Noisy Toric code and random-bond Ising model: The error threshold in a dual picture

It is known that noisy topological quantum codes are related to random-bond Ising models where the order-disorder phase transition in the classical model is mapped to the error threshold of the corresponding topological code. On the other hand, there is a dual mapping between classical spin models and quantum Calderbank-Shor-Steane (CSS) states where the partition function of a classical model defined on a hypergraph $H$ is written as an inner product of a product state and a CSS state on dual hypergraph $\tilde{H}$. It is then interesting to see what the interpretation is of the classical phase transition in the random-bond Ising model within the framework of the above duality, and whether such an interpretation has any connection to the error threshold of the corresponding topological CSS code. In this paper, we consider the above duality relation specifically for a two-dimensional random-bond Ising model. We show that the order parameter of this classical system is mapped to a coherence order parameter in a noisy Toric code model. In particular, a quantum phase transition from a coherent phase to a noncoherent phase occurs when the initial coherent state is affected by two sequences of bit-flip quantum channels where a quenched disorder is induced by measurement of the errors after the first channel. On the other hand, the above transition is directly related to the error threshold of the Toric code model. Accordingly, and since the noisy process can be applied to other topological CSS states, we conclude that the dual correspondence can also provide a useful tool for the study of error thresholds in different topological CSS codes.

Figure 1

(a) Ising spins $s_i$ denoted by black circles live on vertices of a square lattice. Edge spins $S_{ij}=s_i{s}_j$ denoted by green (light) circles are added on links of the lattice. Product of edge spins belonging to a face $F$ of the lattice should be equal to 1. (b) In a quantum formalism, each constraint corresponding to a face of the lattice is mapped to a face operator $B_F={\prod }_{e\in \partial F}{Z}_e$. It is in fact a stabilizer of the Toric code. There is also another kind of stabilizer, $A_v={\prod }_{e\in v}{X}_e$, corresponding to each vertex of the lattice.

Figure 2

Loop operators $T_z^1, T_z^2, T_x^1$, and $T_x^2$ correspond to nontrivial loops around the torus.

Figure 3

A schematic phase diagram of a random-bond Ising model where ferromagnetic and paramagnetic phases are separated by a critical curve. There is also a Nishimori line where $p=\frac{e^{-2\beta J}}{1+e^{-2\beta J}}$.

Figure 4

(a) String ${\gamma }_n$ starts from a spin in the boundary and ends in a spin at site $n$. (b) The Toric code state corresponding to the Ising model with the above boundary condition will have an open boundary where face operators in the boundary are three-local. To each string ${\gamma }_n$ we assign a $Z$-type operator ${\mathrm{\Gamma }}_Z={\prod }_{i\in {\gamma }_n}{Z}_i$.

Figure 5

An $X$ operator on a qubit in the Toric code does not commute with two face operators which are neighbors of that qubit. It leads to two excitations in the neighboring faces. A string of $X$ operators also leads to two excitations in the endpoints of that string.

Figure 6

A Toric code can be defined on a lattice with an open boundary where face operators in the boundary are three-local. A string $\gamma$ which starts from the boundary leads to an excitation in the vertex in the endpoint of $\gamma$. There are two $X$-type loop operators $L_{+}$ and $L_{-}$ which cross the string $\gamma$ for even and odd number of times, respectively.

Figure 7

(a) X-cube model is defined on a cubic lattice with qubits living on links of the lattice. Corresponding to each cell of the lattice denoted by yellow (light) color, an $X$ type stabilizer is defined by the product of $X$ operators for the 12 qubits belonging to the cubic cell. (b) In a dual picture, one should insert spins in the center of each cell. Since each original qubit belongs to four cells of the lattice, each edge of the dual hypergraph involves four spins.

Figure 8

A schematic of a phase diagram for the noisy Toric code model where transition point to the noncoherent phase denoted by $q_{th}$ decreases by increasing quenched noise $p$ in a sense that, corresponding to the transition curve, $q_{th}$ is a decreasing function of $p$. The line $p=q$ refers to the Nishimori line and the transition point on the Nishimori line is called Nishimori point, which determines the error threshold $q_{th}\left(p_c\right)$.