Fibonacci anyons

In condensed matter physics, a Fibonacci anyon is a type of anyon which lives in two-dimensional topologically ordered systems. The Fibonacci anyon $\tau$ is distinguished uniquely by the fact that it satisfies the fusion rule $\tau \otimes \tau =\mathrm{𝟏}\oplus \tau$. Alternatively, the Fibonacci anyon can be defined by fact that it is algebraically described by the unique non-trivial simple object in the Fibonacci category.

Experimentally, it has been proposed that Fibonacci anyons could be hosted in the fractional quantum Hall system. In particular, it is possible that Fibonacci anyons are present in the system with filling factor $\nu =12/5$.

Fibonacci anyons have primary been developed in the context of topological quantum computing. This is because these anyons allow for universal quantum computing based entirely on braiding and performing topological charge measurements, and hence form a natural setting for topological quantum computing. This is in contrast to anyons based on discrete gauge theory, which require a more subtle use of ancillas to perform universal quantum computation.

Method for universal quantum computing

The pipeline for universal quantum computing with Fibonacci anyons can be described as follows. First, one is given an instance of a decision problem which is in the complexity class BQP (for instance, a large integer whose factorization one wishes to determine). Since the problem of additively approximating the Jones polynomial at a third root of unity is BQP complete, this means by definition that there is a polynomial time classical algorithm for taking any efficient quantum circuit an assigning to it a framed link such that the Jones invariant (or really, Kauffman bracket) of that link evaluated at $e^{3\pi i/5}$ encodes the solution of the decision problem. For example, using this procedure, Shor's algorithm for factoring an integer would correspond to some large link. To compute the Kauffman bracket of this link evaluated at $e^{3\pi i/5}$, one would take some material which hosts Fibonacci anyons, and perform a series of creation, braiding, and fusion operators such that the spacetime trajectories of the Fibonacci anyons in this process form the link outputted in the previous step of the process. One would then repeat this experiment polynomially many times, and record the probability that all of the fusion measurements resulted in the vacuum sector. The algebraic properties of the Fibonacci category imply that, after multiplying by a suitable power of the golden ratio, this computed probability is approximately equal to the Kauffman bracket evaluated at $e^{3\pi i/5}$. By construction, there is then a polynomial time classical algorithm for taking this approximation for the Kauffman bracket and using it to deduce the result of the original decision problem with high probability (for instance, in the case of factoring, this algorithm would use the digits of the approximation of the Kauffman bracket to recover the factorization of the input integer). This pipeline is demonstrated below

A pipeline for universal quantum computing with Fibonacci anyons, illustrated with the example of factoring integers using Shor's algorithm.