Systematically generated two-qubit anyon braids

Fibonacci anyons are non-Abelian particles for which braiding is universal for quantum computation. Reichardt has shown how to systematically generate nontrivial braids for three Fibonacci anyons which yield unitary operations with off-diagonal matrix elements that can be made arbitrarily small in a particular natural basis through a simple and efficient iterative procedure. This procedure does not require brute force search, the Solovay-Kitaev method, or any other numerical technique, but the phases of the resulting diagonal matrix elements cannot be directly controlled. We show that despite this lack of control the resulting braids can be used to systematically construct entangling gates for two qubits encoded by Fibonacci anyons.

Figure 1

(a) Three-anyon and (b) four-anyon qubit encoding. The logical qubit states $|0_L\rangle$ and $|1_L\rangle$ are shown in both oval and fusion tree notation.

Figure 2

(a) Hexagon diagram used to find weaving patterns which correspond to matrix products of the form (7). (b) Phase weave corresponding to $F{R}^{3}F{R}^{-2}FRF$. (c) Exchange weave corresponding to $F{R}^{5}F$. (d) Phase weave corresponding to $FRF{R}^{-1}F$ with an $R$ operation added (resulting in $RFRF{R}^{-1}F$). (e) Exchange weave corresponding to $F$ with two $R$ operations added (resulting in $RFR$).

Figure 3

(a) Box representing a phase weave, which in the $k\to \infty$ limit applies a phase factor of $e^{i{\theta }_{\infty }}$ to the state ($\begin{array}{c}★\end{array}{(••)}_0{)}_1$. (b) Controlled-phase gate construction for two four-anyon qubits.

Figure 4

(a) Weave corresponding to phase seed $F{R}^{4}F$. (b) Weave obtained with this seed after one iteration of (10). (c) Weave obtained with this seed after a second iteration of (10). In (b) and (c) dashed lines divide parts of the weave corresponding to $U_k$ and $U_k^{†}$ from those corresponding to the $R^{\prime }$ and ${R^{\prime }}^3$ operations appearing explicitly in (10). For the seed $F{R}^{4}F$ the magnitude of the initial off-diagonal matrix elements is $x_0\simeq 0.571$ and the initial phase is ${\theta }_0\simeq 0.546\pi$. After two iterations $x_2\simeq 8.30\times {10}^{-7}$ and ${\theta }_2\simeq 0.488\pi$. (d) $\delta \theta$, shift in the phase of $\langle 0\left|U\right|0\rangle$ after one iteration of (10), plotted as a function of $x=|\langle 1|U|0\rangle |$, as well as $\mathrm{\Delta }{\theta }^{+}$, $\mathrm{\Delta }{\theta }^{-}$, and $\mathrm{\Delta }{\theta }^{\pm }$ corresponding to different choices for the signs $s_k$ in (15). (e) Controlled-phase gate [see Fig. 3] with ${\theta }_2\simeq {\theta }_{\infty }\simeq 0.488\pi$ obtained using the second iteration phase weave shown in (c).

Figure 5

(a) Box representing an exchange weave, which in the $k\to \infty$ limit maps ($\begin{array}{c}★\end{array}{(••)}_0{)}_1$ to $(•$($\begin{array}{c}★\end{array}$ ${•)}_0{)}_1$. (b) Controlled-$R^2$ gate construction for two three-anyon qubits.

Figure 6

(a) Weave corresponding to the exchange seed $F{R}^{3}F$. (b) Weave obtained with this seed after one iteration of (5). (c) Weave obtained with this seed after a second iteration of (5). For the seed $F{R}^{3}F$ the magnitude of the initial off-diagonal matrix elements is $x_0\simeq 0.300$, and, after two iterations, $x_2=8.67\times {10}^{-14}$. In (b) and (c) dashed lines divide parts of the weave which correspond to $U_k$ and $U_k^{†}$ from those corresponding to the $R$ and $R^3$ operations appearing explicitly in (5). (d) Controlled-$R^2$ gate [see Fig. 5] obtained using the second iteration exchange weave shown in (c). The controlled braid which enacts an $R^2$ operation only if the control qubit is the state 1 is indicated by the dashed box.