Minimal instances for toric code ground states

A decade ago Kitaev's toric code model established the new paradigm of topological quantum computation. Due to remarkable theoretical and experimental progress, the quantum simulation of such complex many-body systems is now within the realms of possibility. Here we consider the question, to which extent the ground states of small toric code systems differ from local-unitary (LU)-equivalent graph states. We argue that simplistic (though experimentally attractive) setups obliterate the differences between the toric code and equivalent graph states; hence we search for the smallest setups on the square and triangular lattices, such that the quasilocality of the toric code Hamiltonian becomes a distinctive feature. To this end, a purely geometric procedure to transform a given toric code setup into a local-Clifford (LC)-equivalent graph state is derived. In combination with an algorithmic computation of LC-equivalent graph states, we find the smallest nontrivial setup on the square lattice to contain five plaquettes and 16 qubits; on the triangular lattice the number of plaquettes and qubits is reduced to four and nine, respectively.

Figure 1

(a) The (nondegenerate) ground state of a single toric code plaquette is a GHZ state and equivalent to the shown graph state. Qubits (unfilled circles) are located on the edges. Vertices (black dots) denote star operators $A_s$ and gray faces are plaquette operators $B_p$ acting on the adjacent qubits. The LC-equivalent graph state connects the qubits directly via bold (red) lines. (b) The eight neighbors (unfilled circles on the [blue] bold rhombus) of the centered qubit (centered, [red] filled circle) in the toric code on a square lattice. Neighbors and the centered qubit have a star and/or plaquette operator in common.

Figure 2

The minimal nonlocal toric code setup on a square lattice (a) and on a triangular lattice (b) with LC-equivalent graph states. The system depicted in (c) was used to prove the nonlocality of the square lattice setup in (a). Notation: Qubits (unfilled circles) are located on the edges. Vertices (black dots) denote star operators $A_s$ and gray faces are plaquette operators $B_p$ acting on the adjacent qubits. The drawn LC-equivalent graph states connect qubits directly via local (bold [red] lines) and nonlocal (bold [red] dotted lines) edges. The graph states are based on the black spanning trees (bold black lines connecting black dots); see Definition 1 and Theorem 1 for details.

Figure 3

The tcm with qubits (unfilled circles) located on the edges of a square lattice embedded into a 2-torus. The ${\sigma }^x$ and ${\sigma }^z$ Pauli operators are denoted by (blue) encircled plus signs and (green) encircled cross signs, respectively. The intersections of $X$- and $Z$-type loop operators, responsible for their commutation relations, are highlighted by bold (red) encircled stars. The upper (left) border is identified with the dashed lower (right) border.

Figure 4

The graph on the left-hand side is simple and not a tree since it features cycles. Deleting edges that support a cycle (illustrated by dotted lines) yields two nonisomorphic spanning trees.

Figure 5

Two simple examples of surface codes and LC-equivalent graph states. Product states (a) are mapped to disjoint graphs and GHZ states (b) are transformed to star graphs. A description of the graphical notation is given in Subsec. 3b.

Figure 6

Interpretation of the bipartite graph $G={\varphi }_{{\mathcal{L}}^{\prime }}\left(\mathcal{L}\right)$ as given in Definition 1. Each vertex $e$ identified with a deleted edge in $\mathcal{E}$ ([blue] pentagon) encodes a fundamental cycle ${\mathcal{C}}_e$ on $\mathcal{L}$ with respect to ${\mathcal{L}}^{\prime }$ via its neighbors in $G$, viz., ${\mathcal{C}}_e=N_e\cup \left\{e\right\}$. Conversely, each vertex $e^{\prime }$ identified with a spanning tree edge ([red] square) encodes a  fundamental cut ${\mathcal{D}}_{e^{\prime }}$ on $\mathcal{L}$ with respect to ${\mathcal{L}}^{\prime }$ via its neighbors in $G$, viz., ${\mathcal{D}}_{e^{\prime }}=N_{e^{\prime }}\cup \left\{e^{\prime }\right\}$. This is a consequence of the duality between fundamental cuts and fundamental cycles.

Figure 7

Construction of $G={\varphi }_{{\mathcal{L}}^{\prime }}\left(\mathcal{L}\right)$ as given in Definition 1 and used in the proof of Theorem 1, given a surface code (a) with arbitrary spanning tree (b). Consider one of the (deleted) edges $e=\{p,q\}$ on which a Hadamard rotation was performed (denoted by a [yellow] encircled star) and its unique path ${\mathcal{C}}_{pq}$ (denoted by [black and yellow] bold dashed lines) on the spanning tree (c). The former $z$-type loop operator along ${\mathcal{C}}_{pq}+e$ acts as ${\sigma }^x$ on the qubit symbolized by a (blue) encircled plus sign (due to the Hadamard rotation) and as ${\sigma }^z$ on the qubits denoted by (green) encircled cross signs. This is a graph state stabilizer of the graph drawn with (red) curved lines (d). On the other hand, consider one of the edges $e^{\prime }$ on the spanning tree (e). The induced bipartition ${\mathbb{V}}_r^{e^{\prime }}\dot{\cup }{\mathbb{V}}_b^{e^{\prime }}$ of the vertices is denoted by (red) triangles and (blue) squares. The product of all (former) star operators labeled by one of these sets acts nontrivially as ${\sigma }^x$ on the qubit symbolized by a (blue) encircled plus sign and as ${\sigma }^z$ on the qubits denoted by (green) encircled cross signs (due to the Hadamard rotations). This is a graph state stabilizer of the graph drawn with (red) curved lines in (e).

Figure 8

The free polyominos for $n=1,2,3,4,5$. Faces with at least one peripheral edge are light colored. The first polyomino featuring a periphery-free cell (dark [blue] colored) is the $+$-shaped pentomino. It is the smallest configuration on the square lattice, such that the geometrical construction yields only nonlocal graph states.

Figure 9

Diagrammatic sketch showing the structure of $\varepsilon -$ LC classes as used in the proof of Proposition 1. Arrows denote different classes of transformations according to their label and color: $\gamma$ operations (magenta), $\xi$ operations (green), $a$ operations (blue), and $b$ operations (red). Vertex $a$ ($b$) is depicted as a (blue) pentagon ([red] square). A detailed description is given in the text.

Figure 10

An arbitrary path from $G_{ab}$ ([red] pentagon in set $A$) to $H$ ([green] square in set $C$) is depicted in (a). The “forbidden” paths (bold [blue] lines) using $a$ operations are swapped to $A$ (bold [red] lines) using $b$ operations instead [see (b)]. Now the whole path from $G_{ab}$ to $H$ is $a$ accessible. A detailed description is given in the text.

Figure 11

Example for the reduction procedure based on Theorem 2. (a) The $+$-shaped pentomino as presented in Fig. 2a. (b), (c) By reducing the upper square twice one obtains successively smaller systems. (d)–(f) This procedure is repeatedly applied to the remaining squares and leads to the system depicted in Fig. 2c. The “removal” of each edge and qubit is justified by Theorem 2; thus the nonlocality of the smallest setup (f) implies the nonlocality of the whole chain of larger setups.