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14
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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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
Spatially coupled quantum LDPC codes
 in "IEEE Information Theory worksop  ITW2012
, 2012
"... Abstract—We propose here a new construction of spatially coupled quantum LDPC codes using a small amount of entangled qubit pairs shared between the encoder and the decoder which improves quite significantly all other constructions of quantum LDPC codes or turbocodes with the same rate. I. ..."
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Abstract—We propose here a new construction of spatially coupled quantum LDPC codes using a small amount of entangled qubit pairs shared between the encoder and the decoder which improves quite significantly all other constructions of quantum LDPC codes or turbocodes with the same rate. I.
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 nocloning 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 nocloning 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 selfdual 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
Families of LDPC Codes Derived from Nonprimitive BCH Codes and Cyclotomic Cosets
"... Abstract—Lowdensity parity check (LDPC) codes are an important class of codes with many applications. Two algebraic methods for constructing regular LDPC codes are derived – one based on nonprimitive narrowsense BCH codes and the other directly based on cyclotomic cosets. The constructed codes hav ..."
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Abstract—Lowdensity parity check (LDPC) codes are an important class of codes with many applications. Two algebraic methods for constructing regular LDPC codes are derived – one based on nonprimitive narrowsense BCH codes and the other directly based on cyclotomic cosets. The constructed codes have high rates and are free of cycles of length four; consequently, they can be decoded using standard iterative decoding algorithms. The exact dimension and bounds for the minimum distance and stopping distance are derived. These constructed codes can be used to derive quantum errorcorrecting codes. Index Terms—LDPC Codes, BCH Codes, Channel Coding, Performance and iterative decoding.
1 Practical Entanglement Distillation Scheme Using Recurrence Method And Quantum Low Density Parity Check Codes
, 907
"... Abstract—Many entanglement distillation schemes use either universal random hashing or breading as their final step to obtain shared almost perfect EPR pairs. Both methods involve random stabilizer quantum errorcorrecting codes whose syndromes can be measured using simple and efficient quantum circ ..."
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Abstract—Many entanglement distillation schemes use either universal random hashing or breading as their final step to obtain shared almost perfect EPR pairs. Both methods involve random stabilizer quantum errorcorrecting codes whose syndromes can be measured using simple and efficient quantum circuits. When applied to high fidelity Werner states, the highest yield protocol among those using local Bell measurements and local unitary operations is the one that uses a certain breading method. And random hashing method losses to breading just by a thin margin. In spite of their high yield, the hardness of decoding random linear code makes the use of random hashing and breading infeasible in practice. In this pilot study, we analyze the performance of recurrence method, a wellknown entanglement distillation scheme, by replacing the final random hashing or breading procedure by various efficiently decodable quantum codes. We find that among all the replacements we have investigated, the one using a certain adaptive quantum low density parity check (QLDPC) code gives the highest yield for Werner states of almost all noise level. In this respect, our finding illustrates the effectiveness of using QLDPC codes in practical entanglement distillation.
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
Stabilizer Quantum Codes: A Unified View based on Forneystyle Factor Graphs
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Quantum LDPC Codes Constructed from PointLine Subsets of the Finite Projective Plane
"... Abstract—Due to their fast decoding algorithms, quantum generalizations of lowdensity parity check, or LDPC, codes have been investigated as a solution to the problem of decoherence in fragile quantum states. However, the additional twisted inner product requirements of quantum stabilizer codes fo ..."
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Abstract—Due to their fast decoding algorithms, quantum generalizations of lowdensity parity check, or LDPC, codes have been investigated as a solution to the problem of decoherence in fragile quantum states. However, the additional twisted inner product requirements of quantum stabilizer codes force fourcycles and eliminate the possibility of randomly generated quantum LDPC codes. Moreover, the classes of quantum LDPC codes discovered thus far generally have unknown or small minimum distance, or a fixed rate. This paper presents several new classes of quantum LDPC codes constructed from finite projective planes. These codes have rates that increase with the block length n and minimum weights proportional to n1/2. Index Terms—error correction codes, quantum error correction, finite geometry. I.