Topological quantum computation may provide a robust approach for encoding and manipulating information utilizing the topological properties of anyonic quasiparticle excitations. We develop an efficient means to map between dense and sparse representations of quantum information (qubits) and a simple construction of multiqubit gates, for all anyon models from Chern-Simons-Witten SU(2)
k theory that support universal quantum computation by braiding ($k\ensuremath{\geqslant}3,\phantom{\rule{4pt}{0ex}}k\ensuremath{\ne}4$). In the process, we show how the constructions of topological quantum memory and gates for
k=2,4 connect naturally to those for $k\ensuremath{\geqslant}3,\phantom{\rule{4pt}{0ex}}k\ensuremath{\ne}4$, unifying these concepts in a simple framework. Furthermore, we illustrate potential extensions of these ideas to other anyon models outside of Chern-Simons-Witten field theory.
Abstract Chern–Simons theory in application to the quantum computing is actively developing at the present. However, most discussed are the questions of using materials with known parameters and building corresponding quantum gates and algorithms. In this paper we discuss opposite problem of finding Chern–Simons level k in the unknown material. For this purpose, we use the previously derived braiding rules for Chern–Simons
SU(2)k S U ( 2 ) k anyons. Using certain operations (turnarounds) on three anyons, one can measure probabilities of annihilation of pairs of anyons, which depend on the parameter of the theory. Therefore, Chern–Simons level k can be found from such an experiment. It is implied that anyons additionally possess certain properties which are required for topological quantum computations.
Majorana zero modes (MZMs) are canonically described by the Ising topological quantum field theory (TQFT), equivalent to SU(2)2 Chern-Simons theory. We investigate whether the braid-group representation governing MZMs could instead arise as the non-Abelian sector of an SO(3)2 Chern-Simons theory (equivalent to SU(2)4), and whether four-point interferometry can experimentally distinguish these topological orders. The non-Abelian sector of SO(3)2 yields a braid-group representation locally indistinguishable from Ising anyons for two-quasiparticle braiding, but differs fundamentally in its fusion rules. Specifically, the SO(3)2 non-Abelian anyon
v (spin-1) obeys
v×v=1+v+ψ, whereas the Ising
σ anyon obeys
σ×σ=1+ψ. We compute the four-point braiding matrices and evaluate the interferometric visibility for a Fabry-Pérot interferometer enclosing four anyons. The presence of the
v×v→v fusion channel introduces an additional interference path with a distinct conformal weight, providing a falsifiable signature distinguishing SO(3)2 from Ising. ## 1. Introduction Majorana zero modes (MZMs) are localized zero-energy excitations in topological superconductors that exhibit non-Abelian statistics. Canonically, MZMs are described by the Ising topological quantum field theory (TQFT), a framework describing topological phases of matter, which corresponds to the SU(2)2 Chern-Simons gauge theory. Chern-Simons theory is a topological gauge theory in 2+1 dimensions whose quantization yields anyonic quasiparticles. We investigate whether the braid-group representation governing MZMs could instead arise as the non-Abelian sector of an SO(3)2 Chern-Simons theory. We demonstrate that while two-quasiparticle braiding is locally indistinguishable from Ising anyons, the theories differ fundamentally in their fusion rules. We propose that four-point interferometry can experimentally distinguish these two topological orders by det