Braiding Majorana fermions and creating quantum logic gates with vortices on a periodic pinning structure

We show how vortices that support Majorana fermions when placed on a periodic pinning array can be used for vortex exchange and independent braiding by performing a series of specific moves with a probe tip. Using these braiding operations, we demonstrate realizations of a Hadamard and a cnot gate. We specifically consider the first matching field at which there is one vortex per pinning site, and we show that there are two basic dynamic operations, move and exchange, from which basic braiding operations can be constructed in order to create specific logic gates. The periodic pinning array permits both control of the world lines of the vortices and freedom for vortex manipulation using a set of specific moves of the probe during which the probe tip strength and height remain unchanged. We measure the robustness of the different moves against thermal effects and show that the three different operations produce distinct force signatures on the moving tip.

Figure 1

Schematic of the system, consisting of a superconductor (pink) coupled to a topological insulator (green). A magnetic field $\mathbf{B}$ is applied perpendicular to the layers. The superconducting layer contains a square array of blind holes (yellow) that each capture one superconducting vortex (pink columns), and a Majorana fermion is localized inside each vortex. An MFM tip is used to manipulate individual vortices.

Figure 2

Pinning site locations (open circles), vortex locations (filled pink circles), vortex trajectories (magenta lines), trapped vortex location (filled green circle), trapped vortex trajectory (red line), and probe tip trajectory (green line) for the fundamental operations. (a) The capture operation in which the probe tip moves along the $\langle 11\rangle$ direction over the pinning site and the vortex inside the pin becomes trapped by the probe tip, followed by the move operation in which the vortex is carried by the probe tip to a new location. In the illustrated trajectory, the probe tip moves the captured vortex a distance $2a$ in the $x$ direction and $2a$ in the $y$ direction, where $a$ is the pinning lattice constant. Only the vortex trapped by the probe tip is depinned. (b) The reposition operation where the empty probe tip moves along the $\langle 10\rangle$ direction over the pinning sites and does not capture a vortex. Individual vortices are temporarily pushed a short distance out of their pinning sites by the probe tip, but fall back into their original pinning sites as the probe tip moves away from the pin. In the illustrated trajectory, the probe is moving counterclockwise.

Figure 3

Plots of the parameter regimes indicating where the fundamental operations are performed successfully (red) or unsuccessfully (blue) as a function of probe tip radius $R_{\mathrm{tr}}$ vs probe tip spring constant $F_{\mathrm{tr}}/R_{\mathrm{tr}}$ in a sample with lattice constant $a=2\lambda$ and pinning strength $F_p=0.3$. (a) The capture operation in which the tip crosses the pin along $\langle 11\rangle$. (b) The reposition operation in which the tip crosses the pins along $\langle 10\rangle$. (c) The combination of capture and reposition, with red indicating the region of parameter space in which both operations can be achieved simultaneously.

Figure 4

The world lines as a function of position and time for an exchange operation of vortex 0 (red) with vortex 1 (blue).

Figure 5

Pinning site locations (open circles), vortex locations (filled circles), vortex trajectories (lines), and probe tip trajectory (green line) for two-vortex exchange operations between vortex 0 (red circle and trajectory) and vortex 1 (blue circle and trajectory). Arrows indicate the motion of the tip in the capture (magenta), move (yellow), and reverse capture (pink) stages. (a) A $\langle 10\rangle$ exchange. (b) A $\langle 11\rangle$ exchange.

Figure 6

The $x$ (a) and $y$ (b) positions vs time in simulation steps for vortex 0 (red), vortex 1 (blue), and the probe tip (green) for the $\langle 10\rangle$ exchange illustrated in Fig. 5. The vertical dashed lines delineate the four stages of probe tip motion. Magenta: Capture; yellow: $\langle 10\rangle$ moves; pink: Reverse capture.

Figure 7

The $x$ (a) and $y$ (b) positions vs time in simulation steps for vortex 0 (red), vortex 1 (blue), and the probe tip (green) for the $\langle 11\rangle$ exchange illustrated in Fig. 5. The vertical dashed lines delineate the four stages of probe tip motion. Magenta: Capture; yellow: $\langle 10\rangle$ move; pink: Reverse capture.

Figure 8

The world lines as a function of position and time for the exchange of non-nearest neighbor vortices 1 (blue) and 3 (pink) without entangling the world line of vortex 2 (green). (a) Type A nonlocal counterclockwise exchange (CCWE), in which the world lines of vortex 1 and 3 are always “above” the world line of vortex 2. (b) Type B nonlocal CCWE, in which the world lines of vortex 1 and 3 are always “below” the world line of vortex 2. (c) Two consecutive nonlocal CCWEs of type A of vortices 1 and 3. (c) A nonlocal CCWE of type A of vortices 1 and 3 followed by a nonlocal CCWE of type B of vortices 1 and 3.

Figure 9

The world lines as a function of position and time for a braiding operation of vortex 0 (red) and vortex 1 (blue).

Figure 10

Pinning site locations (open circles), vortex locations (filled circles), vortex trajectories (lines), and probe tip trajectory (green line) for braiding operations between vortex 0 (red circle and trajectory) and vortex 1 (blue circle and trajectory). Arrows indicate the motion of the tip in the capture (magenta), move (yellow), and reverse capture (pink) stages. (a) A $\langle 10\rangle$ braid. (b) A $\langle 11\rangle$ braid.

Figure 11

The $x$ (a) and $y$ (b) positions vs time in simulation steps for vortex 0 (red), vortex 1 (blue), and the probe tip (green) for the $\langle 10\rangle$ braid operation illustrated in Fig. 10. The vertical dashed lines denote the five stages of probe tip motion. Magenta: Capture; yellow: $\langle 10\rangle$ moves; pink: Reverse capture.

Figure 12

Pinning site locations (open circles), vortex locations (filled circles), vortex trajectories (lines), and probe tip trajectory (green line) for braiding operations between vortex 0 (red circle and trajectory) and a non-neighboring vortex (blue circle and trajectory). (a) Braiding of vortices that are separated by two lattice constants. (b) Braiding of distant vortices.

Figure 13

The world lines as a function of position and time for a Hadamard gate involving vortex 0 (red), 1 (blue), and 2 (green), along with the world line of vortex 3 (pink) which is part of the qubit definition but is not directly involved in the manipulations.

Figure 14

Pinning site locations (open circles), vortex locations (filled circles), vortex trajectories (lines), and probe tip trajectory (green line) for a Hadamard gate involving vortex 0 (red circle and trajectory), vortex 1 (blue circle and trajectory), and vortex 2 (dark green circle and trajectory). Vortex 3, which is part of the qubit definition, is one of the pink vortices that is not directly involved in the manipulations.

Figure 15

Pinning site locations (open circles), vortex locations (filled circles), vortex trajectories (lines), and probe tip trajectory (green line) for the individual stages of operation of the Hadamard gate in Fig. 14 involving vortex 0 (red circle and trajectory), vortex 1 (blue circle and trajectory), and vortex 2 (dark green circle and trajectory). Vortex 3, which is part of the qubit definition, is one of the pink vortices that is not directly involved in the manipulations. Arrows indicate the motion of the tip in the capture (magenta), move (yellow), repositioning (blue), and reverse capture (pink) stages. (a) First four stages consisting of a $\langle 10\rangle$ clockwise exchange between vortices 0 and 1. (b) Second four stages consisting of a $\langle 10\rangle$ counterclockwise exchange between vortices 0 and 2. (c) Repositioning of probe tip. (d) Final four stages consisting of a $\langle 10\rangle$ clockwise exchange between vortices 2 and 1.

Figure 16

The $x$ (a) and $y$ (b) positions vs time in simulation steps for vortex 0 (red), vortex 1 (blue), vortex 2 (dark green), and the probe tip (light green) for the Hadamard gate operation illustrated in Fig. 14. Vortex 3 is stationary and is not illustrated here. The vertical dashed lines indicate the 13 stages of probe tip motion. Magenta: Capture; yellow: $\langle 10\rangle$ moves; blue: Repositioning of probe; pink: Reverse capture.

Figure 17

The world lines as a function of position and time for a two-qubit cnot gate involving vortex 0 (red), 1 (blue), 2 (dark green), 3 (pink), 4 (orange), and 5 (violet).

Figure 18

Pinning site locations (open circles), vortex locations (filled circles), vortex trajectories (lines), and probe tip trajectory (green line) for a cnot gate involving vortex 0 (red circle and trajectory), 1 (blue circle and trajectory), 2 (green circle and trajectory), 3 (pink circle and trajectory), 4 (orange circle and trajectory), and 5 (violet circle and trajectory).

Figure 19

Pinning site locations (open circles), vortex locations (filled circles), vortex trajectories (lines), and probe tip trajectory (green line) for the individual stages of operation of the cnot gate in Fig. 18 involving vortex 0 (red circle and trajectory), 1 (blue circle and trajectory), 2 (green circle and trajectory), 3 (pink circle and trajectory), 4 (orange circle and trajectory), and 5 (violet circle and trajectory). Arrows indicate the motion of the tip in the capture (magenta), move (yellow), repositioning (blue), and reverse capture (pink) stages. (a) $\langle 10\rangle$ exchange between vortices 0 and 1 followed by repositioning of the probe tip to the pinning site occupied by vortex 2. (b) $\langle 10\rangle$ exchange between vortices 2 and 3 followed by a $\langle 10\rangle$ exchange between vortices 2 and 4. (c) $\langle 10\rangle$ exchange between vortices 2 and 5 followed by reposition of the probe tip to the pinning site originally occupied by vortex 4. (d) $\langle 10\rangle$ exchange between vortices 3 and 4. (e) $\langle 10\rangle$ exchange between vortices 3 and 5. (f) $\langle 10\rangle$ exchange between vortices 4 and 5.

Figure 20

The $x$ (a) and $y$ (b) positions vs time in simulation steps for vortex 0 (red), vortex 1 (blue), vortex 2 (dark green), vortex 3 (pink), vortex 4 (orange), vortex 5 (violet), and the probe tip (black) for the cnot gate operation illustrated in Fig. 18. The vertical dashed lines indicate the 31 stages of probe tip motion. Magenta: Capture; yellow: $\langle 10\rangle$ moves; blue: Repositioning of probe; pink: Reverse capture.

Figure 21

The fidelity ${\epsilon }_{\mathrm{op}}$ of the operations vs the magnitude $F_v^T$ of the thermal noise on the vortices for varied probe tip fluctuation magnitudes $V_{\mathrm{tr}}^T=$ 0.02 (black), 0.04, 0.06, 0.08, 0.10, 0.12, 0.14, 0.16, 0.18, and 0.20 (magenta). (a) The capture operation. (b) The reposition operation. (c) The $\langle 10\rangle$ exchange. (d) The $\langle 11\rangle$ exchange, which shows a strong sensitivity to noise.

Figure 22

Pinning site locations (open circles), vortex locations (filled circles), vortex trajectories (lines), and probe tip trajectory (green line) for the $\langle 11\rangle$ exchange operation performed at a finite temperature of $F_v^T=0.1.$ Vortex 0 (red circle and trajectory) and vortex 1 (blue circle and trajectory) should be the only two vortices participating in the exchange; however, due to the thermal fluctuations, vortex 2 (dark green circle and trajectory) is able to depin and interfere with the operation.

Figure 23

Probe tip force signatures $F_x$ (a), $F_y$ (b), and $F_{\mathrm{tot}}$ (c) vs time in simulation steps for the capture (magenta) and $\langle 10\rangle$ move (yellow) operations illustrated in Fig. 2.

Figure 24

Probe tip force signatures $F_x$ (a), $F_y$ (b), and $F_{\mathrm{tot}}$ (c) vs time in simulation steps for the reposition (blue) operations illustrated in Fig. 2.

Figure 25

Probe tip force signatures $F_x$ (a), $F_y$ (b), and $F_{\mathrm{tot}}$ (c) vs time in simulation steps for the $\langle 10\rangle$ exchange operation illustrated in Figs. 5 and 6. Magenta: Capture; yellow: $\langle 10\rangle$ moves; pink: Reverse capture.

Figure 26

Probe tip force signatures $F_x$ (a), $F_y$ (b), and $F_{\mathrm{tot}}$ (c) vs time in simulation steps for the $\langle 10\rangle$ braiding operation illustrated in Figs. 10 and 11. Magenta: Capture; yellow: $\langle 10\rangle$ moves; pink: Reverse capture.

Figure 27

Probe tip force signatures $F_x$ (a), $F_y$ (b), and $F_{\mathrm{tot}}$ (c) vs time in simulation steps for the Hadamard gate operation illustrated in Figs. 14–16. Magenta: Capture; yellow: $\langle 10\rangle$ moves; pink: Reverse capture; blue: Repositioning of probe.

Figure 28

Probe tip force signatures $F_x$ (a), $F_y$ (b), and $F_{\mathrm{tot}}$ (c) vs time in simulation steps for the cnot gate operation illustrated in Figs. 18–20. Magenta: Capture; yellow: $\langle 10\rangle$ moves; pink: Reverse capture; blue: Repositioning of probe.