Constructing functional braids for low-leakage topological quantum computing

We discuss how to significantly reduce leakage errors in topological quantum computation by introducing an irrelevant error in phase, using the construction of a controlled-NOT (CNOT) gate in the Fibonacci anyon model as a concrete example. To be specific, we construct a functional braid in a six-anyon Hilbert space that exchanges two neighboring anyons while conserving the encoded quantum information. The leakage error is $\sim {10}^{-10}$ for a braid of $\sim 100$ interchanges of anyons. Applying the braid greatly reduces the leakage error in the construction of generic controlled-rotation gates.

Figure 1

(a) Basis states for the two-dimensional Hilbert space (a qubit) spanned by two pairs of anyons with total topological charge 0, where the black dots denote the Fibonacci anyons in the Fibonacci model. (b) Braiding matrices for the qubit, where $\tau =(\sqrt{5}-1)∕2$. The anyons are labeled 1–4 from left to right in (a) and from bottom to top in (b).

Figure 2

(Color online) (a) A braid with 99 anyon interchanges that approximates a phase gate, where $\mid {\delta }_1\mid =\mid {\delta }_2\mid =\sqrt{1-r^2}\simeq 6.2\times {10}^{-10}$. (b) The exchange braid in the six-anyon braid space, in which ${\sigma }_4$ and ${\sigma }_5$ are the corresponding $5\times 5$ braiding matrices. Note that the sequence of interchanges for the top three anyons is exactly the same as in (a). The phase factor $e^{i\theta }$ inherited from (a) is not important and can be canceled out by the inverse exchange braid. The upper $2\times 2$ block is exactly proportional to the identity matrix, and $1-r\approx 2\times {10}^{-19}$. The lower $3\times 3$ block $A$ (in the noncomputational space) is irrelevant if the quantum state remains in the computational space. The leakage to the noncomputational space spanned by $\mid {\left({\mathbf{\left(}{\left(11\right)}_1{\left(11\right)}_0\mathbf{\right)}}_1{\left(11\right)}_1\right)}_0⟩$, $\mid {\left({\mathbf{\left(}{\left(11\right)}_0{\left(11\right)}_1\mathbf{\right)}}_1{\left(11\right)}_1\right)}_0⟩$, and $\mid {\left({\mathbf{\left(}{\left(11\right)}_1{\left(11\right)}_1\mathbf{\right)}}_1{\left(11\right)}_1\right)}_0⟩$ is determined by $\mid {ϵ}_i\mid ⩽6.2\times {10}^{-10}$ for $i=1–6$.

Figure 3

(Color online) (a) A braid that approximates a controlled-${\sigma }_3^2$ gate to a distance $\simeq 1.2\times {10}^{-9}$. The control (top) qubit can be treated as two composite anyons and each composite anyon can move as a single anyon. In the six-anyon braid space, we apply the exchange braid [see Fig. 2b], ${\sigma }_3^2$ (two counterclockwise interchanges of the third and the fourth anyons from the bottom), and the inverse exchange braid. If the control qubit is in the state ∣0⟩, the encoded qubit remains unchanged; otherwise, a ${\sigma }_3^2$ rotation is performed. (b) A braid with 108 interchanges that approximates an $iX$ gate to a distance $\simeq 1.5\times {10}^{-6}$. Replacing the ${\sigma }_3^2$ sandwiched in (a) by the $iX$ braid, we obtain a braid that approximates a CNOT gate (up to a single-qubit rotation) to a distance $\sim {10}^{-6}$.

Figure 4

(Color online) Distance versus braid length in approaching the identity matrix (squares) or any phase gate, i.e., unitary diagonal matrix (dots). The decays can be fit by $1.6e^{-L∕7.3}$ (dashed line) and $0.76e^{-L∕4.3}$ (solid line), respectively. The significant reduction in distance due to the freedom in phase is key to the low-leakage realization of topological quantum computation based on the exchange braid.