Universal quantum computation with the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>ν</mml:mi><mml:mo>=</mml:mo><mml:mn>5</mml:mn><mml:mo>∕</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math>fractional quantum Hall state

We consider topological quantum computation (TQC) with a particular class of anyons that are believed to exist in the fractional quantum Hall effect state at Landau-level filling fraction $\ensuremath{\nu}=5∕2$. Since the braid group representation describing the statistics of these anyons is not computationally universal, one cannot directly apply the standard TQC technique. We propose to use very noisy nontopological operations such as direct short-range interactions between anyons to simulate a universal set of gates. Assuming that all TQC operations are implemented perfectly, we prove that the threshold error rate for nontopological operations is above $14%$. The total number of nontopological computational elements that one needs to simulate a quantum circuit with $L$ gates scales as $L(\mathrm{ln}\phantom{\rule{0.2em}{0ex}}L{)}^{3}$.

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