Lifting Topological Codes: Three-Dimensional Subsystem Codes from Two-Dimensional Anyon Models

Topological subsystem codes in three spatial dimensions allow for quantum error correction with no time overhead, even in the presence of measurement noise. The physical origins of this single-shot property remain elusive, in part due to the scarcity of known models. To address this challenge, we provide a systematic construction of a class of topological subsystem codes in three dimensions built from Abelian quantum double models in one fewer dimension. Our construction not only generalizes the recently introduced subsystem toric code [Kubica and Vasmer, Nat. Commun. 13, 6272 (2022)] but also provides a new perspective on several aspects of the original model, including the origin of the Gauss law for gauge flux, and boundary conditions for the code family. We then numerically study the performance of the first few codes in this class against phenomenological noise to verify their single-shot property. Lastly, we discuss Hamiltonians naturally associated with these codes, and argue that they may be gapless.

Popular Summary

In order to reliably store and process quantum information, one needs to protect it against the deleterious effects of noise. Ideally, one would construct a “self-correcting quantum memory,” a many-body system whose thermodynamic properties passively protect the encoded information. Classically, magnetic materials can be used for this purpose. Unfortunately, it is not known how to realize such a quantum material in fewer than four spatial dimensions. The typical solution to the problem of protecting quantum information is to use active quantum error-correcting codes. Such codes allow measurements to be made that do not collapse the encoded quantum state but still allow for errors to be inferred. Traditional codes are designed to suppress physical errors corrupting the information but not errors in the readout of the error syndromes themselves.

In this work, we construct an infinite family of topological subsystem codes in three spatial dimensions that protects against measurement errors in a single shot. These codes are obtained by taking a new perspective on older codes, the subsystem toric code and gauge color code.

Our new understanding of these codes allows us to lift two-dimensional codes to codes with the single-shot property. We provide a physical understanding of this important error-correction property in terms of the physics of the lower-dimensional codes.

Figure 1

(a) The three-dimensional subsystem toric code can be defined on the cubic lattice. Qubits are placed on edges. Pauli-$X$- and $Z$-type gauge generators are depicted as triangular red and blue faces, respectively, whereas stabilizer generators are associated with volumes. (b) Gauge generators can be partitioned into two commuting sets, associated to green and yellow vertices, respectively. Around each vertex, the gauge generators form a two-dimensional sphere supporting a toric code.

Figure 2

Given a cubic lattice with checkerboard red and blue volumes, and bicolored (green and yellow) vertices, the code lattice is defined by the following: (a) Joining the centers of all blue volumes by directed edges. Code qudits are associated to the blue edges. (b) Inflating the vertices to form 2-spheres until they touch, with directed triangulation induced by the blue edges. An interactive version of this figure is available [45].

Figure 3

Stabilizer operators are those that can be recovered from gauge generators of either color. They are associated to the volumes of the cubic lattice. (a),(b) show the faces participating in a given $X$-type stabilizer. In (c),(d), we show how to recover the stabilizers by multiplying certain gauge generators in two different ways. An interactive version of this figure is available [45].

Figure 4

${\mathbf{T}}^2\times \mathbf{I}$ with class (I) boundaries. Two-dimensional AQD models are on the top and bottom boundaries, with lattice periodic in other directions. (b) shows the view from above. All spheres incident on the top boundary are colored yellow, and the AQD model on the surface is colored green. Colors are reversed at the bottom boundary. (c) Gauge generators assigned to the top surface, realizing a class (I) boundary. To aid discussion, we fix the coordinate axes in (d) for the remainder of the paper. (e) Logical operators on the SAQD are sheetlike operators that intersect the boundary. They can be dressed to stringlike operators supported on the boundary, corresponding to the logical operators of the boundary AQD.

Figure 5

Class (II) boundaries. At the rough (a) and smooth (b) boundaries, the half spheres are decorated with a quantum double model with the appropriate type of boundary.

Figure 6

Gauge flux configurations. Unlike the subsystem toric code, but similarly to the gauge color code, gauge flux can split for general SAQDs. In addition, the flux lines can carry $g\in \mathcal{G}$ units of flux (indicated by shading). (a) Typical gauge flux configuration in the absence of measurement errors. Flux obeys a Gauss law defined by the input AQD model. (b) When measurement errors occur, flux lines do not obey the required Gauss law. An interactive version of this figure is available [45].

Figure 7

(a) A plot of the logical error rate, $p_{\mathrm{fail}}$, as a function of the qudit and measurement error rate, $p$, over $t=2$ QEC cycles for various ${\mathbb{Z}}_d$ SAQDs. The error bars show the Agresti-Coull 95% confidence intervals [66]. For each value of $d$, the error threshold is the point where the curves for different $L$ intersect. (b) A plot analogous to (a) but for $t=16$ QEC cycles. Although the logical error rates increase, the error thresholds are essentially the same. (c) A plot of the error thresholds, $p_{\mathrm{th}}$, as a function of the number of QEC cycles for different values of $d$. The error thresholds are estimated from data such as those shown in (a) and (b) using finite-size scaling analysis [67], and the error bars are derived using bootstrapping with 100 estimates. (d) A plot of the rescaled error thresholds, $p_{\mathrm{th}}^{\ast }$, as a function of the number of QEC cycles, where the error thresholds from (c) are rescaled according to Eq. (13). In this case the advantage for larger $d$ disappears.

Figure 8

${\mathbf{T}}^2\times \mathbf{I}$, two-dimensional AQD models are on the top and bottom boundaries, with lattice periodic in other directions. We regard the “dangling” edges as having length $1/2$, meaning this lattice has linear size $6\times 6\times 6$. (b) shows the view from above. All spheres incident on the top boundary are colored yellow, and the AQD model on the surface is colored green. Colors are reversed at the bottom boundary.

Figure 9

On ${\mathbf{T}}^2\times \mathbf{I}$, the code supports two qudits. Bare logical operators are given by sheets of $X$ and $Z$ operators in the $x$-$y$ and $y$-$z$ planes. Minimal weight dressed logical operators are given by the intersection of the plane with the upper and lower faces. Here we indicate one logical pair. An interactive version of this figure is available [45].

Figure 10

Boundary (green) gauge generators. Left and right faces are smooth, front and back are rough. (a) shows a top view, (b–e) show the green tetrahedra incident on the boundaries. At the left and right boundaries, partial vertex terms (two-body) are included, but partial faces are not. At the front and back boundaries, partial faces are included, but partial vertex terms are omitted. Yellow gauge generators are analogous.

Figure 11

Example lattice on ${\mathbf{I}}^3$. We regard the “dangling” edges as having length $1/2$, meaning this lattice has linear size $6\times 6\times 6$.

Figure 12

Given a fixed coordinate system, the edges of the lattice can be colored using the rule indicated here.