Emulating Quantum Teleportation of a Majorana Zero Mode Qubit

Topological quantum computation based on anyons is a promising approach to achieve fault-tolerant quantum computing. The Majorana zero modes in the Kitaev chain are an example of non-Abelian anyons where braiding operations can be used to perform quantum gates. Here we perform a quantum simulation of topological quantum computing, by teleporting a qubit encoded in the Majorana zero modes of a Kitaev chain. The quantum simulation is performed by mapping the Kitaev chain to its equivalent spin version and realizing the ground states in a superconducting quantum processor. The teleportation transfers the quantum state encoded in the spin-mapped version of the Majorana zero mode states between two Kitaev chains. The teleportation circuit is realized using only braiding operations and can be achieved despite being restricted to Clifford gates for the Ising anyons. The Majorana encoding is a quantum error detecting code for phase-flip errors, which is used to improve the average fidelity of the teleportation for six distinct states from $70.76\pm 0.35\%$ to $84.60\pm 0.11\%$, well beyond the classical bound in either case.

Figure 1

Experimental configuration and the mapping of Kitaev chain to its spin and Majorana representations. (a) The superconducting quantum processor. We choose eight adjacent qubits labelled with $Q_1$ to $Q_8$ from the 12-qubit processor to perform the experiment. Qubits $Q_1$ to $Q_4$ and $Q_5$ to $Q_8$ are held by Alice and Bob, respectively. Pairs of qubits form a spin-mapped Kitaev chain ($KC$), each which encodes a single logical qubit. Each qubit couples to a resonator for state readout, marked by $R_1$ to $R_8$. After decoding, the resonators marked by “syn” are syndrome measurements to detect phase-flip errors in the qubits. An encoded qubit is teleported from $K{C}_1$ to $K{C}_3$. (b) Mapping between spin, fermions, and Majorana modes. The pairing of Majorana modes in the topologically nontrivial regime are indicated by the dotted ovals. In the topologically nontrivial phase, the MZMs are present at the ends of the chain.

Figure 2

Quantum circuits for simulating the teleportation of a MZM encoded qubit. (a) The modified quantum teleportation scheme. (b) The braiding sequence for MZMs that performs the quantum circuit in (a). (c) The corresponding spin-mapped qubit circuit of the MZM braiding sequence shown in (b). All measurements are performed in the $|0⟩$, $|1⟩$, which the exception of the measurement on qubit 6, where tomography (“tomo”) is performed. The measurements marked with syn are syndrome measurements, where single-qubit phase errors are detected for a measurement outcome of $|1⟩$. (d) The gate decompositions for the braiding, encoding, and decoding gates in (c). In all figures, $|\psi ⟩$ is the state to be teleported. Black lines connecting the quantum gates denote qubits, dark blue lines denote MZMs, and orange lines denote classical information transfer.

Figure 3

Teleportation fidelities with and without error syndrome detection. The fidelity is calculated according to $F=⟨\psi |\rho |\psi ⟩$, where $|\psi ⟩=\{|0⟩,|1⟩,|+⟩,|-⟩,|+i⟩,|-i⟩\}$ are the ideal states to be teleported. The dashed line denotes the $F=2/3$ threshold. The error bars denote 1 standard deviation.

Figure 4

Tomography of the final teleported state after using the error syndrome measurements. The initial state prepared on qubit 2 is (a) $|0⟩$, (b) $|1⟩$, (c) $|+⟩$, (d) $|-⟩$, (e) $|+i⟩$, (f) $|-i⟩$. Frames show ideal teleportation states; colored bars show the experimentally determined state.