Graph states as ground states of many-body spin-1∕2Hamiltonians

Abstract

We consider the problem whether graph states can be ground states of local interaction Hamiltonians. For Hamiltonians acting on n qubits that involve at most two-body interactions, we show that no n-qubit graph state can be the exact, non-degenerate ground state. We determine for any graph state the minimal d such that it is the non-degenerate ground state of a d-body interaction Hamiltonian, while we show for d ′ -body Hamiltonians H with d ′ < d that the resulting ground state can only be close to the graph state at the cost of H having a small energy gap relative to the total energy. When allowing for ancilla particles, we show how to utilize a gadget construction introduced in the context of the k-local Hamiltonian problem, to obtain n-qubit graph states as non-degenerate (quasi-)ground states of a two-body Hamiltonian acting on n ′ > n spins.

Remarks on duality transformations and generalized stabilizer states

“…The transition between those two regimes is continuous and thus we observe that the states |Ψ can realize local entropies ranging continuously in the interval [0,1]. This is in contrast to standard stabilizer states whose local entropies are quantized in integer units which in itself is enough to see that standard stabilizer states will be ground states to a very limited set of Hamiltonians (see [24] for a much more detailed discussion).…”

Quantum computational capability of a 2D valence bond solid phase

“…[11][12][13][14][15][16], are constructed to have such a convenient yet artificial property -as often referred as one of peculiar properties of the correlations of the 2D cluster state -that it is possible to decouple deterministically (by measurements of only neighboring sites) a 1D-chain structure that encodes the direction of a simulated time as a quantum logical wire of the quantum circuit model. This peculiarity is said to be artifact of another less realistic feature of the 2D cluster state in that it cannot be the exact ground state of any two-body Hamiltonian of spin 1 2 's [17,18]. Thus one cannot expect such convenience for the correlations of a genuine 2D ground state of a naturally-occurring spin system.…”

Measurement-Based Quantum Computing With Valence-Bond-Solids

“…These results herald bad news for the practical realization of MBQC based on VBS resource states since spin-1/2 systems appear to be the most prevalent systems in nature. One could try to overcome this situation by relaxing the requirement for "frustration-free" or "uniqueness" and resort to some other physical mechanisms, such as topological protection [34,35] or perturbation [36,37,38]. Finally it is worth noting that it is still an open question whether one can find a realistic spin-1 VBS resource state for MBQC with a two-body Hamiltonian which is both frustration-free and gapped.…”