Quantum Walks with Non-Abelian Anyons

Abstract

We study the single particle dynamics of a mobile non-Abelian anyon hopping around many pinned anyons on a surface, by modeling it with a discrete time quantum walk. During the evolution, the spatial degree of freedom of the mobile anyon becomes entangled with the fusion degrees of freedom of the collective system. Each quantum trajectory makes a closed braid on the world lines of the particles establishing a direct connection between statistical dynamics and quantum link invariants. We find that asymptotically a mobile Ising model anyon becomes so entangled with its environment that its statistical dynamics reduces to a classical random walk with linear dispersion in contrast to particles with Abelian statistics which have quadratic dispersion.

Transport properties of anyons in random topological environments

“…The fusion Hilbert space H fusion of the non-Abelian anyons acts as a nonlocal and highly non-Markovian environment whose dimension grows exponentially as d | m| . The entanglement between the quantum walk DOFs and fusion DOFs is known to erase the quantum effects in the case of uniform filling [8]. Furthermore, on comparing (26) to the Abelian expression (6) for N = 8, we find they are identical except for the prefactor T a a .…”

“…The dispersion of the walker was found to be quadratic for Abelian anyons, as in the usual quantum walk [7], while for nonAbelian anyons it has been shown to be asymptotically linear [8,9], just like in classical random walks. The essential reason is that entanglement resulting from braiding non-Abelian anyons is sufficient to suppress the quantum correlations responsible for the quadratic speedup.…”

“…There are three kinds of particles in this model: vacuum, fermion, and non-Abelian anyon. In analogy with the notation in [8], the initial fusion state | 0 describes the vacuum configuration of pairs of anyons with half the members braided to the right. The trace over the fusion degree of freedom can be related to the Kauffman bracket polynomial L a a [18] of a link L a a (see Fig.…”

“…The trace over the fusion degree of freedom can be related to the Kauffman bracket polynomial L a a [18] of a link L a a (see Fig. 2), which arises from the Markov closure of the braid word B † a B a [6,8,19]. This link corresponds to the world lines (strands) of all the anyons.…”

“…The term in the denominator is a normalization factor, which is defined to be the value of the term in the numerator in the perfectly correlated case t = 0. For simplicity, the correlator is calculated for only the uniform filling (m s = 1 ∀ s) using the same method as described in [8]. defined in our method, in which case we set τ (s ,s ) = 0.…”