  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.
