Learning and testing quantum states of fermionic systems

Abstract: The experimental realization of increasingly complex quantum states in quantum devices underscores the pressing need for new methods of state learning and verification. Among the various classes of quantum states, fermionic systems hold particular significance due to their crucial roles in physics. Despite their importance, research on learning quantum states of fermionic systems remains surprisingly limited. In our work, we aim to present a comprehensive rigorous study on learning and testing states of fermionic systems. We begin by analyzing arguably the simplest important class of fermionic states—free-fermionic states—and subsequently extend our analysis to more complex fermionic states. We meticulously delineate scenarios in which efficient algorithms are feasible, providing experimentally practical algorithms for these cases, while also identifying situations where any algorithm for solving these problems must be inherently inefficient. At the same time, we present novel fundamental results of independent interest on fermionic systems, with additional applications beyond learning and characterizing quantum devices, such as many-body physics, resource theory of non-Gaussianity, and circuit compilation strategies. (Talk based on https://arxiv.org/pdf/2409.17953 , https://arxiv.org/pdf/2402.18665)