We investigate the capability of multiple-braiding in
SU(2)k anyon models to realize universal quantum computation. The multiple elementary braiding matrices (MEBMs) are derived from the
q-deformed representation theory of
SU(2). Through a general analysis of the MEBMs, we find that multiple-braiding loses its universality only when the braiding multiplicity
m causes the MEBMs to collapse to a scalar (up to a global phase). Our numerical analysis of the MEBMs of
SU(2)k anyon models with
k=3,5,6,7, and
m∈[2,9], shows that the values of
m at which the braid group representation fails to be dense agree exactly with the theoretical prediction. For those
m that support universality, single-qubit gates are compiled to high precision by a genetic algorithm-enhanced Solovay-Kitaev algorithm (GA-enhanced SKA), and a genetic algorithm (GA) search with progressively increasing braid length yields an approximately the local equivalence class
[CNOT]. Notably, even-order braiding operations offer a physical advantage by reducing the number of non-Abelian anyons required in braiding-based topological quantum computation (TQC). Our analytical and numerical results together provide strong evidence for identifying which braiding multiplicities
m support universal quantum computation in
SU(2)k anyon models.
In topologically protected quantum computation, quantum gates can be carried out by adiabatically braiding two-dimensional quasiparticles, reminiscent of entangled world lines. Bonesteel et al. [Phys. Rev. Lett. 95, 140503 (2005)], as well as Leijnse and Flensberg [Phys. Rev. B 86, 104511 (2012)], recently provided schemes for computing quantum gates from quasiparticle braids. Mathematically, the problem of executing a gate becomes that of finding a product of the generators (matrices) in that set that approximates the gate best, up to an error. To date, efficient methods to compute these gates only strive to optimize for accuracy. We explore the possibility of using a generic approach applicable to a variety of braiding problems based on evolutionary (genetic) algorithms. The method efficiently finds optimal braids while allowing the user to optimize for the relative utilities of accuracy and/or length. Furthermore, when optimizing for error only, the method can quickly produce efficient braids.