Bounds on universal quantum computation with perturbed two-dimensional cluster states

Motivated by the possibility of universal quantum computation under noise perturbations, we compute the phase diagram of the two-dimensional (2D) cluster state Hamiltonian in the presence of Ising terms and magnetic fields. Unlike in previous analysis of perturbed 2D cluster states, we find strong evidence of a very well-defined cluster phase, separated from a polarized phase by a line of first- and second-order transitions compatible with the 3D Ising universality class and a tricritical end point. The phase boundary sets an upper bound for the amount of perturbation in the system so that its ground state is still useful for measurement-based quantum computation purposes. Moreover, we also compute the local fidelity with the unperturbed 2D cluster state. Besides a classical approximation, we determine the phase diagram by combining series expansion and variational infinite projected entangled-pair states methods. Our work constitutes an analysis of the nontrivial effect of few-body perturbations in the 2D cluster state, which is of relevance for experimental proposals.

Figure 1

Second-order phase transition for $J_z=h\sin \left(\alpha \right)$, $h_x=h\cos \left(\alpha \right)$, with $\alpha =1.0$ and $J=1$. We detect the transition at $h^{\mathrm{crit}}\approx 0.36$ (dashed line).

Figure 2

First-order phase transition for $J_z=h\sin \left(\alpha \right)$, $h_x=h\cos \left(\alpha \right)$, with $\alpha =0.2$ and $J=1$. We detect the transition at $h^{*}\approx 0.82$ (dashed line).

Figure 3

Phase diagram as a function of variables $X=(1-J_n+h_{x,n})/\sqrt{2}$ and $Y=\sqrt{3/2}(1-J_n-h_{x,n})$ for $J=1$ obtained by the combined series expansion plus iPEPS approach. The phase boundaries obtained within the CA are shown as dot-dashed lines. Additionally, dashed lines refer to fidelities per site $d=0.999$ (black) and $d=0.99$ (green). Left inset shows the critical exponent $\nu$ for the divergence of the correlation length along the second-order line in the phase diagram. The notation $\nu [p,q]$ indicates the type of resummation of the series expansion results in order to get different fittings to the exponent. Right inset is a zoom of the first-order transition points close to the self-dual line (vertical red line).

Figure 4

Fidelity per site, $d$, shown for the cases $J_z=0$ (upper panel) and $h_x=0$ (lower panel). The inset is a zoom of the region close to the second-order transition.