Universal measurement-based quantum computation in two-dimensional symmetry-protected topological phases

Recent progress in the characterization of gapped quantum phases has also triggered the search for a universal resource for quantum computation in symmetric gapped phases. Prior works in one dimension suggest that it is a feature more common than previously thought, in that nontrivial one-dimensional symmetry-protected topological (SPT) phases provide quantum computational power characterized by the algebraic structure defining these phases. Progress in two and higher dimensions so far has been limited to special fixed points. Here we provide two families of two-dimensional $Z_2$ symmetric wave functions such that there exists a finite region of the parameter in the SPT phases that supports universal quantum computation. The quantum computational power appears to lose its universality at the boundary between the SPT and the symmetry-breaking phases.

Figure 1

Lattice and qubits: each physical site (illustrated by a filled circle) contains multiple qubits: (a) three on a honeycomb lattice, (b) four on a square lattice, and (c) either three or four on a random planar graph of mixed degrees 3 and 4. Each dotted loop indicates the constraint on the state of qubits connected by the loop; they need to be the same, either all $|0\rangle$ or all $|1\rangle$, i.e., a GHZ constraint, $|00\cdots 0\rangle +|11\cdots 1\rangle$.

Figure 2

Phase diagram vs $g$. $\mathrm{Tr}\left(T^2\right)$ is represented by open black squares and the order parameter $\langle {\sigma }_z\rangle$ is represented by open blue circles. (a) Honeycomb lattice (see also Ref. [43]); (b) square lattice. Transitions are estimated to be at $|g_c|=0.759\left(1\right)$ on the honeycomb lattice and at $|g_c|=0.633\left(1\right)$ on the square lattice.

Figure 3

Illustration of POVM outcomes and GHZ loops. Larger, green circles represent outcomes of $E_2$, while smaller, red circles represent $E_1$ outcomes. Individual qubits are not shown. (a) A cluster is defined as a loop composed of one or multiple hexagons, with the latter indicated by a curved loop. (b) Simplified rendition of (a).

Figure 4

Illustration of POVM outcomes on a honeycomb lattice of 800 hexagons. (a) $g=0.74$ (macroscopic loops) and (b) $g=0.78$ (no macroscopic loops). Loops are suppressed and $E_1$ outcomes are indicated by red circles and $E_2$ outcomes by green circles. But the existence of a macrosopic loop can be seen from the existence of a pathway (like in a maze) from the left side to the right or from the top to the bottom in (a).

Figure 5

Probability $P_{\mathrm{span}}$ of a spanning loop (i.e., a macroscopic GHZ loop) vs parameter $g$ for the case $g<1$ for various linear sizes $L$ on the honeycomb lattice. The total number of hexagons is $2L^2$. Each data point is averaged over 2000–4000 samples and the statistical error is less than 1%. A percolation transition is estimated at $g_{c1}\approx 0.760\left(2\right)$ in the thermodynamic limit. In the region $g_{c1}<g\le 1$, there is no macroscopic GHZ loop ($P_{\mathrm{span}}\to 0$ as $L\to \infty$), and hence a sufficiently large valence-bond state with the number of qubits being proportional to the original system size can be obtained. This means that the system provides a universal resource for MBQC.

Figure 6

(a) Local tensors for the wave functions. (b) The $Z_2$ symmetry action on the physical spins is equivalent to the local MPO action on the inner indices. In the drawing, the operator $\widehat{O}$ is applied first, followed by $X\otimes X$. But in the equation for the local MPO, one usually writes $\widehat{O}X\otimes X$, reversing the order. As one can see the neighboring local MPOs cancel and for a closed surface the wave functions are indeed $Z_2$ symmetric.

Figure 7

(a) Local tensors for the wave functions. (b) The $Z_2$ symmetry action on the physical spins is equivalent to the local MPO action on the inner indices. In the drawing, the operator $\widehat{O}$ is applied first, followed by $X\otimes X$. But in the equation for the local MPO, one usually writes $\widehat{O}X\otimes X$, reversing the order. As one can see the neighboring local MPOs cancel and for a closed surface the wave functions are indeed $Z_2$ symmetric.

Figure 8

Probability $P_{\mathrm{span}}$ of a spanning loop (i.e., a macroscopic GHZ loop) vs parameter $g$ for the case $g>1$ on the honeycomb lattice. (a) Estimate of the upper bound; (b) lower bound on the threshold value $g_{c2}$ that separates regions with and without quantum computational universality. It is estimated that $1.205\left(5\right)\lesssim g_{c2}\lesssim 1.390\left(2\right)$.

Figure 9

Probability $P_{\mathrm{span}}$ of a spanning loop (i.e., a macroscopic GHZ loop) vs parameter $g$ for the case $g<1$ for various linear sizes $L$ on a square lattice. The total number of squares is $L^2$. A percolation transition is estimated at $g_{c1}\approx 0.635\left(3\right)$ in the thermodynamic limit. In the region $g_{c1}<g\le 1$, there is no macroscopic GHZ loop ($P_{\mathrm{span}}\to 0$ as $L\to \infty$), and hence a sufficiently large valence-bond state with the number of qubits being proportional to the original system size can be obtained. This means that the system provides a universal resource for MBQC.