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Physical Review A

Universal quantum computation with the nu=5/2 fractional quantum Hall state (16 pages)

Sergey B. Bravyi

Published Jan 1, 2006• 1 citation
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Universal quantum computation with the ν=5∕2 fractional quantum Hall state

Sergey B. Bravyi

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 $L(\mathrm{ln}\phantom{\rule{0.2em}{0ex}}L{)}^{3}$.

arXiv:quant-ph/0511178•Physical Review A•Apr 12, 2006•251 citations

Towards universal topological quantum computation in the ν=52 fractional quantum Hall state

Michael Hartley Freedman, Chetan Nayak, Kevin P. Walker

The Pfaffian state, which may describe the quantized Hall plateau observed at Landau level filling fraction $\ensuremath{\nu}=\frac{5}{2}$, can support topologically-protected qubits with extremely low error rates. Braiding operations also allow perfect implementation of certain unitary transformations of these qubits. However, in the case of the Pfaffian state, this set of unitary operations is not quite sufficient for universal quantum computation (i.e. is not dense in the unitary group). If some topologically unprotected operations are also used, then the Pfaffian state supports universal quantum computation, albeit with some operations which require error correction. On the other hand, if certain topology-changing operations can be implemented, then fully topologically-protected universal quantum computation is possible. In order to accomplish this, it is necessary to measure the interference between quasiparticle trajectories which encircle other moving trajectories in a time-dependent Hall droplet geometry [cond-mat/0512072].

arXiv:cond-mat/0512066•Physical Review B•Jun 7, 2006•125 citations

Quantum computation with quasiparticles of the fractional quantum Hall effect

D. V. Averin, V.J. Goldman

arXiv:cond-mat/0110193•Solid State Communications•Dec 1, 2001•32 citations
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 1414%14. The total number of nontopological computational elements that one needs to simulate a quantum circuit with LLL gates scales as L(lnL)3L(\mathrm{ln}\phantom{\rule{0.2em}{0ex}}L{)}^{3}L(lnL)3.
The Pfaffian state, which may describe the quantized Hall plateau observed at Landau level filling fraction $\ensuremath{\nu}=\frac{5}{2}$, can support topologically-protected qubits with extremely low error rates. Braiding operations also allow perfect implementation of certain unitary transformations of these qubits. However, in the case of the Pfaffian state, this set of unitary operations is not quite sufficient for universal quantum computation (i.e. is not dense in the unitary group). If some topologically unprotected operations are also used, then the Pfaffian state supports universal quantum computation, albeit with some operations which require error correction. On the other hand, if certain topology-changing operations can be implemented, then fully topologically-protected universal quantum computation is possible. In order to accomplish this, it is necessary to measure the interference between quasiparticle trajectories which encircle other moving trajectories in a time-dependent Hall droplet geometry [cond-mat/0512072].