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Projected least squares is an intuitive and numerically cheap technique for quantum state tomography: compute the least-squares estimator and project it onto the space of states. The main result of this paper equips this point estimator with rigorous, non-asymptotic convergence guarantees expressed in terms of the trace distance. The estimator’ssample complexityis comparable to the strongest convergence guarantees available in the literature and—in the case of the uniform POVM—saturates fundamental lower bounds. Numerical simulations support these competitive features.

 

Interest in high-spin organic materials is driven by opportunities to enable far-reaching fundamental science and develop technologies that integrate light element spin, magnetic, and quantum functionalities. Although extensively studied, the intrinsic instability of these materials complicates synthesis and precludes an understanding of how fundamental properties associated with the nature of the chemical bond and electron pairing in organic materials systems manifest in practical applications. Here, we demonstrate a conjugated polymer semiconductor, based on alternating cyclopentadithiophene and thiadiazoloquinoxaline units, that is a ground-state triplet in its neutral form. Electron paramagnetic resonance and magnetic susceptibility measurements are consistent with a high-to-low spin energy gap of 9.30 × 10 −3 kcal mol −1 . The strongly correlated electronic structure, very narrow bandgap, intramolecular ferromagnetic coupling, high electrical conductivity, solution processability, and robust stability open access to a broad variety of technologically relevant applications once thought of as beyond the current scope of organic semiconductors.