One-way quantum computing achieves the full power of quantum computation by

performing single particle measurements on some many-body entangled state,

known as the resource state. As single particle measurements are relatively

easy to implement, the preparation of the resource state becomes a crucial

task. An appealing approach is simply to cool a strongly correlated quantum

many-body system to its ground state. In addition to requiring the ground state

of the system to be universal for one-way quantum computing, we also want the

Hamiltonian to have non-degenerate ground state protected by a fixed energy

gap, to involve only two-body interactions, and to be frustration-free so that

measurements in the course of the computation leave the remaining particles in

the ground space. Recently, significant efforts have been made to the search of

resource states that appear naturally as ground states in spin lattice systems.

The approach is proved to be successful in spin-5/2 and spin-3/2 systems. Yet,

it remains an open question whether there could be such a natural resource

state in a spin-1/2, i.e., qubit system. Here, we give a negative answer to

this question by proving that it is impossible for a genuinely entangled qubit

states to be a non-degenerate ground state of any two-body frustration-free

Hamiltonian. What is more, we prove that every spin-1/2 frustration-free

Hamiltonian with two-body interaction always has a ground state that is a

product of single- or two-qubit states, a stronger result that is interesting

independent of the context of one-way quantum computing.