Z. K. Minev

K. Najafi

S. Majumder

J. Wang

 A. Stern

E. Kim

C. Jian

G. Zhu

With IBM Quantum, we braid non-abelian Fibonacci anyons in string-net condensates to realise fault-tolerant universal quantum computation.

Topologically ordered quantum systems host anyons that, when braided, support fault-tolerant quantum computation. Yet creating Fibonacci string-net condensates, which support universal quantum gates and sample classically hard graph polynomials, has remained elusive. We demonstrate a scalable method of creating such condensates using superconducting qubits to dynamically create, measure and braid Fibonacci anyons. 

The remarkable complexity of the vacuum state of a topologically-ordered many-body quantum system encodes the character and intricate braiding interactions of its emergent particles, the anyons.} Quintessential predictions exploiting this complexity use the Fibonacci string-net condensate (Fib-SNC) and its Fibonacci anyons to go beyond classical computing. Sampling the Fib-SNC wavefunction is expected to yield estimates of the chromatic polynomial of graph objects, a classical task that is provably hard. At the same time, exchanging anyons of Fib-SNC is expected to allow fault-tolerant universal quantum computation. Nevertheless, the physical realization of Fib-SNC and its anyons remains elusive. Here, we introduce a scalable dynamical string-net preparation (DSNP) approach, suitable even for near-term quantum processors, which dynamically prepares Fib-SNC and its anyons through reconfigurable graphs. Using a superconducting quantum processor, we couple the DSNP approach with composite error-mitigation on deep circuits to successfully create, measure, and braid anyons of Fib-SNC in a scalable manner. We certify the creation of anyons by measuring their `anyon charge', finding an average experimental accuracy of 94%. Furthermore, we validate that exchanging these anyons yields the { expected} golden ratio~ϕ with~98% average accuracy and~8% measurement uncertainty. Finally, we sample the Fib-SNC to estimate the chromatic polynomial at~ϕ+2 for {several} graphs. Our results establish the proof of principle for using Fib-SNC and its anyons for fault-tolerant universal quantum computation and {for aiming at} a classically-hard problem.

Realizing string-net condensation: Fibonacci anyon braiding for universal gates and sampling chromatic polynomials