Interface between Topological and Superconducting Qubits

We propose and analyze an interface between a topological qubit and a superconducting flux qubit. In our scheme, the interaction between Majorana fermions in a topological insulator is coherently controlled by a superconducting phase that depends on the quantum state of the flux qubit. A controlled-phase gate, achieved by pulsing this interaction on and off, can transfer quantum information between the topological qubit and the superconducting qubit.

Figure 1

On the surface of the TI, patterned SC islands can form (a) STIS quantum wire, (b) flux qubit, and (c) a hybrid system of topological and flux qubits. (a) Two MFs (red dots) are localized at two SC trijunctions, connected by an STIS quantum wire (dashed blue line). The coupling between the MFs is controlled by the SC phases ${\varphi }_d=\epsilon$ and ${\varphi }_u=-\pi$. (b) A flux qubit consists of four JJs connecting four SC islands ($a$, $b$, $c$, $d$) in series, enclosing an external magnetic flux $f{\Phi }_0$. (c) The hybrid system consists of an STIS wire and a flux qubit. The STIS wire (between islands $d$ and $u$) couples the MFs, with coupling strength controlled by the flux qubit. The SC phase ${\varphi }_c$ can be tuned by a phase controller (not shown), and ${\varphi }_d={\varphi }_c\pm {\theta }_4^{*}$ with the choice of $\pm$ sign depending on the state of the flux qubit.

Figure 2

(a) The energy splitting $E(\epsilon )$ (in units of $\Delta E=v_F/L$) as a function of ${\Lambda }_{\epsilon }=\frac{{\Delta }_{0}L}{v_F}\sin \epsilon /2$. (b) A contour plot of potential energy $U$ as a function of $\{{\theta }_1,{\theta }_2,{\theta }_4\}$ with ${\theta }_3=\pi -{\theta }_1-{\theta }_2-{\theta }_4$. There are two potential minima associated with flux qubit states $|0⟩$ and $|1⟩$. (c) A contour plot of $U$ as a function of $\{{\theta }_1,{\theta }_4\}$ with ${\theta }_1={\theta }_2$ and ${\theta }_3=\pi -2{\theta }_1-{\theta }_4$. (d) Marginal probability distributions of ${\theta }_4$ associated with states $|0⟩$ (blue solid line) and $|1⟩$ (red dashed line). The parameters are $E_J/E_C=80$ and $\{E_{J,i}/E_J{\}}_{i=1,2,3,4}=\{1,1,\alpha =0.8,\beta =10\}$.

Figure 3

Comparison between the phase separation $\Delta \epsilon \propto {\beta }^{-1}$ (black solid line) and the magnitude of quantum fluctuations $\zeta \propto {\beta }^{-1/2}$ (purple dashed line), assuming $E_J/E_C=80$.