Consistent Simulation of Fibonacci Anyon Braiding within a Qubit Quasicrystal Inflation Code

TL;DR

This paper presents a scalable, physically realizable method for simulating Fibonacci anyon braiding using a qubit quasicrystal code, validated through experiments on IBM Quantum hardware and numerical analysis up to 17 qubits.

Contribution

It introduces a local 3-qubit gate framework for Fibonacci anyon braiding within a quasicrystal code, enabling scalable and accurate topological quantum simulations.

Findings

Successfully executed braiding of Fibonacci anyons on IBM Quantum hardware.

Constructed a single $8 imes 8$ gate ($B_{gate}$) for local braiding operations.

Confirmed the code's adherence to Temperley-Lieb and braid group relations up to 17 qubits.

Abstract

The simulation of non-Abelian anyon braiding is a critical step towards fault-tolerant quantum computation. We introduce a framework for this task based on a one-dimensional Quasicrystal Inflation Code (QIC). The code is defined by a local Hamiltonian whose ground state manifold enforces Fibonacci tiling constraints and possesses the correct Fibonacci degeneracy. We derive the corresponding braid operators and demonstrate that, while they are formally non-local, they possess an exact, local 3-qubit structure. This allows us to distill their action into a single, physically realizable $8 \times 8$ gate, which we term the $B_{gate}$. We prove through comparative compilation for systems with different numbers of qubits that constructing circuits by composing these local gates is dramatically more scalable than compiling the equivalent global unitary, showing a greater than tenfold reduction in circuit depth. To validate the framework, we successfully executed a braiding algorithm for the Jones polynomial, a topological invariant of knots, on an IBM Quantum processor, quantifying the signal degradation due to hardware noise. Finally, we numerically confirm for systems up to 17 qubits that our construction rigorously satisfies the required Temperley-Lieb and braid group relations while preserving the code subspace. This work establishes a validated and physically grounded pathway for the scalable simulation of anyonic braiding on programmable quantum systems.