Non-Abelian braiding of Fibonacci anyons with a superconducting processor

The study presents the fixed-point wavefunction in the string-net model, which captures essential properties of topologically ordered states in (2+1) dimensions. This wavefunction is a superposition of various string-net configurations, characterized by their geometry and types of strings. The wavefunction is determined by four local constraints and can be calculated for any string-net configurations.

In the quantum circuit scheme, the linear relations described by the symbol F, corresponding to multi-controlled unitary gates, are simulated. The F move changes the type of one string according to its four connected strings, represented by a five-qubit gate. A simplified quantum circuit is used to realize the F move when there is prior information on the string-net configuration.

The exact ground-state wavefunction in the string-net picture and the corresponding quantum state simulated on physical qubits are presented. For the Fibonacci string-net model, the normalized quantum state that simulates the state Φ with the geometry of one loop is given. The wavefunction Φ of two independent loops can also be simulated.

An exactly solvable lattice spin Hamiltonian, with the fixed-point wavefunction Φ as the ground state, is introduced. The Hamiltonian is defined by the Qv operator and local constraints (Bp) that uniquely specify the wavefunction. The quasiparticle excitations live at the endpoints of the string operators, and their creation, movement, and fusion are described in the string-net picture.

The closed string operator \({B}_{p}^{s}\) is defined, which can be regarded as creating a pair of type-s excitations from vacuum, winding them around in this plaquette, and then annihilating them to the vacuum. The creation and fusion operation for closed string operators introduce a constant factor related to the quantum dimension of type-s string.

Non-Abelian anyons have multiple fusion outputs and can be used for constructing topologically protected logical qubits. In this work, four Fibonacci anyons with the vacuum total charge are used to encode one logical qubit. The measurement results of the logical qubit can be obtained by measuring the fusion outcomes of the first or last two anyons.

The braiding operator σ1 and σ2 are calculated in the encoding scheme of \(\left\vert \bar{0}\right\rangle\) and \(\left\vert \bar{1}\right\rangle\), and their matrix representations are associated with the encoding scheme. The process (5) expresses the elements Mab of the monodromy matrix, which is a real negative value taking the form of −1/d2, where d is the quantum dimension of the quasiparticle excitation.

The circuits for ground-state preparation and anyon braiding are composed of scalable modules and can be optimized by blocks. The experimental circuits for preparing the ground state are explicitly displayed. Randomized measurement is adopted to obtain the second-order Rényi entropies and calculate the topological entanglement entropy.

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