Negative Hybridization: a Potential Cure for Braiding with Imperfect Majorana Modes

Abstract

Majorana zero modes, the elementary building blocks for the quantum bits of topological quantum computers, are known to suffer from hybridization as their wavefunctions begin to overlap. This breaks the ground state degeneracy, splitting their energy levels and leading to an accumulation of error when performing topological quantum gates. Here we show that the energy splitting of the Majorana zero modes can become negative, which can be utilized to reduce the average hybridization energy of the total gate. We present two illustrative examples where negative hybridization suppresses gate errors to such an extent that they remain below the fault-tolerance threshold. As an intrinsic property of Majorana zero modes, negative hybridization enables systems based on imperfect Majorana zero modes to regain functionality for quantum information processing.

Figures (5)

• Figure 1: Energy hybridization of two MZMs performing a $\mathbf{\sqrt{\rm Z}}$ gate. (a) Sketch of two MZMs (colored dots) being exchanged on a T-junction geometry, performing a $\sqrt{\rm Z}$ gate. Light and dark red sections correspond to topological and trivial parameters, respectively. (b) Example of energy splitting between the two MZMs as a function of time $t$ (with total braid time $T$). (c) Fidelity $\mathcal{F}$ corresponding to (b) for different braid times $T$. The braid times where the fidelity exceeds the threshold of 99% to be error correctable are highlighted in yellow. (d, e) Same as (b, c) but with slightly different parameters, leading to weak negative hybridization. $\mathcal{F}$ vs. $T$ shows enlarged yellow region. (f, g) Same as (b, c) and (d, e) but with parameters which induce significant negative hybridization and thus vanishing $\bar{E}$. As a consequence, Majorana oscillations are absent and $\mathcal{F}$ exceeds 99% irrespective of $T$.
• Figure 2: Negative hybridization induced by a local gate potential. (a) The hybridization energy between the MZMs throughout the braid, for varying applied voltage strengths, $V$, located at the site below the junction. As $V$ increases, the main parity swaps occur earlier in the braid, leading to greater amounts of negative hybridization. (b) Integrated energy, $\bar{E}$, for the braid as a function of $V$. The dashed line at $V=1.27\tilde{t}$ marks the potential where the braid has $\bar{E}=0$. Inset: T-junction with the additional local gate highlighted in black. (c) Fidelity of the braid for various $V$ and $T$. Around $V=1.27\tilde{t}$ the fidelity remains close to 1, even for large values of $T$.
• Figure 3: Symmetric braids: errors of corrected and uncorrected single-qubit gates. (a) Schematic of the corrected $\sqrt{X}$ gate showing MZMs (colored) at the borders of the topological (light red and light blue) and trivial (dark red and dark blue) regions. The red regions correspond to positive chemical potentials, while the blue ones indicate that the chemical potential is negative. (b) Energy spectrum of the Majorana bound states for the corrected $\sqrt{X}$ gate as function of time, labeled with the braid operations taking place. (c) Error in fidelity for $\sqrt{X}$ gate vs. braid times. The red region covers the fidelities which exceed the 1% error threshold. The green curve is the uncorrected result, all other lines have been corrected by means of symmetric braids. $\delta=0$ (solid black) is perfectly symmetric, and finite $\delta$ values (dashed) indicate deviation from perfect symmetry. (d) Same as (c) but for the $\sqrt{Z}$ gate.
• Figure 4: CNOT gate error. (a) Loss of fidelity for the corrected and uncorrected CNOT gates vs. braid time, compared to the natural time evolution of the system. Results slightly depend on initial states, leading to the spread of the uncorrected curve. (b) Average fidelities for the corrected and uncorrected gates over a range of chemical potential values, demonstrating the robustness of the protocol. Results are for the braid time $T=4000\hbar/\tilde{t}$, indicated by the dashed line in (a) The dots indicate common data between (a) and (b).
• Figure 5: Schematic of extended T-junction system. The sites are represented by dots, with adjacent ones being connected by hopping and superconducting terms. The color denotes the chemical potential at the site in the initial configuration, where light red is $\mu_{\rm topo}$ and dark red is $\mu_{\rm triv}$. The sites enclosed within each box make up a T-junction, 3T-junction and 5T-junction for $M=1$, $M=2$ and $M=3$, respectively. The horizontal leg had $L_0$ sites, while the vertical legs have $L_i$, with the $i^{th}$ vertical leg attaches to the horizontal backbone at site $x_i$. The system has four buffer sites at each end, labeled by $l_{\rm buffer}$, while the spacing between MZMs on the horizontal leg is given by $L_h$ as shown.