Classical simulation versus universality in measurement-based quantum computation

We investigate for which resource states an efficient classical simulation of measurement-based quantum computation is possible. We show that the Schmidt-rank width, a measure recently introduced to assess universality of resource states, plays a crucial role in also this context. We relate Schmidt-rank width to the optimal description of states in terms of tree tensor networks and show that an efficient classical simulation of measurement-based quantum computation is possible for all states with logarithmically bounded Schmidt-rank width (with respect to the system size). For graph states where the Schmidt-rank width scales in this way, we efficiently construct the optimal tree tensor network descriptions, and provide several examples. We highlight parallels in the efficient description of complex systems in quantum information theory and graph theory.

Figure 1

(Color online) (a) Example of a subcubic tree $T$ with six leaves [indicated in blue (darker dots)]. (b) Tree $T\backslash e$ obtained by removing edge $e$ and induced bipartition $(A_T^e,B_T^e)$.

Figure 2

(Color online) Tensor network with three tensors $A_{ajk}^{\left(1\right)},A_{bjl}^{\left(2\right)},A_{ckl}^{\left(3\right)}$ and three open indices $a,b,c$ corresponding to a cycle graph. (b) Tensor network with six tensors $A_{abi}^{\left(1\right)},A_{ijk}^{\left(2\right)},\dots ,A_{ghk}^{\left(6\right)}$ and eight open indices $a,b,c,d,e,f,g,h$ corresponding to a tree graph.

Figure 3

(Color online) Subcubic tree with root $r$, where leaves (corresponding to the $n=13$ parties of the system) are indicated in blue (dark), and inner vertices are indicated in red (light). The tree is arranged in such a way that all inner vertices of same depth $\delta$ are on the same horizontal line.

Figure 4

(Color online) (a) Same subcubic tree as depicted in Fig. 3, where vertices are rearranged. We consider an inner vertex $v$ of depth $\delta =1$ with lower edges $e_1$, $e_2$ and upper edge $e_3$, and the corresponding tripartition of the system into groups $X_1,X_2,X_3$. (b) Subtree $T_v$ and (c) tree $T_v^{*}$ (for definition see text).