Towards universal topological quantum computation in the $\nu =\frac{5}{2}$ fractional quantum Hall state

The Pfaffian state, which may describe the quantized Hall plateau observed at Landau level filling fraction $\nu =\frac{5}{2}$, can support topologically-protected qubits with extremely low error rates. Braiding operations also allow perfect implementation of certain unitary transformations of these qubits. However, in the case of the Pfaffian state, this set of unitary operations is not quite sufficient for universal quantum computation (i.e. is not dense in the unitary group). If some topologically unprotected operations are also used, then the Pfaffian state supports universal quantum computation, albeit with some operations which require error correction. On the other hand, if certain topology-changing operations can be implemented, then fully topologically-protected universal quantum computation is possible. In order to accomplish this, it is necessary to measure the interference between quasiparticle trajectories which encircle other moving trajectories in a time-dependent Hall droplet geometry [cond-mat/0512072].

Figure 1

A system with $n$ quasihole pairs (held at pairs of antidots, depicted as shaded circles) supports $n$ qubits. Additional antidots (hatched) can be used to move the quasiparticles, as described in Sec. 3.

Figure 2

(Color online) The exchange of two qubits through a third antidot.

Figure 3

Using an AFM tip to braid quasiparticles.

Figure 4

(Color online) Taking a quasiparticle from the edge around one of the quasiparticles in a qubit implements a logical NOT on the qubit.

Figure 5

(Color online) The state of a qubit can be determined from a measurement of the longitudinal conductance when interedge tunneling is allowed at two interfering junctions.

Figure 6

The interference between the two trajectories in Fig. 5 can be obtained from the Jones polynomial (operator) evaluated on the two diagrams in this figure. In (a) the qubit is in the state 0, while in (b) it is in state 1.

Figure 7

(Color online) When there is a charge $e∕4$ qusiparticle at the antidot in the middle of the device, then there is no interference between the two trajectories from $X$ to $Y$ contributing to the longitudinal conductance.

Figure 8

A quasiparticle trajectory that winds around the antidot (straight line) can be deformed into a single time slice or stretched over a long time.

Figure 9

Two antidots are merged for a short while and then separated again. The spacetime curve $\gamma$ encircles the merger region. The interference between the trajectories $p$ and $p^{\prime }$ measure the topological charge around this curve.

Figure 10

(Color online) Two antidots are merged for a short while and then separated again. By putting delays into the possible quasiparticle trajectories (depicted in red and blue), we can enable them to interfere. In order to give the two antidots enough time to merge, we might wish to make the velocities $v_1$, $v_2$ small.

Figure 11

(Color online) In (a), we have two pairs of quasiparticles. The dashed lines connect quasiparticles that fuse to form the (topologically) trivial particle. In (b), the second and third quasiparticles are exchanged. A measurement of the topological charge around the blue curve or an operation that takes another quasiparticle around this curve will give a nontrivial result as a result of this exchange. The same result is clearly obtained even if the final position of the quasiparticles is slightly different from those shown. If the exchange is incomplete, as in (c), then a measurement around the blue curve will give a non-trivial result but a measurement around the red curve will give a trivial result. As long as we are careful to measure the system with the blue curves in (b) and (c), we will find the correct result.

Figure 12

By bringing together quasiparticles 1 and 2, which form a qubit, we can apply the gate $g_1$. This operation is unprotected and requires control of the distance between the quasiparticles and the length of time that they are brought together. However, the required precision might not be very stringent as a result of the availability of topologically protected operations such as $g_3$.

Figure 13

(Color online) A nondemolition measurement of the total topological charge of two neighboring qubits (comprised of 1, 2 and 3, 4) can be done with the interference measurement shown here. Together with $g_3$ and the operation $U_P$ shown in Fig. 12, this forms a universal gate set.

Figure 14

Quasiparticles 1 and 2 form a qubit; 3 and 4 form a second qubit. Exchanging 2 and 3 applies the gate $g_3$.

Figure 15

By adding an overpass, we connect two antidots. The dashed curve labeled $B_1$ is the combined boundary of the antidots-plus-overpass. In the implementation of both $g_1$ and $g_2$, we need to check that $W\left(B_1\right)=1$. In the case of $g_2$, the controlled qubit must be taken around $C$ in order to implement a controlled phase gate on it. To enact $g_1$, a double Dehn twist must be performed in $C$. The two antidots joined by an overpass in the middle of the figure are topologically equivalent to a torus with two punctures corresponding to $B_0$ and $B_1$ as shown at the bottom. The meridian of the torus is $C$.

Figure 16

A nonplanar overpass configuration can be mimicked by breaking the region $A$ and adding $B$. This is done by merging the two antidots between times time $t_1$ and $t_2$. During this merger interval, we allow the quantum Hall fluid the fill the region $B$. This faux pass splits the merged antidot in the perpendicular direction. These changes are then undone to return the system to its initial configuration. In order to verify that removing $A$ was harmless, we have to perform a tilted interferometry measurement to check that the topological charge around $\gamma$ is trivial. See Fig. 20, later, for a time slicing of this process.

Figure 17

Topologically, the process depicted here is equivalent to that depicted in Fig. 16. However, as a result of the tilt of the faux pass, the triviality of the charge around the hole in $A$ can be measured with an untilted measurement. Consequently, the curve $\gamma$ can now be deformed into a single time slice. The curve labelled $C$ in this figure is equivalent to the curve $C$ that goes over the overpass in Fig. 15. This is the curve on which we wish to perform a Dehn twist. Measuring the topological charge around this curve allows us to effect the same transformation on the quantum state of the system without any surgery. See Fig. 24, later, for a time slicing of this process.

Figure 18

The curve in (a) can be framed, for example, as shown in (b) and (c). The self-linking number in (c) is greater by $+1$ than in (b).

Figure 19

If we measure the topological charge around $({\zeta }_1,-1)$ and it is not 1, then we have performed a Dehn filling on $({\zeta }_1,-1)$ and left behind a Wilson loop $\beta$, carrying this topological charge, inside the solid torus. The state of the system is given by the functional integral in Eq. (24) for this manifold. The 3-D mapping cylinder of a Dehn twist can be described as follows: delete a solid torus, then replace it with a twisted Dehn filling. For convenience, we have drawn the torus in a nonstandard way so that $L-M$ appears simple.

Figure 20

(Color online) A nonplanar overpass configuration can be mimicked by breaking the region $A$ and adding $B$. This is done by merging the two antidots at time $t_1$ and splitting them in the other direction. The faux pass $B$ is then used to either apply $g_1$ or $g_2$, as described in Sec. 7. These changes are then undone to return the system to its initial configuration. In order to verify that removing $A$ was harmless, we have to perform a tilted interferometry measurement in which the red and yellow trajectories interfere.

Figure 21

(Color online) The interference between the red and yellow trajectories is a measurement of the topological charge around $B_1$. In order to apply $g_2$, we take the controlled qubit across the faux pass.

Figure 22

(Color online) The quasiparticle trajectories shown in Figs. 20, 21 may actually look more like the trajectory on the right above: a series of hops from one small antidot to another.

Figure 23

(Color online) In order to apply $g_1$, we must create a pair of auxiliary quasiparticles, take one over the faux pass (not shown), and then annihilate them again. While this is occurring, we must measure the topological charge around a curve that follows the auxiliary quasiparticle over the faux pass, while encircling it. By encircling the auxiliary quasiparticle (presumably while hopping along a bucket brigade of antidots), this trajectory obtains a nontrivial framing: it is the framed curve $({\xi }_i,-1)$. A measurement of the interference between the red and yellow curves can determine the topological charge around this (tilted) framed curve.

Figure 24

(Color online) By sending an annular intermediary between the two antidots, we can construct the spacetime history of Fig. 17. The shuttle between the two antidots is annular and the island of quantum Hall fluid in the middle of the annulus is the faux pass $B$. A measurement of the interference between the red and yellow trajectories can check that the tilted hole in region $A$ is topologically trivial as far as Chern-Simons theory is concerned.

Figure 25

By evaluating the Jones polynomial at $q=\exp (\pi i∕4)$ for these links, we can obtain the desired matrix elements for braiding operations manipulating the qubit. The boxed 1 is a projector on the pair of quasiparticles that puts them in the state ∣1⟩.

Figure 26

(a) A quasiparticle trajectory that winds around a quasiparticle at an antidot (b) a quasiparticle trajectory that does not wind around the antidot; (c) the matrix element between the states resulting from these two trajectories.

Figure 27

The spacetime history of a process in which an overpass connecting two antidots is added and then removed. Warning: this history does not embed in $2+1$ dimensions, so the $z$ coordinate in this picture cannot be literally interpreted as time. The handle must move into a fourth direction.

Figure 28

The spacetime history of Fig. 27 is equivalent to the Feynman diagrams above in the notation introduced in Appendix .