PIRSA:12100130 Scientific Series

DOI10.48660/12100130

Series: Perimeter Institute Quantum Discussions

Kovalev, A. (2012). Fault tolerance of "bad" quantum low-density parity check codes. Perimeter Institute. https://pirsa.org/12100130

Kovalev, Alexey. Fault tolerance of "bad" quantum low-density parity check codes. Perimeter Institute, Oct. 29, 2012, https://pirsa.org/12100130 

@misc{ pirsa_PIRSA:12100130,
  doi = {10.48660/12100130},
  url = {https://pirsa.org/12100130},
  author = {Kovalev, Alexey},
  keywords = {},
  language = {en},
  title = {Fault tolerance of "bad" quantum low-density parity check codes},
  publisher = {Perimeter Institute},
  year = {2012},
  month = {oct},
  note = {PIRSA:12100130 see, \url{https://pirsa.org}}
}

Abstract

In my talk, I will discuss various families of quantum low-density parity check (LDPC) codes and their fault tolerance. Such codes yield finite code rates and at the same time simplify error correction and encoding due to low-weight stabilizer generators. As an example, a large family of   hypergraph-product codes is considered. Of particular interest are families of quantum LDPC codes with finite rate and distance scaling as square root of blocklength since this represents the best known exponent in distance scaling, even for codes of dimensionality 1. In relation to such codes, we show that any family of LDPC codes, quantum or classical, where distance scales as a positive power of the block length, $d \propto n^\alpha$, $\alpha>0$ ($\alpha<1$ for "bad" codes), can correct all errors with certainty if the error rate per qubit is sufficiently small. We specifically analyze the case of LDPC version of the quantum hypergraph-product codes recently suggested by Tillich and Z\'emor. These codes are a finite-rate generalization of the toric codes, and, for sufficiently large quantum computers, offer an advantage over the toric codes.

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