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We theoretically investigate the nature of the state at quarter filled lowestLandau level and predict that, as the quantum well width is increased, atransition occurs from the composite fermion Fermi sea into a novel non-Abelianfractional quantum Hall state that is topologically equivalent to $f$-wavepairing of composite fermions. This state is topologically distinct from thefamiliar $p$-wave paired Pfaffian state. We compare our calculated phasediagram with experiments and make predictions for many observable quantities.

Prediction of a non-Abelian fractional quantum Hall state with $f$-wave pairing of composite fermions in wide quantum wells