Braid Topologies for Quantum Computation

In topological quantum computation, quantum information is stored in states which are intrinsically protected from decoherence, and quantum gates are carried out by dragging particlelike excitations (quasiparticles) around one another in two space dimensions. The resulting quasiparticle trajectories define world lines in three-dimensional space-time, and the corresponding quantum gates depend only on the topology of the braids formed by these world lines. We show how to find braids that yield a universal set of quantum gates for qubits encoded using a specific kind of quasiparticle which is particularly promising for experimental realization.

Figure 1

Basis states for the three-dimensional Hilbert space of three quasiparticles and qubit encoding. The ovals enclose groups of quasiparticles in $q$-spin eigenstates labeled by the corresponding eigenvalues. The states $|0_L⟩$ and $|1_L⟩$ (denoted, respectively, $|\mathbf{(}(•,•{)}_0,•{\mathbf{)}}_1⟩$ and $|\mathbf{(}(•,•{)}_1,•{\mathbf{)}}_1⟩$ in the text) span the computational qubit space, while the state $|\mathrm{NC}⟩$ (denoted $|\mathbf{(}(•,•{)}_1,•{\mathbf{)}}_0⟩$ in the text) is a noncomputational state.

Figure 2

(a) Elementary three-quasiparticle braids. The pictures represent quasiparticle world lines in $2+1$-dimensional space-time, with time flowing from left to right. The matrices ${\sigma }_1$ and ${\sigma }_2$ are the transition matrices produced by these elementary braids which act on the three-dimensional Hilbert space shown in Fig. 1. Here $\tau =(\sqrt{5}-1)/2$ is the inverse of the golden mean. The upper $2\times 2$ blocks of these matrices act on the computational qubit space (total $q$-spin 1) and are used to perform single-qubit rotations, while the lower right element is a phase which multiplies $|\mathrm{NC}⟩$. (b) A general three-quasiparticle braid and the corresponding matrix expression for the transition matrix it produces. Here $|{\psi }_i⟩$ is the initial state and $|{\psi }_f⟩$ the final state after braiding. Note that these (and subsequent) figures only represent the topology of the braid. In any actual implementation, quasiparticles will have to be kept sufficiently far apart to keep from lifting the topological degeneracy.

Figure 3

(a) A three-quasiparticle braid in which one quasiparticle is woven around two static quasiparticles and returns to its original position (left), and yields approximately the same transition matrix as braiding the two stationary quasiparticles around each other twice (right). The corresponding matrix equation is also shown. To characterize the accuracy of this approximation, we define the distance between two matrices, $U$ and $V$, to be $ϵ=\Vert U-V\Vert$, where $\Vert O\Vert$ is the operator norm of $O$ equal to the square-root of the highest eigenvalue of $O^{†}O$. The distance between the matrices resulting from the actual braiding (left) and the desired effective braiding (right) is $ϵ\simeq 2.3\times {10}^{-3}$. (b) A two-qubit braid constructed by weaving a pair of quasiparticles from the control qubit (top) through the target qubit (bottom) using the weaving pattern from (a). The result of this operation is to effectively braid the upper two quasiparticles of the target qubit around each other twice if the control qubit is in the state $|1_L⟩$, and otherwise do nothing. This is an entangling two-qubit gate which can be used for universal quantum computation. Since all effective braiding takes place within the target qubit, any leakage error is due to the approximate nature of the weave shown in (a). By systematically improving this weave using the Solovay-Kitaev construction, leakage error can be reduced to whatever level is required for a given computation.

Figure 4

(a) An injection weave for which the product of elementary braiding matrices, also shown, approximates the identity to a distance of $ϵ\simeq 1.5\times {10}^{-3}$. This weave injects a quasiparticle (or any $q$-spin 1 object) into the target qubit without changing any of the underlying $q$-spin quantum numbers. (b) A weaving pattern which approximates a NOT gate to a distance of $ϵ\simeq 8.5\times {10}^{-4}$. (c) A controlled-NOT gate constructed using the weaves shown in (a) and (b) to inject the control pair into the target qubit, perform a NOT operation on the injected target qubit, and then eject the control pair from the target qubit back into the control qubit. The distance between the gate produced by this braid acting on the computational two-qubit space and an exact controlled-NOT gate is $ϵ\simeq 1.8\times {10}^{-3}$ and $ϵ\simeq 1.2\times {10}^{-3}$ when the total $q$-spin of the six quasiparticles is 0 and 1, respectively. Again, the weaves shown in (a) and (b) can be made as accurate as necessary using the Solovay-Kitaev theorem, thereby improving the controlled-NOT gate to any desired accuracy. By replacing the central NOT weave, arbitrary controlled-rotation gates can be constructed using this procedure.