Constraints on Measurement-Based Quantum Computation in Effective Cluster States

The aim of this work is to study the physical properties of a one-way quantum computer in an effective low-energy cluster state. We calculate the optimal working conditions as a function of the temperature and of the system parameters. The central result of our work is that any effective cluster state implemented in a perturbative framework is fragile against special kinds of external perturbations. Qualitative aspects of our work are important for any implementation of effective low-energy models containing strong multisite interactions.

Figure 1

(a) Physical qubits (black dots) on a CaVO lattice. The four physical qubits of a lattice site $\mu \in \mathcal{L}$ (gray circles) are named with the double indices $(\mu ,1),\dots ,(\mu ,4)$. The two physical qubits of a bond $m\in \mathcal{M}$ are called $m(1)$ and $m(2)$. To a physical qubit ($\mu$, $i$), the neighboring qubit on the bond is $\xi (\mu ,i)$. The $ZZ$ interactions of $H_0$ are solid lines inside the gray circles. (b) Energy spectrum of $H_{\mu }^{\mathrm{loc}}$ [Eq. (3)].

Figure 2

Fidelity per site $d$ [Eq. (5)] in dependence of ${\lambda }_{xz}$ for different temperatures $T$. Inset: Contribution $c({\lambda }_{xz}/g)$ of a physical $Z$ operator to the effective $\tilde{X}$ operator.

Figure 3

(a) Ground-state fidelity per site $d=|⟨0_z|R_z|+⟩{|}^2$ of $H_z$ as a function of ${\lambda }_{xz}$ and $h_z$ for $T=0$. (b) Fidelity per site $d$ [Eq. (8)] of $H_{zz}$ in dependence of ${\lambda }_{xz}$ and ${\lambda }_{zz}$ for $T=0.001g/k_B$.