Identifying Abelian and non-Abelian topological orders in the string-net model using a quantum scattering circuit

Abstract

Realizing universal topological quantum computers requires the manipulation of non-Abelian topological orders in a physical system, which presents great challenges. Conversely, the rapid development in circuit-based quantum computing offers a reliable quantum simulation approach to study these topological orders. The preliminary problem is how to identify distinct topological orders. Here, we develop a framework based on the quantum scattering circuit to directly and efficiently measure the modular transformation matrix, which is widely deemed as the fingerprint of a given topological order. The information of the modular transformation matrix is encoded in the probe qubit, and the readout merely requires single-qubit Pauli measurements. We further implement the scheme in a nuclear magnetic resonance quantum simulator to emulate the string-net model, where an Abelian ${\mathbb{Z}}_{2}$ toric code and a non-Abelian Fibonacci order emerge. In particular, the latter is predicted to be the simplest candidate for universal topological quantum computers. The two topological orders are unambiguously distinguished by the experimentally measured modular transformation matrices. As an experimental demonstration of a non-Abelian topological order with efficient readout, our work may open avenues toward investigating topological orders in circuit-based quantum simulators.