Optimal error correction in topological subsystem codes

A promising approach to overcome decoherence in quantum computing schemes is to perform active quantum error correction using topology. Topological subsystem codes incorporate both the benefits of topological and subsystem codes, allowing for error syndrome recovery with only 2-local measurements in a two-dimensional array of qubits. We study the error threshold for topological subsystem color codes under very general external noise conditions. By transforming the problem into a classical disordered spin model, we estimate using Monte Carlo simulations that topological subsystem codes have an optimal error tolerance of $5.5\left(2\right)\%$. This means there is ample space for improvement in existing error-correcting algorithms that typically find a threshold of approximately $2\%$.

Figure 1

Graphical representation of the qubit arrangement for topological subsystem color codes on a regular triangular lattice. Each of the triangular unit cells (large light gray triangles) contains three physical qubits (balls). The two-qubit gauge generators ${\sigma }^w\otimes {\sigma }^w$ are shown in green (gray, $w=x$), yellow (light gray, $w=y$) and blue (dark gray, $w=z$). These are the lines connecting the qubits (balls). They are arranged such that each physical qubit has two generators of $z$ type, one of $x$ type and one of $y$ type. See main text for details.

Figure 2

(a) A regular triangular lattice satisfies the vertex three-colorability requirement (indicated by A, B, C). (b) To construct a topological subsystem code, we place three qubits (filled large circles) inside each of the triangular unit cells and connect them with ${\sigma }^z\otimes {\sigma }^z$ gauge generators (dotted lines). The links between these triangles are assigned ${\sigma }^x\otimes {\sigma }^x$ and ${\sigma }^y\otimes {\sigma }^y$ gauge generators (yellow/light gray and green/dark grey solid lines, respectively). (c) For the mapping, gauge generators represented by colored lines in (b) are associated with Ising spins $s^{x,y,z}$ and the qubits with interactions. (d) Introducing new Ising spin variables $s^{zz}=s^z{s}^{{}^{\prime }z}$ allows for the removal of local ${\mathbb{Z}}_2$ symmetries.

Figure 3

Crossing of the correlation function ${\xi }_L/L$ with a disorder rate of $p=0.048$. The data exhibit a clear crossing at a transition temperature of $T_c\left(p\right)\approx 1.251\left(8\right)$ [30]. The shaded area corresponds to the error bar in the estimate of $T_c\left(p\right)$. Note that error bars are calculated using a bootstrap analysis of 500 resamplings. Corrections to scaling are minimal at this disorder rate, but increase closer to the error threshold.

Figure 4

Computed phase diagram for the classical disordered spin model shown in Eq. (3). Each data point $T_c\left(p\right)$ on the phase boundary (dashed curve separating white and shaded regions) is calculated by locating the crossing in correlation function ${\xi }_L/L$ for different system sizes $L$ at a fixed disorder rate $p$. The Nishimori line (blue solid line) indicates where the requirement for the mapping [Eq. (6)] holds. The error threshold $p_c\approx 0.055\left(2\right)$ is found where the Nishimori line intersects the phase boundary between the ordered phase (shaded) and the disordered phase (not shaded, larger $T$ and $p$). Below $p_c\approx 0.055\left(2\right)$ error correction is feasible. The red (dark gray) shaded vertical bar corresponds to the statistical error estimate for $p_c$.