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

The remarkable complexity of a topologically ordered many-body quantum system is encoded in the characteristics of its anyons. Quintessential predictions emanating from this complexity employ the Fibonacci string net condensate (Fib SNC) and its anyons: sampling Fib-SNC would estimate chromatic polynomials while exchanging its anyons would implement universal quantum computation. However, physical realizations remained elusive. We introduce a scalable dynamical string net preparation (DSNP) that constructs Fib SNC and its anyons on reconfigurable graphs suitable for near-term superconducting processors. Coupling the DSNP approach with composite error-mitigation on deep circuits, we create, measure, and braids Fibonacci anyons; charge measurements show 94% accuracy, and exchanging the anyons yields the expected golden ratio ϕ with 98% average accuracy. We then sample the Fib SNC to estimate 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 aim at a classically hard problem.