Towards Realistic Implementations of a Majorana Surface Code

Surface codes have emerged as promising candidates for quantum information processing. Building on the previous idea to realize the physical qubits of such systems in terms of Majorana bound states supported by topological semiconductor nanowires, we show that the basic code operations, namely projective stabilizer measurements and qubit manipulations, can be implemented by conventional tunnel conductance probes and charge pumping via single-electron transistors, respectively. The simplicity of the access scheme suggests that a functional code might be in close experimental reach.

Figure 1

Structure of a surface code built on MBSs [red dot in panel (a)], a few of which define a Majorana-Cooper box [(b)]. Ring exchange of MBSs via tunneling bridges [(c)] defines a physical qubit through the eigenvalues $\pm 1$ of the plaquette operator ${\mathcal{O}}_n$, i.e., the product of all eight Majorana operators in the loop. An assembly of qubits [(d)] can encode information qubits, e.g., the unit indicated by a dark shaded area. For further discussion, see text.

Figure 2

Surface code setup. Pairs of normal leads (indicated by vertical lines for clarity) are tunnel coupled to MBSs ${\gamma }_1$ and ${\gamma }_2$ located on different boxes. The two-terminal conductance measurement provides information about plaquettes ${\mathcal{O}}_A$ and ${\mathcal{O}}_B$, cf. Eq. (4). Using SETs in the external circuit, plaquettes can be flipped as discussed in the text.

Figure 3

Concrete design of a 2D on-chip implementation. The Majorana-Cooper box hosts four MBSs (red dots) and is implemented as a 90 deg rotated $H$ structure from a pair of second-generation topological nanowires [17] joined by a mesoscopic superconducting slab (“SC”). This slab ensures a common charging energy for the box. One normal lead (“$|$”) and one SET (along with its gate electrode, ‘‘$g$”) are tunnel coupled to each MBS by employing $T$ junctions (see text), such that there are eight contacts per box. MBSs on neighboring boxes are connected by short wires via tunnel contacts (‘‘$t$”).