Universal quantum computation with the $\nu =5∕2$ fractional quantum Hall state

We consider topological quantum computation (TQC) with a particular class of anyons that are believed to exist in the fractional quantum Hall effect state at Landau-level filling fraction $\nu =5∕2$. Since the braid group representation describing the statistics of these anyons is not computationally universal, one cannot directly apply the standard TQC technique. We propose to use very noisy nontopological operations such as direct short-range interactions between anyons to simulate a universal set of gates. Assuming that all TQC operations are implemented perfectly, we prove that the threshold error rate for nontopological operations is above $14\%$. The total number of nontopological computational elements that one needs to simulate a quantum circuit with $L$ gates scales as $L(\ln L{)}^3$.

Figure 1

Braid group generators.

Figure 2

The elementary purification round.

Figure 3

A braid gate $C$ implementing a cyclic shift of stabilizers $S_1\to S_2\to S_3\to S_1$.

Figure 4

A single round of $a_8$-purification protocol. Time flows upwards. Each line represents one copy of the noisy $\mid a_8⟩$ state (eight $\sigma$ particles). Each triangle $E$ corresponds to the elementary purification round shown in Fig. 2. Each rectangle $C$ represents a braid gate that shifts the generators $S_1$, $S_2$, and $S_3$ cyclically; see Fig. 3. Finally, a circle $W$ stands for the syndrome whirling transformation.

Figure 5

Input and output error rates for a single round of $a_8$-purification protocol.

Figure 6

The graph shows the number of level-$0$ ancillas, $n_0$, that one needs to prepare one level-$k$ ancilla $(k=1,\dots ,5)$ with a success probability $1∕2$. The success probability has been evaluated using Monte Carlo simulation of the protocol with ${10}^5$ trials. Solid lines show the dependence $n_0\left({ϵ}_0\right)$ that one gets using a naive equation $n_0=8^k{\prod }_{j=0}^{k-1}Z({ϵ}_j{)}^{-1}$ (it neglects fluctuations of $n_k$).

Figure 7

Implementation of a nondestructive measurement of ${\widehat{c}}_1{\widehat{c}}_2{\widehat{c}}_3{\widehat{c}}_4$ that consumes one copy of $\mid a_8⟩$.

Figure 8

Preparation of $\mid a_8⟩$ via eigenvalue measurement of ${\widehat{c}}_5{\widehat{c}}_6{\widehat{c}}_7{\widehat{c}}_8$.

Figure 9

Teleportation.

Figure 10

A preparation of $\mid a_4⟩$ based on the short-range two-particle interaction.

Figure 11

A two-point contact interferometer. Four $\sigma$ particles are trapped at antidots inside the interferometer loop. Electric current propagates along the top and bottom edges of the FQH electron gas (the unshaded region). Tunneling of $\sigma$ particles occurs at the constrictions.

Figure 12

A braid $b$ describing interference of the two tunneling paths in the two-point contact interferometer.