Realizing universal Majorana fermionic quantum computation

Majorana fermionic quantum computation (MFQC) was proposed by S. B. Bravyi and A. Yu. Kitaev [Ann. Phys. (NY) 298, 210 (2002)], who indicated that a (nontopological) fault-tolerant quantum computer built from Majorana fermions may be more efficient than that built from distinguishable two-state systems. However, until now scientists have not known how to realize a MFQC in a physical system. In this paper we propose a possible realization of MFQC. We find that the end of a line defect of a $p$-wave superconductor or superfluid in a honeycomb lattice traps a Majorana zero mode, which becomes the starting point of MFQC. Then we show how to manipulate Majorana fermions to perform universal MFQC, which possesses possibilities for high-level local controllability through individually addressing the quantum states of individual constituent elements by using timely cold-atom technology.

Figure 1

(a) Illustration of the honeycomb lattice of two interpenetrating triangular lattices. Along the red (gray) links, the SC or SF order parameter is finite, while along the black links, the SC or SF order parameter is zero. (b) The order parameter $\Delta$ vs the interaction strength $U$ for the case of $t^{\prime }=0.01t$ at half filling at zero temperature. (c) The order parameter $\Delta \left(T\right)$ vs the temperature $T$ for the case of $t^{\prime }=0.01t$, $U=5.24t$ at half filling.

Figure 2

Edge states of the $p$-wave SC or SF on a honeycomb lattice for the system with an armchair open boundary. The parameters are chosen as $\Delta =1.34t$, $t^{\prime }=0.01t$, $\mu =0.0136t$. Along the red (gray) links, the SC or SF order parameter is finite, while along the black links, the SC or SF order parameter is zero.

Figure 3

(a) The lattice vacancy is represented by the shaded dashed circle, and the two green (light gray) dots denote the two Majorana modes, ${\gamma }_L,$ ${\gamma }_R$. (b) Half of the energy splitting ${\varepsilon }_i/\left(2t\right)$ vs on-site potential $V_{i_0}/t$. The parameters are chosen as $\Delta =1.34t,$ $t^{\prime }=0.01t,$$\mu =0.0136t$.

Figure 4

(a) Illustration of an LD. Two Majorana modes, ${\gamma }_L$ and ${\gamma }_R$, denoted by green (light gray) dots, are localized at ends of the LD. (b) Illustration of an LD created by an addressing line-laser beam in an optical lattice.

Figure 5

(a) Illustration of two line defects. The two Majorana fermions, ${\gamma }_{jL}$ and ${\gamma }_{iR}$, are denoted by the green (light gray) dots located at the ends of LD $j$ and LD $i$, respectively. (b) Half of the energy splitting $\Delta E/t$ vs the distance between the ends of the two LDs shown in (a). The parameters are chosen as $\Delta =1.34t,$ $t^{\prime }=0.01t,$ $\mu =0.0136t$.

Figure 6

Illustration of four Majorana modes induced by two LDs that can be combined into two complex fermions ($c_i,$ $c_{i+{\mathbf{a}}_3}$).

Figure 7

The random potential effect for the energy gap $\Delta E_g$ of the SF or SC. The energy gap will be closed at about $V_a\ge 3.5t.$

Figure 8

(a) The DOS of the line defects for $\sqrt{\langle {\left(V_{rp,i}\right)}^2\rangle }=0$. (b) The DOS of the line defects for $\sqrt{\langle {\left(V_{rp,i}\right)}^2\rangle }=0.5$.

Figure 9

Illustration of $p$-wave SF order for three different cases. (a) Case I, with $p$-wave SF order along the ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ directions, which are denoted by red (gray) links. (b) Case II, with $p$-wave SF order along the ${\mathbf{a}}_2$ and ${\mathbf{a}}_3$ directions, which are denoted by red (gray) links. (c) Case III, with p-wave SF order along the ${\mathbf{a}}_1$ and ${\mathbf{a}}_3$ directions, which are denoted by red (gray) links.

Figure 10

The per unit cell ground-state energy difference $\Delta E_{\mathrm{uc}}=E_{\text{unit-cell}}^{\left(1\right)}-E_{\text{unit-cell}}^{\left(2\right)}=E_{\text{unit-cell}}^{\left(1\right)}-E_{\text{unit-cell}}^{\left(3\right)}$ between the SF of case I and that of case II and between the SF of case I and that of case III.