Strong Resilience of Topological Codes to Depolarization

The inevitable presence of decoherence effects in systems suitable for quantum computation necessitates effective error-correction schemes to protect information from noise. We compute the stability of the toric code to depolarization by mapping the quantum problem onto a classical disordered eight-vertex Ising model. By studying the stability of the related ferromagnetic phase via both large-scale Monte Carlo simulations and the duality method, we are able to demonstrate an increased error threshold of 18.9(3)% when noise correlations are taken into account. Remarkably, this result agrees within error bars with the result for a different class of codes—topological color codes—where the mapping yields interesting new types of interacting eight-vertex models.

Popular Summary

Quantum computers are much more vulnerable to noise than classical computers, because the quantum states of the tiny qubits in them can be altered by the smallest noise, easily leading to errors. Error correction is then obviously an issue of paramount importance for the success of quantum computation. One very promising approach, indeed, currently the best candidate for practical implementations, uses topological error-correction codes. The error-correcting performance of a topological code is captured essentially by its error threshold: As long as the noise intensity is below the threshold, noise-induced errors can be fully corrected by well-designed manipulations that involve only a few qubits. For some of the landmark topological codes, however, working out what this threshold is for the most generic form of noise disturbance is a difficult technical challenge. In this paper, we have accomplished this feat, for several of such codes and for a very general form of noise, by recasting the study of their stabilities as the study of existence of ferromagnetic phases in certain classes of classical models of interacting spins.

In general, a quantum error-correction code works by first defining a set of error-identifying quantum measurements (or “check operators”), then making the measurements to identify the error (establishing a so-called “error syndrome”), and finally prescribing and executing a set of error-correction quantum operations on the qubits. A topological code is distinguished by two key features: First, all the quantum measurements needed for error correction are “local,” involving only a few qubits that can be viewed as “neighbors.” Second, no local operation on its own can change the encoded state of the whole computer. In its essence, “topological” signifies this robustness against local disturbances. The two families of topological codes that we have focused on in this work are the most-studied toric codes and the color codes. In the former, the physical qubits are placed on the square-lattice-like grid on the surface of a torus, and in the latter, on the vertices of a trivalent, e.g., hexagonal, lattice—the architectures of the qubit connectivity dictated by the nature of the quantum measurements involved in the codes.

A previous work by Dennis, Kitaev, Landahl, and Preskill in 2004 pioneered the conceptual approach of determining the error threshold by mapping the quantum problem onto a classical spin model. The form of the noise investigated there was, however, only one of the three possible fundamental types. Our study explores the case of the most generic noise form, which includes not only all three noise types, but also any correlations among them. Finding the mapping becomes, therefore, considerably more challenging technically. We have succeeded in demonstrating that, for the noise form we considered, the classical counterpart of the toric code is an 8-vertex spin model. The error threshold then corresponds to the point in the classical model where a magnetic ordering transition is lost due to the underlying disorder in the classical spins, which is equivalent to the presence of faulty qubits. Using Monte Carlo simulations and duality arguments, we are able to find the error threshold to be at 19% approximately—higher than what was previously thought. Remarkably, the mapping of the color codes leads to new types of classical 8-vertex models, but at the same time, their error thresholds are also at 19%.

We believe that this interdisciplinary effort should bring us a step closer toward the ultimate goal of building high-error-tolerance and large-scale quantum computers, and we also anticipate that this work will be of interest to the statistical mechanics community as well.

Figure 1

Phase boundary estimated from Monte Carlo simulations for the estimation of the error threshold of the toric code, as well as two realizations of color codes (see text). The error threshold $p_c$ corresponds to the point at which the Nishimori lines intersects the phase boundary. Remarkably, the phase boundaries for all three codes agree within error bars. The stable ordered phase corresponds to the regime where quantum error correction is feasible.

Figure 2

When computing the stability of the toric code to depolarization, the problem maps onto a classical statistical Ising model on two stacked square lattices. In addition to the standard two-body interactions for both top (a) and bottom (b) layers, the resulting Hamiltonian also includes four-body terms (c) that introduce correlations between the layers.

Figure 3

For the hexagonal arrangement, there is a stabilizer operator $Z^6$ for each of the hexagon plaquettes (a). In the mapping, these stabilizer operators translate to classical Ising spins, which are placed on the dual lattice [regular triangular lattice, (b)]. The square-octagonal setup (c) has wider computing capabilities because it allows for a larger class of quantum gates to be implemented. There are stabilizers $Z^4$ ($Z^8$) on the rectangles (octagons). The corresponding dual lattice in the mapping is the union-jack lattice (d).

Figure 4

For topological color codes, qubits are arranged on trivalent lattices (hexagonal or square-octagonal). These codes are mapped to triangular lattices (triangular and union jack, respectively) with plaquette interactions (a),(b) on each layer, as well as six-body interactions correlating the two layers (c).

Figure 5

Crossing of the correlation function ${\xi }_L/L$ for the toric code with a disorder rate of $p=0.170$. The data exhibit a clear crossing at a transition temperature of $T_c(p)\approx 2.14(2)$. Corrections to scaling are still minimal at this disorder rate, but they increase closer to the error threshold. Inset: Close-up of the area where the crossing occurs. The conservative estimate for the transition temperature is indicated by the vertical shaded area.

Figure 6

Clusters used to estimate the error threshold for the depolarizing channel. The blue lines and triangles denote quenched random variables ${\tau }^Z$, and the red lines and triangles correspond to ${\tau }^X$. The central site is the spin variable summed over. The outer sites represent the spin variables fixed in the up direction.