Entangling spins in double quantum dots and Majorana bound states

We study the coupling between a singlet-triplet qubit realized in a double quantum dot to a topological qubit realized by spatially well-separated Majorana bound states. We demonstrate that the singlet-triplet qubit can be leveraged for readout of the topological qubit and for supplementing the gate operations that cannot be performed by braiding of Majorana bound states. Furthermore, we extend our setup to a network of singlet-triplet and topological hybrid qubits that paves the way to scalable fault-tolerant quantum computing.

Figure 1

Two possible physical realizations of our proposal. (a) The linear arrangement with two quantum dots (blue cylinders), $L$-QD and $R$-QD, and two TNWs (red) hosting MBSs at their ends, ${\gamma }_L$ and ${\gamma }_R^{\prime }$, which are realized in the same NW. There are two additional NWs that allow for braiding of the MBSs. The amplitude of the coupling between ${\gamma }_L\left({\gamma }_R^{\prime }\right)$ and $L$-QD $(R$-QD), $t_L\left(t_R\right)$, is controlled by a local backgate (green). The interdot tunneling $t_M$ is similarly controlled by a back gate (orange). (b) The same model can be realized by four NWs forming a hash tag. In this case, the QDs are realized at the intersection of the two vertical NWs with the upper horizontal NW. The TNWs are on the left and right sides of the lower wire (red). We assume a large controllable barrier between the left and right TNWs so there is no overlap between ${\gamma }_L$ and ${\gamma }_R^{\prime }$. The physical parameters are defined analogously to the first setup. We note that the “hash-tag” geometry requires two tunnel barriers to control the coupling between the dots and the MBS.

Figure 2

Schematic energy diagram demonstrating two possible regimes of the coupling between QDs and MBSs. (a) In the first regime, the single-particle levels of the QDs are tuned close to MBS energies, ${\epsilon }_0,\tilde{V}\ll E_{R,L}^Z/2\ll \mathrm{\Delta }$. (b) In the second regime, the single-particle levels of QDs are tuned away from zero, $E_{R,L}^Z\ll {\epsilon }_0,\tilde{V}\ll \mathrm{\Delta }$. Here, $\mathrm{\Delta }$ is the proximity induced gap, ${\epsilon }_0$ is the detuning of the DQD levels with respect to the zero of energy of MBSs, and $E_j^{\mathrm{Z}}$ the Zeeman splittings on the QDs.

Figure 3

The protocol to generate additional two-qubit gates utilizing the $YY(\pi /4)$ gate and single-qubit gates. The subscripts $S$ and $M$ referring to a gate acting on the $S{T}_0$ and MBS qubits in the main text are omitted in this figure for simplicity. Operations by $X\left(\varphi \right),Y\left(\varphi \right)$, or $Z\left(\varphi \right)$ on the $S{T}_0$ or MBS qubit branch corresponds to a rotation about the $x,y$, or $z$ axis of the Bloch sphere, respectively, of the appropriate qubit. (a) An Ising $XX(\pi /4)$ gate can be obtained from an Ising $YY(\pi /4)$ gate using single-qubit operations which can be further manipulated into a $U_{\text{CNOT}}^{\mathrm{MBS}}$ gate. (b) The Ising $YY(\pi /4)$ gate is transformed into a $U_{\text{CNOT}}^{S{T}_0}$ gate by similar single-qubit operations. (c) A $U_{\text{SWAP}}$ gate can be obtained by successive application of three CNOT gates. (d) An Ising $XY(\pi /4)$ can be made from the Ising $YY(\pi /4)$ gate by involving an appropriate exchange of MBSs.

Figure 4

(a) A network of $S{T}_0$-MBS hybrid qubits in the case when the TNWs and DQD are formed within the same NW [see Fig. 1]. Qubits are selectively, long-range coupled by moving the DQDs in the neighborhood of a floating gate, e.g., qubit (1) is coupled to qubit (2). Braiding of the MBSs can be performed by using the double T-junction as in qubit (3). (b) A network of $S{T}_0$-MBS hybrid qubits formed within a hashtag of NWs [see Fig. 1]; a single hybrid qubit is drawn within the dashed blue line. Long-distance coupling between qubits is mediated by jelly bean quantum dots (JBs) that are tunably coupled to the DQDs. (c) The legend: DQD denotes a double quantum dot, TNW denotes a topological nanowire, JB denotes jelly bean quantum dot, FG denotes the floating gate, and NW denotes a nanowire in the nontopological regime.