Universal quantum computation with weakly integral anyons

Abstract

Harnessing non-abelian statistics of anyons to perform quantum computational tasks is getting closer to reality. While the existence of universal anyons by braiding alone such as the Fibonacci anyon is theoretically a possibility, accessible anyons with current technology all belong to a class that is called weakly integral---anyons whose squared quantum dimensions are integers. We analyze the computational power of the first non-abelian anyon system with only integral quantum dimensions---$D(S_3)$, the quantum double of $S_3$. Since all anyons in $D(S_3)$ have finite images of braid group representations, they cannot be universal for quantum computation by braiding alone. Based on our knowledge of the images of the braid group representations, we set up three qutrit computational models. Supplementing braidings with some measurements and ancillary states, we find a universal gate set for each model.Comment: add 2 references and corrected many minor typo

Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter

“…The modular S and T matrices of D(S 3 ) are given by [20]: where all rows and columns are ordered alphabetically, A − H, and ω = e 2πi/3 is the primitive third root of unity. and hence have no nontrivial splitting idempotents.…”

An entropic invariant for 2D gapped quantum phases

“…More precisely, we are interested in models where the square of the quantum dimensions is not an integer (compare with Theorem IV.1). Models for which the square of the quantum dimensions are integers are called weakly integral, and include quantum double models coming from groups (including the so-called twisted quantum double models) [48]. Unfortunately we do not have direct answer to this.…”

Congruence subgroups from representations of the three-strand braid group

“…For r = 2, 3, and 4 there are two-dimensional irreducible unitary representations ρ with po(ρ(σ 1 )) = r arising from weakly integral modular tensor categories. The modular tensor category structure of C = Rep D(S 3 ) was described completely in [CHW15]. The authors label the simple objects of this category with the letters A through G. It was shown that the anyon C corresponding to the standard representation of the S 3 gives rise to a unitary representation ρ C : B 3 → U(Hom(C, C ⊗3 )).…”