Ternary logic design in topological quantum computing
Abstract
A quantum computer can perform exponentially faster than its classical counterpart. It works on the principle of superposition. But due to the decoherence effect, the superposition of a quantum state gets destroyed by the interaction with the environment. It is a real challenge to completely isolate a quantum system to make it free of decoherence. This problem can be circumvented by the use of topological quantum phases of matter. These phases have quasiparticles excitations called anyons. The anyons are chargeflux composites and show exotic fractional statistics. When the order of exchange matters, then the anyons are called nonabelian anyons. Majorana fermions in topological superconductors and quasiparticles in some quantum Hall states are nonabelian anyons. Such topological phases of matter have a ground state degeneracy. The fusion of two or more nonabelian anyons can result in a superposition of several anyons. The topological quantum gates are implemented by braiding and fusion of the nonabelian anyons. The faulttolerance is achieved through the topological degrees of freedom of anyons. Such degrees of freedom are nonlocal, hence inaccessible to the local perturbations. In this paper, the Hilbert space for a topological qubit is discussed. The Ising and Fibonacci anyonic models for binary gates are briefly given. Ternary logic gates are more compact than their binary counterparts and naturally arise in a type of anyonic model called the metaplectic anyons. The mathematical model, for the fusion and braiding matrices of metaplectic anyons, is the quantum deformation of the recoupling theory. We proposed that the existing quantum ternary arithmetic gates can be realized by braiding and topological charge measurement of the metaplectic anyons.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 July 2022
 DOI:
 10.1088/17518121/ac7b55
 arXiv:
 arXiv:2204.01000
 Bibcode:
 2022JPhA...55D5302I
 Keywords:

 topological quantum computation;
 ternary logic design;
 ternary arithmetic circuits;
 metaplectic anyons;
 Quantum Physics
 EPrint:
 J. Phys. A: Math. Theor. 55 (2022) 305302 (54pp)