The present invention is directed to systems and methods of providing universal quantum computation that avoid certain external control fields that either are hard or impossible to implement, or are serious sources of decoherence (errors). The systems and methods extend the set of scalable physical platforms suitable for implementing quantum computation in solid state, condensed matter and atomic and molecular physics systems. The invention includes identifying of suitable encodings of logical qubits into three physical qubits--i.e. three quantum mechanical systems of two levels--and performing quantum computing operations by changing the quantum states of physical qubits making up one or more logical qubits using only generalized anisotropic exchange interactions. This includes performing a quantum unitary operation over a single logical qubit or a non-local (entangling) two-qubit unitary operation. An exemplary embodiment of the invention uses a physical qubit represented by two electronic quantum levels of a nanoparticle supported in an electromagnetic cavity. The physical qubit, i.e. its quantum levels, can be tuned by electromagnetic field from a pulsed laser, so that the qubit can be brought into the interaction (resonance) with another physical qubit via exchange of cavity quantum electromagnetic modes.