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A Construction of Quantum LDPC Codes from Cayley Graphs
- In Proc. of IEEE International Symposium on Information Theory, ISIT 2011
, 2011
"... We study a construction of Quantum LDPC codes proposed by MacKay, Mitchison and Shokrollahi. It is based on the Cayley graph of F n 2 together with a set of generators regarded as the columns of the parity–check matrix of a classical code. We give a general lower bound on the minimum distance of the ..."
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Cited by 8 (5 self)
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We study a construction of Quantum LDPC codes proposed by MacKay, Mitchison and Shokrollahi. It is based on the Cayley graph of F n 2 together with a set of generators regarded as the columns of the parity–check matrix of a classical code. We give a general lower bound on the minimum distance of the Quantum code in O(dn 2) where d is the minimum distance of the classical code. When the classical code is the [n, 1, n] repetition code, we are able to compute the exact parameters of the associated Quantum code which are [[2 n, 2 n+1 2, 2 n−1
Upper Bounds on the Rate of Low Density Stabilizer Codes for the Quantum Erasure Channel
"... Using combinatorial arguments, we determine an upper bound on achievable rates of stabilizer codes used over the quantum erasure channel. This allows us to recover the no-cloning bound on the capacity of the quantum erasure channel, R ≤ 1 − 2p, for stabilizer codes: we also derive an improved upper ..."
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Cited by 3 (3 self)
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Using combinatorial arguments, we determine an upper bound on achievable rates of stabilizer codes used over the quantum erasure channel. This allows us to recover the no-cloning bound on the capacity of the quantum erasure channel, R ≤ 1 − 2p, for stabilizer codes: we also derive an improved upper bound of the form R ≤ 1 − 2p − D(p) with a function D(p) that stays positive for 0 < p < 1/2 and for any family of stabilizer codes whose generators have weights bounded from above by a constant – low density stabilizer codes. We obtain an application to percolation theory for a family of self-dual tilings of the hyperbolic plane. We associate a family of low density stabilizer codes with appropriate finite quotients of these tilings. We then relate the probability of percolation to the probability of a decoding error for these codes on the quantum erasure channel. The application of our upper bound on achievable rates of low density stabilizer codes gives rise to an upper bound on the critical probability for these tilings. 1
unknown title
, 2009
"... Quantum LDPC codes with positive rate and minimum distance proportional to n 1/2 ..."
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Quantum LDPC codes with positive rate and minimum distance proportional to n 1/2
New constructions of CSS codes obtained by
"... Abstract—We generalize a construction of non-binary quantum LDPC codes over F2m due to [KHIK11] and apply it in particular to toric codes. We obtain in this way not only codes with better rates than toric codes but also improve dramatically the performance of standard iterative decoding. Moreover, t ..."
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Abstract—We generalize a construction of non-binary quantum LDPC codes over F2m due to [KHIK11] and apply it in particular to toric codes. We obtain in this way not only codes with better rates than toric codes but also improve dramatically the performance of standard iterative decoding. Moreover, the new codes obtained in this fashion inherit the distance properties of the underlying toric codes and have therefore a minimum distance which grows as the square root of the length of the code for fixed m. I.
