The ground states of some many-body quantum systems can serve as resource states for the one-way quantum computing model, achieving the full power of quantum computation. Such resource states are found, for example, in spin-$\frac{5}{2}$ and spin-$\frac{3}{2}$ systems. It is, of course, desirable to have a natural resource state in a spin-$\frac{1}{2}$, that is, qubit system. Here, we give a negative answer to this question for frustration-free systems with two-body interactions. In fact, it is shown to be impossible for any genuinely entangled qubit state to be a nondegenerate ground state of any two-body frustration-free Hamiltonian. What is more, we also prove that every spin-$\frac{1}{2}$ frustration-free Hamiltonian with two-body interaction always has a ground state that is a product of single- or two-qubit states. In other words, there cannot be any interesting entanglement features in the ground state of such a qubit Hamiltonian.

• Received 30 May 2010

DOI:https://doi.org/10.1103/PhysRevA.83.050301