Parallelized quantum error correction with fracton topological codes

Fracton topological phases have a large number of materialized symmetries that enforce a rigid structure on their excitations. Remarkably, we find that the symmetries of a quantum error-correcting code based on a fracton phase enable us to design decoding algorithms. Here we propose and implement decoding algorithms for the three-dimensional X-cube model. In our example, decoding is parallelized into a series of two-dimensional matching problems, thus significantly simplifying the most time-consuming component of the decoder. We also find that the rigid structure of its point excitations enables us to obtain high threshold error rates. Our decoding algorithms bring to light some key ideas that we expect to be useful in the design of decoders for general topological stabilizer codes. Moreover, the notion of parallelization unifies several concepts in quantum error correction. We conclude by discussing the broad applicability of our methods and we explain the connection between parallelizable codes and other methods of quantum error correction. In particular we propose that our concept represents a generalization of single-shot error correction.

Figure 1

The support of stabilizer generators $A_v,B_c^{\mathbf{r}}$, and $B_c^{\mathbf{g}}$ are shown in (a), (b), and (c), respectively. The faces of the lattice can be three-colored as in (d). (e) Lineon excitations which are created by applying Pauli-X errors over dual lines. The color of the excitation is determined by the orientation of the line operator that creates the excitation. (f) Fracton excitations of the vertex stabilizers caused by Pauli-Z operators supported on the red faces.

Figure 2

A Pauli-X error. On the plane marked by the dotted grid we match all the green and red excitations. We use the blue excitations to help find the correct matching. Panel (a) shows an error where two green defects are caused by a stringlike error with blue defects at its corners. Nearby errors can lead to confusion. At (b) an error moves a blue defect away from the plane, and at (c) an additional error introduces a blue defect that could also mislead the decoder into finding an incorrect matching along the red dashed line.

Figure 3

Pauli-Z errors on a plane of the lattice. By comparison with the toric code we see this error cannot be corrected.

Figure 4

The process of finding a correction operator for a correctable collection of cell defects. (a) An odd parity of each of the differently colored subsets of defects is shown in the bulk of the lattice. (b) Each of the defects is moved freely onto a plane. (c) The defects are then moved onto a one-dimensional line where the planes meet. Finally, (d) the defects are moved along their respective one-dimensional lines to neutralize at a common point.

Figure 5

Showing that the parity of the number of defects of the same color is equal. Each red defect has one incident edge from the green matching and one from the blue matching (a), and the green and blue defects have only one incident edge from the blue and green matching; see (b) and (c), respectively. These form one-dimensional chains where an even number of red defects can form a one-dimensional cycle (d), two greens or two blues can be connected directly (e) and (f), a string of an odd number of connected red defects terminates at two differently colored defects (g), or an even number of red defects can terminate at two defects of the same color (h). The number of differently colored defects in each of the possible combinations that are shown explicitly between (d) and (h) are of equal parity.

Figure 6

The support of the one dimensional logical operators of the X-cube model. The logical Pauli-X (Pauli-Z) operators are supported on the blue (green) qubits, and both logical operators intersect at the pink face.

Figure 7

Logical failure rate using the decoder to correct cell defects where errors are independent and identically distributed bit flips at rate $p$. We find a threshold $\sim 9.4\%$ at the crossing point where we compare system sizes $L=72,84,96,108$.

Figure 8

Correctable clusters of vertex defects. (Left) We can move defects paired on a plane of constant $z$ onto other planes of constant $z$ with the operator shown. (Right) The projection of defects that were paired over planes of constant $z$ is shown on a single plane. Given that each defect is paired to another defect on the $xz$ and $yz$ planes, the projected edges mark a boundary with projected defects on its corners. The defects on this plane can be corrected by Pauli-Z operators acting on the faces enclosed by the boundary.

Figure 9

Threshold error rate for the vertex excitations where we perform matching over planes that satisfy a ${\mathbb{Z}}_2$ symmetry, and we assign weights to edges according to the separation of two vertices by their Manhattan distance. We find a threshold $\sim 3.7\%$ using system sizes $L=36,42,48,54,60$.

Figure 10

An error configuration marked by red faces that cannot be decoded by pairing defects using minimum-weight perfect matching where edges are weighted according to their separation. The defects may be paired incorrectly via the red dashed lines. We improve the decoder by choosing weights such that we also account for the separation between defects that have paired with the defects of the edge we are trying to assign a weight to. We look at the defects that have been paired via the $x$-matching that are connected to the defects of interest via the blue edges. Due to the difference in orientation of one of the errors, if we assign weights that account for the separation of the defects of interest, and the separation of their other partners, a high weight will be assigned to the incorrect edges. We mark the large separation between the paired defects with green lines.

Figure 11

Threshold error rate using the vertex defect decoder after five iterations of belief propagation. We find a threshold $\sim 4.3\%$ using system sizes $L=36,42,48,54,60$. We use $\sim {10}^4$ samples to find each data point.

Figure 12

A single stabilizer $S\left(T\right)$ that is measured sequentially at times $T=t-1,t,t+1$, where time is represented along the vertical direction. A measurement error occurs at time $t$. This creates defects for measurements $\tilde{S}(t-1)=S(t-1)S\left(t\right)$ and $\tilde{S}\left(t\right)=S\left(t\right)S(t+1)$. These defects are marked by red points. The measurement error can be viewed as a string segment that connects these defects along the temporal direction.

Figure 13

A syndrome configuration for the toric code in the case where measurement errors are noisy. Measurement errors appear as strings that align with the temporal direction of the spacetime diagram, where time is running vertically along the page, and physical errors are strings that align along planes that are orthogonal to the temporal direction. Defects appear at the end points of the strings.

Figure 14

(Left) A spacetime diagram of a two-dimensional topological code, for instance, the toric code, where time runs up along the page. We imagine an error that persists over a long time that will effectively introduce a puncture that extends through the time dimension. One can imagine a situation where a small error will have a catastrophic effect on the logical information. We show two small errors that introduce a defect into the extended puncture at the top and the bottom of the figure. These defects could easily be mismatched. (Middle) We imagine a code where decoding can be parallelized on planes of constant time, i.e., a single-shot code. Again time runs up the page. In this situation the puncture that extends through time appears as a small puncture on each plane. Such a puncture is not disastrous to the decoding procedure. (Right) We show the decoding problem on a single plane. The small puncture does not extend across the breadth of the plane, so it is easy for a decoder to determine if a defect is absorbed by the small puncture.