Braiding Majoranas in a linear quantum dot-superconductor array: Mitigating the errors from Coulomb repulsion and residual tunneling

Abstract

Exchanging the positions of two non-Abelian anyons transforms between many-body wavefunctions within a degenerate ground-state manifold. This behavior is fundamentally distinct from fermions, bosons and Abelian anyons. Recently, quantum dot-superconductor arrays have emerged as a promising platform for creating topological Kitaev chains that can host non-Abelian Majorana zero modes. In this work, we propose a minimal braiding setup in a linear array of quantum dots consisting of two minimal Kitaev chains coupled through an ancillary, normal quantum dot. We focus on the physical effects that are peculiar to quantum dot devices, such as interdot Coulomb repulsion and residual single electron tunneling. We find that the errors caused by either of these effects can be efficiently mitigated by optimal control of the ancillary quantum dot that mediates the exchange of the non-Abelian anyons. Moreover, we propose experimentally accessible methods to find this optimal operating regime and predict signatures of a successful Majorana braiding experiment.

Figures (9)

• Figure 1: (a) Schematic of the minimal setup required for braiding in a linear array of quantum dots. Yellow squares are normal quantum dots, blue regions are superconducting leads mediating normal and Andreev tunneling. Purples lines are electrostatic gates to control the parameters. (b) Majorana representation of the Hamiltonian: The grey ovals with filled circles represent the Majorana operators $\gamma_{a, A/B}$ for dot $a$, lines represent effective couplings. By tuning its chemical potential, the ancillary dot D supplies two Majoranas forming a virtual trijunction together with dots $L2$ and $R1$. (c) Schematic of the single Majorana exchange protocol. A full braid is implemented by varying $\mu_D$, $\Gamma_{L}$, and $\Gamma_R$ in sequence twice. $\mu_D, \Gamma_L$, and $\Gamma_R$ can be experimentally controlled via three electrostatic gates (dark purple). (d) Occupation probability of the $|oo\rangle$ state depending on $t$ over a full braiding operation. At times $3T$ and $6T$ the protocol implements exchange and full braid of the two Majoranas neighboring the ancillary dot respectively. The line highlights the change in parity the system undergoes during the protocol.
• Figure 2: Effects of interdot Coulomb interaction between ancillary dot and adjacent Kitaev chain dots. (a) and (d) show the infidelity in dependence of symmetric and asymmetric Coulomb energy respectively. (b) and (e) show local conductance spectroscopy through the ancillary dot. Due to the interaction, the excitation minimum shifts in chemical potential to a lower value corresponding to Eq. \ref{['eq:mu_Dstar']}. Retuning $\mu_{D,\mathrm{min}}$ to this value corrects the adverse effect of the Coulomb interaction. This is supported by (c) and (f) showing the infidelity in dependence of $\mu_{D,\mathrm{min}}$. In line with the excitation minimum, the infidelity reduces to zero when $\mu_{D,\mathrm{min}}=\mu_D^*$. The discontinuity at $\mu_D^*-\Gamma_0$ is due to our choice of $\mu_{D,\mathrm{max}}-\mu_{D,\mathrm{min}}=\Gamma_0$ where the occupied state on the dot becomes resonant with the states in the Kitaev chains. Measuring the traces as those presented in c) and f) experimentally can be considered a signature of Majorana braiding.
• Figure 3: (a) Infidelity, $1-F$, in the $(\Delta \mu_D, \Gamma_{\text{min}})$-plane. (b) Infidelity as a function of $\Delta \mu_D$ for different cuts of $\Gamma_{\mathrm{min}}$ in (a). (c) Infidelity as a function of $\Gamma_{\mathrm{min}}$ for different cuts of $\Delta \mu_D$ in (a).
• Figure 4: (a) and (c) Infidelity, $1-F$, in the $(\mu_{D,\text{min}}, \Gamma_{\text{min}})$ plane for $U_a=0$ and $U_a=10\Gamma_0$ respectively. For both, we choose $\Delta \mu_D=10\Gamma_0$. The dotted lines show the numerical minimum of the infidelity, solid lines correspond to the expectation of eqns. \ref{['eq:mu_D_opt']} and \ref{['eq:mu_Dstar']}. (b) and (d) Infidelity as a function of $\mu_{D,\text{min}}$ for different cuts in $\Gamma_{\text{min}}$ through (a) and (c) showing that for increasingly negative values of $\mu_{D,\mathrm{min}}$ the infidelity vanishes regardless of residual tunnel coupling. Note that panels a) and b) share the same label for $x$ axes, and so do panels c) and d).
• Figure 5: (a) Tunnel spectroscopy, $G_{DD}$, over the ancillary dot in the $(V, \varphi)$ plane. Only at odd integer multiples of $\pi$ the conductance indicates the necessary degeneracy at $V=0$. Additionally, the linear splitting of that degeneracy with phase indicates the lack of protection of the protocol against phase noise. (b) Infidelity, $1-F$, as a function of $\varphi$ for a single $T$. The oscillations indicate Rabi oscillations between the states in the ground-state manifold. (c) Infidelity of the double braid protocol averaged over multiple $T$. Since the outcome of the non-Abelian exchange does not depend on any specific choice of $T$, the perfect fidelities at odd interger multiples of $\pi$ persist while the Rabi oscillations present in (b) average away.