This article reviews the physics of multipolar interactions and multipolar order in $f$-electron systems, using the actinide dioxides as a paradigm. In the past few years, these apparently simple cubic compounds have been studied intensively, and many new phenomena have been discovered. Here the experimental results are discussed together with the current theoretical understanding of multipolar interactions.

The substitution of one metal ion in a Cr-based molecular ring with dominant antiferromagnetic couplings allows the engineering of its level structure and ground-state degeneracy. Here we characterize a ${\mathrm{Cr}}_7\mathrm{Ni}$ molecular ring by means of low-temperature specific-heat and torque-magnetometry measurements, thus determining the microscopic parameters of the corresponding spin Hamiltonian. The energy spectrum and the suppression of the leakage-inducing $S$ mixing render the ${\mathrm{Cr}}_7\mathrm{Ni}$ molecule a suitable candidate for the qubit implementation, as further substantiated by our quantum-gate simulations.

The physical implementation of quantum information processing relies on individual modules—qubits—and operations that modify such modules either individually or in groups—quantum gates. Two examples of gates that entangle pairs of qubits are the controlled NOT-gate (CNOT) gate, which flips the state of one qubit depending on the state of another, and the "Equation missing"gate that brings a two-qubit product state into a superposition involving partially swapping the qubit states. Here we show that through supramolecular chemistry a single simple module, molecular {Cr₇Ni} rings, which act as the qubits, can be assembled into structures suitable for either the CNOT or "Equation missing"gate by choice of linker, and we characterize these structures by electron spin resonance spectroscopy. We introduce two schemes for implementing such gates with these supramolecular assemblies and perform detailed simulations, based on the measured parameters including decoherence, to demonstrate how the gates would operate.

We propose a model based on quadrupolar ordering of localized $f$ electrons to explain the 17.5 K phase transition of U${\mathrm{Ru}}_2$ ${\mathrm{Si}}_2$. The tiny staggered magnetic moment observed by neutron scattering is interpreted as a weak secondary effect associated to the symmetry-breaking perturbation. The model is able to account for the observed behavior of the linear and nonlinear susceptibilities throughout the transition. A connection with the quadrupolar Kondo theory is proposed.

We report the results of resonant x-ray scattering experiments performed at the Np $M_{4,5}$ edges in $\mathrm{N}\mathrm{p}{\mathrm{O}}_{\mathrm{2}}$. Below $T_0=25\mathrm{K}$, the development of long-range order of Np electric quadrupoles is revealed by the growth of superlattice Bragg peaks. The polarization and azimuthal dependence of the intensity of the resonant peaks are well reproduced assuming anisotropic tensor susceptibility scattering from a triple-$\overrightarrow{q}$ longitudinal antiferroquadrupolar structure. Electric-quadrupole order in $\mathrm{N}\mathrm{p}{\mathrm{O}}_{\mathrm{2}}$ could be driven by the ordering at $T_0$ of magnetic octupoles of ${\Gamma }_5$ symmetry, splitting the Np ground state quartet and leading to a singlet ground state with zero dipole-magnetic moment.