave function networks - amathematical framework to describe wave function snapshots based on networktheory. For many-body systems, these networks can become scale free - amathematical structure that has found tremendous success in a broad set offields, ranging from biology to epidemics to internet science. We demonstratethe potential of applying these techniques to quantum science by introducingprotocols to extract the Kolmogorov complexity corresponding to the output of aquantum simulator, and implementing tools for fully scalable cross-platformcertification based on similarity tests between networks. We demonstrate theemergence of scale-free networks analyzing data from Rydberg quantum simulatorsmanipulating up to 100 atoms. We illustrate how, upon crossing a phasetransition, the system complexity decreases while correlation length increases- a direct signature of build up of universal behavior in data space. Comparingexperiments with numerical simulations, we achieve cross-certification at thewave-function level up to timescales of 4 μ s with a confidence level of90accuracy. Our framework is generically applicable to the output of quantumcomputers and simulators with in situ access to the system wave function, andrequires probing accuracy and repetition rates accessible to most currentlyavailable platforms.