Stacked codes: Universal fault-tolerant quantum computation in a two-dimensional layout

We introduce a class of three-dimensional color codes, which we call stacked codes, together with a fault-tolerant transformation that will map logical qubits encoded in two-dimensional (2D) color codes into stacked codes and back. The stacked code allows for the transversal implementation of a non-Clifford $\pi /8$ logical gate, which when combined with the logical Clifford gates that are transversal in the 2D color code give a gate set that is both fault-tolerant and universal without requiring nonstabilizer magic states. We then show that the layers forming the stacked code can be unfolded and arranged in a 2D layout. As only Clifford gates can be implemented transversally for 2D topological stabilizer codes, a nonlocal operation must be incorporated in order to allow for this transversal application of a non-Clifford gate. Our code achieves this operation through the transformation from a 2D color code to the unfolded stacked code induced by measuring only geometrically local stabilizers and gauge operators within the bulk of 2D color codes together with a nonlocal operator that has support on a one-dimensional boundary between such 2D codes. We believe that this proposed method to implement the nonlocal operation is a realistic one for 2D stabilizer layouts and would be beneficial in avoiding the large overheads caused by magic state distillation.

Figure 1

Two instances of the 2D hexagonal color code of distance (a) $d=3$ and (b) $d=5$. In each case, a set of independent edges $\left\{H_{e_i}\right\}$, shown in red, can be chosen as the set that will form the $Z$ gauge operators when paired with the identical edge from another code copy, thus forming weight-4 gauge operators $\left\{H_{e_i}^{(2k-1)}{H}_{e_i}^{\left(2k\right)}\right\}$.

Figure 2

(a) Graphical representation of the primal lattice of the $(d-1)+1$ stacked code formed by stacking different copies of 2D color codes, shown here for $d=5$. The copies of the 2D code are coupled either by measuring gauge operators or logical operator pairs (shown in blue) between the different layers. (b) Dual lattice for the 3D stacked code ($d=5$). Vertices represent cell stabilizers in the primal lattice and edges represent faces shared by connected stabilizers.

Figure 3

Examples of the different representations of equivalent logical error strings that exist in the 3D stacked code. The color of the logical strings are chosen according to the color of the edges they follow. The curved lines represent joining of edges through a stabilizer of the same color. In (a), because the string lies on the green-yellow boundary, it can be chosen to be either of the complementary colors, blue or purple. In (b), the error string connects the bottom blue boundary to the joint boundary of the other three colors at the ancilla qubit, following blue edges. Example (c) shows how multiple colored boundaries can fuse in the bulk, thus negating the excitation that would otherwise be present.

Figure 4

A 2D layout for the implementation of the stacked code ($d=5$ shown). Pairs of copies of the 2D hexagonal color code are layered on top of one another in a single 2D layer, in such a way as to keep the gauge operators geometrically local. (a) Initial layout of the stacked code transformation in 2D. The 2D layers $\left(2k\right)$ and $(2k+1)$ are coupled by measuring joint logical $X$ and $Z$ operators (Bell stabilizers), with supporting qubits shown in blue. Although Bell stabilizers for the stacked code are high weight, involving all blue qubits, the only required measurements are those associated with local 2D stabilizer/gauge operators together with one-dimensional operators of weight $O\left(d\right)$ (shaded blue). The only 2D plane that is not initially coupled to another layer (or ancilla qubit) is the bottom $k=1$ layer, which stores the encoded qubit. (b) Measurement of the weight-4 $Z$-type gauge operators, shown in red. $X$-type stabilizers from individual layers are combined to form cell-like stabilizers by stabilizer evolution. Original joint logical $X$ measurements, given by blue shaded region, are mapped to all blue qubits.

Figure 5

A two-dimensional layout of the construction presented in Fig. 4. The two originally superimposed lattices have respective gray and white lattice qubits. Only one of the color code stabilizers (per pair) have been colored, for clarity. Gauge measurement operators are given by red faces. Here, we have identified three individual gauge measurements per pair of codes for clarity, there are actually $3(d^2-1)/8$ such gauge measurements for each pair of distance $d$ codes.

Figure 6

Initial coupling of split regions of a 2D color code. The original code is split into multiple color code copies by turning off and modifying certain stabilizer measurements. Different patches are coupled to form a Bell pair by measuring joint logical $X$ and $Z$ stabilizers between them, shown in blue. The patch that is not coupled in this way retains the quantum information that was originally stored in the code. The different patches are then further joined together by measuring gauge operators by matching up weight-2 edges from the different patches (forming weight-4 gauge operators), shown by red and cyan edges.