Quantum computational universality of Affleck-Kennedy-Lieb-Tasaki states beyond the honeycomb lattice

Universal quantum computation can be achieved by simply performing single-spin measurements on a highly entangled resource state, such as cluster states. The family of Affleck-Kennedy-Lieb-Tasaki (AKLT) states has recently been explored; for example, the spin-1 AKLT chain can be used to simulate single-qubit gate operations on a single qubit, and the spin-3/2 two-dimensional AKLT state on the honeycomb lattice can be used as a universal resource. However, it is unclear whether such universality is a coincidence for the specific state or a shared feature in all two-dimensional AKLT states. Here we consider the family of spin-3/2 AKLT states on various trivalent Archimedean lattices and show that in addition to the honeycomb lattice, the spin-3/2 AKLT states on the square octagon $(4,8^2)$ and the “cross” $(4,6,12)$ lattices are also universal resource, whereas the AKLT state on the “star” $(3,{12}^2)$ lattice is likely not due to geometric frustration.

Figure 1

The four trivalent Archimedean lattices: (a) the honeycomb $\left(6^3\right)$, (b) the square octagon $(4,8^2)$, (c) the cross $(4,6,12)$, and (d) the star $(3,{12}^2)$.

Figure 2

Illustrations of the AKLT state on a square-octagon lattice and a graph state on a graph. (a) AKLT state. Each site is spin 3/2 and can be regarded as three virtual spin-1/2 particles in their symmetric subspace. Two virtual qubits connected by an edge form a spin-singlet state. (b) A graph state. One qubit resides at each vertex, and one stabilizer generator of the form $X\otimes Z\cdots Z$ is shown (with $X\equiv {\sigma }_x$ at a vertex and $Z\equiv {\sigma }_z$ on neighboring vertices). Cluster states are graph states defined on regular lattices.

Figure 3

Illustration of POVM outcomes and construction of the graph for the resulting graph state on a square-octagon lattice. (a) POVM outcomes: $x$ (red), $y$ (green), and $z$ (blue). (b) The graph after formation of domains and edge modulo-2 operation.

Figure 4

Illustration of POVM outcomes and construction of the graph for the resulting graph state on a star lattice. (a) POVM outcomes: $x$ (red), $y$ (green), and $z$ (blue). (b) The graph after formation of domains and edge modulo-2 operation. One can also see the effect of the geometric frustration, which causes two edges in some triangles to be removed.

Figure 5

Illustrations of the counting of $e_I$ and $v_c$ (see text). The POVM types are shown in the circles: (a) before the type at $c$ was flipped and (b) after it was flipped. In (a), $e_I=2$, arising from edges $(c,1)$ and $(c,2)$. In (b) the type on $c$ is flipped from $x$ to $z$, and it is easy to see that $e_I=0$. In (a), there are two distinct domains ($v_c=2$) that vertices $c$, 1, 2, and 3 belong to. In (b) $v_c=3$ if vertices 1 and 2 belong to the same domain; otherwise, $v_c=3$.

Figure 6

Average domain size, average degree of a vertex, and the largest domain size (inset) in the typical graphs vs $L=20,40,60,80,100,120,500$, with $N=L^2$ being the total number of sites for the square-octagon lattice. The inset shows the largest domain size scales with $N$ logarithmically.

Figure 7

Percolation study of the random graphs of domains resulting from the original square-octagon lattice: probability of a spanning cluster $p_{\mathrm{span}}$ vs that to delete an edge $p_{\mathrm{delete}}$ with $N=L^2$. (a) Site percolation and (b) bond percolation.

Figure 8

Average domain size, average degree of a vertex, and the largest domain size (inset) in the typical graphs vs $L=24,48,72,96,120,504$, with $N=L^2$ being the total number of sites for the cross lattice. The inset shows the largest domain size scales with $N$ logarithmically.

Figure 9

Percolation study of the random graphs of domains resulting from the original cross lattice: probability of a spanning cluster $p_{\mathrm{span}}$ vs that to delete an edge $p_{\mathrm{delete}}$ with $N=L^2$. (a) Site percolation and (b) bond percolation.

Figure 10

Illustration of the key local measurement in converting the cluster states on (a) the square-octagon and (b) the cross lattices to a cluster state on the honeycomb lattice. The white circles indicate the locations where Pauli $Z$ measurement is performed. The picture on the right-hand side in (a) shows the resulting graph after removing the $Z$-measured sites, which is topologically equivalent to a honeycomb lattice, except with additional sites of vertex degree 2. In (b) the resultant graph is shown by removing the edges due to $Z$ measurements on the sites indicated by white circles. In both cases, Pauli $X$ measurements can be applied to those sites of vertex degree 2 to further reduce to the honeycomb lattice.

Figure 11

Illustration of the key local measurement in converting the cluster state on the star lattice to a cluster state on the honeycomb lattice. The white squares indicate the locations where Pauli $Y$ measurement is performed. The action of Pauli $Y$ measurement is to perform local complementation followed by removal of the vertex measured. The resultant graph is shown on the right-hand side, which is topologically equivalent to a honeycomb lattice. Pauli $X$ measurements can be applied to those sites of vertex degree 2 to further reduce to the honeycomb lattice.