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Arrays of trapped ultracold molecules represent a promising platform forimplementing a universal quantum computer. DeMille has detailed a prototypedesign based on Stark states of polar $^1\Sigma$ molecules as qubits. Herein,we consider an array of polar $^2\Sigma$ molecules which are, in addition,inherently paramagnetic and whose Hund's case (b) free-rotor states are Bellstates. We show that by subjecting the array to combinations of concurrenthomogeneous and inhomogeneous electric and magnetic fields, the entanglement ofthe array's Stark and Zeeman states can be tuned and the qubit sites addressed.Two schemes for implementing an optically controlled CNOT gate are proposed andtheir feasibility discussed in the face of the broadening of spectral lines dueto dipole-dipole coupling and the inhomogeneity of the electric and magneticfields.

Prospects for Quantum Computing with an Array of Ultracold Polar Paramagnetic Molecules