Figure 1

The schematics of global Hadamard operation at every second row in red dotted atoms.Reuse & Permissions

Figure 2

To build 3D cluster states, seven CZ and two Hadamard operations are globally required in 2D optical lattices. Solid lines indicate the horizontal and vertical directions in the CZ gates, while dashed lines show the CZ operations in diagonal directions (small black arrows show the direction of moving atoms). In (d), the colored yellow (gray) edges present a unit cell of 3D cluster states in the lattices.Reuse & Permissions

Figure 3

How to build a surface code directly in 2D optical lattices. The surface code is directly created by Pauli ${\sigma }^x$ measurements (denoted by X) at all red dotted (Li) atoms in (c) [12, 18]. Then, red dotted atoms are removed by turning off their optical lattices.Reuse & Permissions

Figure 4

(a) and (b) Simulation of the application of the two-qubit verification protocol in the basic blocks shown in Figs. 2d and 3b, respectively, where the simulations are computed with a fixed dephasing error parameter $\theta =\pi /5$. The color (gray) links show a positive result of the two-qubit entanglement witness. Shaded atoms represent the border of the unit blocks, which has to be measured in order to compute the local fidelity. Values of the fidelity to the graph state as a function of $\theta$ for the 3D cubic block (solid) and the surface-code block (dashed) for (c) dephasing and (d) unitary CZ (Ising) errors. The horizontal line shows the minimum value for the observable to show genuine multipartite entanglement. Notice that the plots are roughly similar for both blocks due to the similar size of the blocks considered.Reuse & Permissions