Performance of Planar Floquet Codes with Majorana-Based Qubits

Quantum error correction is crucial for any quantum computing platform to achieve truly scalable quantum computation. The surface code and its variants have been considered the most promising quantum error correction scheme due to their high threshold, low overhead, and relatively simple structure that can naturally be implemented in many existing qubit architectures, such as superconducting qubits. The recent development of Floquet codes by Hastings and Haah offers another promising approach. By going beyond the usual paradigm of stabilizer codes, Floquet codes achieve similar performance while being constructed entirely from two-qubit measurements. This makes them particularly suitable for platforms where two-qubit measurements can be implemented directly, such as the measurement-only topological qubits based on Majorana zero modes (MZMs) proposed by Karzig et al. Here, we explain how two variants of Floquet codes can be implemented on MZM-based architectures without any auxiliary qubits for syndrome measurement and with shallow syndrome-extraction sequences. We then numerically demonstrate their favorable performance. In particular, we show that they improve the threshold for scalable quantum computation in MZM-based systems by an order of magnitude and significantly reduce space and time overheads below threshold.

Popular Summary

Quantum error correction is crucial for any quantum computing platform to achieve truly scalable quantum computation. However, leading quantum error-correction schemes dramatically increase, by factors of hundreds or thousands, the space and time required to implement useful algorithms compared to non-error-corrected implementations. A new family of “Floquet” codes seems promising for balancing overhead with structural requirements on physical qubits. We quantify the performance of these codes, finding that their space and time requirements can be substantially lower compared to the state of the art.

Floquet codes offer a structure that is particularly suitable for platforms where two-qubit measurements can be implemented directly, such as qubits based on “Majorana zero modes” (MZMs). We explain how two variants of Floquet codes can be implemented efficiently on MZM-based architectures. We then numerically demonstrate their favorable performance compared to other quantum error-correcting codes. For an MZM architecture, the Floquet codes tolerate more noise, require fewer (physical) qubits, and need less time than schemes based on other codes.

Figure 1

(a) The tiling unit for the honeycomb code. Qubits are indicated by black dots. We assign a check operator $ZZ$ for each horizontal edge, $XX$ for each edge with positive slope, and $YY$ for each edge with negative slope. (b) The parallelogram boundary conditions for the honeycomb code. The solid-color edges are measured in a period-6 sequence (yellow, blue, green, yellow, green, blue). The curved edges are Pauli operators, chosen such that the triple of check operators meeting at a vertex are pairwise anticommuting. (The degree-4 vertex in the top left corner is treated specially; see Table 1.) For example, the curved green edges along the bottom are assigned $XY$ operators (from left to right). The striped black-yellow edges, bordering green faces, are measured during the first yellow round, while the dashed black-yellow edges, bordering blue faces, are measured during the second yellow round.

Figure 2

(a) The tiling unit for the 4.8.8 Floquet code. As in Fig. 1, qubits are indicated by black dots. Every diagonal edge is $YY$, every horizontal edge $ZZ$, and every vertical edge $XX$. (b) The rectangular boundary conditions for the 4.8.8 Floquet code. As in the honeycomb code, the solid-color edges are measured in a period-6 sequence (yellow, blue, green, yellow, green, blue). The dashed and striped black-yellow edges are the same Pauli type as the solid yellow edges but are measured at different rounds in the measurement sequence. The dashed edges, incident to green faces, are measured during the first yellow round. The striped edges, incident to blue faces, are measured during the second yellow round.

Figure 3

The square lattice of tetrons used to implement the honeycomb and 4.8.8 Floquet codes. The topological superconducting wires (dark blue) have a MZM at either end. The qubit islands (light-blue shaded) correspond to two parallel topological wires joined by a trivial superconducting backbone (teal), with MZMs (red stars) labeled according to the box in the upper left. Rows of tetrons are separated by coherent links (floating topological wires). Neighboring qubit islands are connected by semiconducting segments (tan), with two semiconducting columns separating each column of qubits.

Figure 4

(a) The vertical brick-wall unit of the honeycomb lattice. (b) The vertical brick-wall unit of the 4.8.8 lattice. (c) The bulk-measurement loops. The colors refer to the measurement time steps, rather than the Pauli operators.

Figure 5

The boundary measurements for the 4.8.8 Floquet code. (a) 2-gons along vertical boundaries require $ZZ$ between vertically adjacent qubit islands, which can be implemented using both semiconducting columns to the left or right of the qubit. (b) 2-gons along horizontal boundaries require $ZX$ and $XZ$ between horizontally adjacent qubit islands, which can be implemented using the coherent links above or below the qubit islands. The white dot indicates that, for convenient physical implementation, the roles of the $X$ and $Z$ operators are swapped for that qubit as compared to the bulk. For example, a horizontal edge is assigned $ZZ$ in the bulk but becomes either $XZ$ or $ZX$ when an end point is marked by the white dot.

Figure 6

The boundary measurements for the honeycomb Floquet code. (a) 2-gons along vertical boundaries require $YZ$ or $ZY$ between vertically adjacent qubit islands, which can be implemented using both semiconducting columns to the left or right of the qubit and the coherent link between the qubits. (b) 2-gons along horizontal boundaries require $ZY$ and $XZ$ or $YZ$ and $ZX$ measurements between horizontally adjacent qubit islands, which can be implemented using the coherent links above or below the qubit islands. As in Fig. 5, the white dot indicates that the roles of the $X$ and $Z$ operators are swapped for that qubit as compared to the bulk. Note that the $XZ$ and $ZX$ measurement loops are shown in Fig. 5.

Figure 7

Logical error rate estimates for the honeycomb and 4.8.8 Floquet codes near threshold: (a) honeycomb torus; (b) honeycomb parallelogram; (c) 4.8.8 torus; (d) 4.8.8 rectangle. In the legend, $d$ is the effective distance of the code. The dashed gray line indicates Pr[logical error]$=\mathrm{Pr}$[physical error].

Figure 8

(a) Empirical logical error estimates for the honeycomb and 4.8.8 Floquet codes with planar boundaries. Effective distances (3,5,7,9) are shown. (b) Heuristic estimates of $0.25(p/0.012{)}^{(d+1)/2}$ as a function of the effective distance $d$ for the parallelogram honeycomb code. Empirical estimates are shown as solid lines for reference. (c) Heuristic estimates of $0.07(p/0.01{)}^{(d+1)/2}$ for the rectangular 4.8.8 Floquet code.

Figure 9

Overhead estimates for a target logical error rate of ${10}^{-12}$ with respect to (a) the number of qubits (space), (b) the measurement depth (time), and (c) the space-time volume (number of qubits $\times$ measurement depth). The solid lines are based on empirical estimates from Fig. 8, while the dashed lines are based on heuristic estimates from Figs. 8 and 8. For reference, corresponding surface-code estimates from Ref. [27] are shown in blue. The gray lines along the bottom show the improvement factor of the Floquet codes relative to the surface code. They are obtained by taking the minimum of the two surface-code values (double or windmill), the minimum of the two Floquet code values (honeycomb or 4.8.8), and computing the ratio surface/Floquet.

Figure 10

The six-step measurement sequence for the planar 4.8.8 code patch shown in Fig. 2, as implemented in an array of tetron qubits.