Preserving universal resources for one-way quantum computing

The common spin Hamiltonians such as the Ising, $XY$, and Heisenberg models do not have eigenstates that are suitable resources for measurement-based quantum computation. Various highly entangled many-body states have been suggested as a universal resource for this type of computation; however, it is not easy to preserve these states in solid-state systems due to their short coherence times. To solve this problem, we propose a scheme for generating a Hamiltonian that has a cluster state as a ground state. Our approach employs a series of pulse sequences inspired by established NMR techniques and holds promise for applications in many areas of quantum information processing.

Figure 1

(a) Switching on and off the parts $H_s$ and $H_{\mathrm{int}}$ of the Hamiltonian $H=H_s+H_{\mathrm{int}}$ gives rise to an effective time evolution described by the stabilizer Hamiltonian $H_{\mathrm{stab}}$ [see Eq. (4)] whose ground state is a cluster state. (b) Graph representation of a one-dimensional four-qubit (squares) cluster state stabilized by the Ising interaction. (c) Two-dimensional $4\times 4$ twisted cluster state stabilized by the $XY$ interaction, a universal resource for one-way quantum computation. The dashed lines and Pauli operators in each direction illustrate the twistedness of the state and the corresponding stabilizer Hamiltonian.

Figure 2

Dependence of the cluster-state fidelity for $\tau =\pi /\left(4\Omega \right)$ on random errors in the pulse duration for one-dimensional $XY$ qubit arrays of length $N=6$, 8, and 10. The width of the Gaussian distribution is denoted by $\sigma$; the error bars indicate the standard deviation.