IEEE International Symposium on Information Theory (ISIT)

Quantum LDPC codes with positive rate and minimum distance proportional to n½

Jean–Pierre Tillich, Gilles Zémor

Published Jun 1, 2009• 46 citations

Abstract

The current best asymptotic lower bound on the minimum distance of quantum LDPC codes with fixed non-zero rate is logarithmic in the block length. We build quantum LDPC codes with fixed non-zero rate and prove that their minimum distance grows proportionally to the square root of the block length.

DOI: 10.1109/isit.2009.5205648

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IEEE International Symposium on Information Theory (ISIT)

Quantum LDPC codes with positive rate and minimum distance proportional to n½

Jean–Pierre Tillich, Gilles Zémor

Published Jun 1, 2009• 46 citations

Publisher

Abstract

DOI: 10.1109/isit.2009.5205648

Related papers

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arXiv:0903.0566IEEE Transactions on Information TheoryJan 24, 2014345 citations

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Apr 16, 20205 citations

We give a construction of quantum LDPC codes of dimension

\Theta (\log N)

and distance

\Theta (N/\log N)

as the code length$N\to \infty

. Using a product of chain complexes this construction also provides a family of quantum LDPC codes of distance

\Omega (N^{1-\alpha /2}/\log N)

and dimension

\Omega (N^\alpha \log N)$, where

0 \le \alpha < 1

. We also introduce and study a new operation called lifted product, which naturally generalizes the product operations for quantum codes and chain complexes. Moreover, as a simple byproduct of our results on quantum codes, we obtain a new result on classical codes. We show that for any fixed

R < 1

there exists an asymptotically good family of classical quasi-cyclic LDPC codes of rate at least

R

with, in some sense, optimal circulant size

\Omega (N/\log N)

as the code length$N\to \infty $.

X

-distance but a much smaller

Z

-distance. However, together with a classical expander LDPC code and a tensoring method that generalises a construction of Hastings and also the Tillich-Zemor construction of quantum codes, we obtain quantum LDPC codes whose minimum distance exceeds the square root of the code length and whose dimension comes close to a square root of the code length. When the ingredient is a 3-dimensional Ramanujan complex, we show that its 2-systole behaves like a square of the log of the complex size, which results in an overall quantum code of minimum distance

n^{1/2}\log n

, and sets a new record for quantum LDPC codes. When we use a 2-dimensional Ramanujan complex, or the 2-skeleton of a 3-dimensional Ramanujan complex, we obtain a quantum LDPC code of minimum distance

n^{1/2}\log^{1/2}n

. We then exploit the expansion properties of the complex to devise the first polynomial time algorithm that decodes above the square root barrier for quantum LDPC codes.