Resources required for topological quantum factoring

Abstract

We consider a hypothetical topological quantum computer composed of either Ising or Fibonacci anyons. For each case, we calculate the time and number of qubits (space) necessary to execute the most computationally expensive step of Shor's algorithm, modular exponentiation. For Ising anyons, we apply Bravyi's distillation method [S. Bravyi, Phys. Rev. A 73, 042313 (2006)] which combines topological and nontopological operations to allow for universal quantum computation. With reasonable restrictions on the physical parameters we find that factoring a 128-bit number requires approximately ${10}^{3}$ Fibonacci anyons versus at least $3\ifmmode\times\else\texttimes\fi{}{10}^{9}$ Ising anyons. Other distillation algorithms could reduce the resources for Ising anyons substantially.