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Codeword Stabilized Quantum Codes
, 2007
"... We present a unifying approach to quantum error correcting code design that encompasses additive (stabilizer) codes, as well as all known examples of nonadditive codes with good parameters. We use this framework to generate new codes with superior parameters to any previously known. In particular, ..."
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Cited by 17 (3 self)
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We present a unifying approach to quantum error correcting code design that encompasses additive (stabilizer) codes, as well as all known examples of nonadditive codes with good parameters. We use this framework to generate new codes with superior parameters to any previously known. In particular, we find ((10,18, 3)), ((10, 20, 3)) and ((11, 48, 3)) codes.
Grassmannian packings from operator Reed-Muller codes
- IEEE Trans. Info. Theory
"... Abstract—This paper introduces multidimensional general-izations of binary Reed–Muller codes where the codewords are projection operators, and the corresponding subspaces are widely separated with respect to the chordal distance on Grassmannian space. Parameters of these Grassmannian packings are de ..."
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Cited by 6 (0 self)
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Abstract—This paper introduces multidimensional general-izations of binary Reed–Muller codes where the codewords are projection operators, and the corresponding subspaces are widely separated with respect to the chordal distance on Grassmannian space. Parameters of these Grassmannian packings are derived and a low complexity decoding algorithm is developed by mod-ifying standard decoding algorithms for binary Reed–Muller codes. The subspaces are associated with projection operators determined by Pauli matrices appearing in the theory of quantum error correction and this connection with quantum stabilizer codes may be of independent interest. The Grassmannian pack-ings constructed here find application in noncoherent wireless communication with multiple antennas, where separation with respect to the chordal distance on Grassmannian space guarantees closeness to the channel capacity. It is shown that the capacity of the noncoherent multiple-input–multiple-output (MIMO) channel at both low and moderate signal-to-noise ratio (SNR) (under the constraint that only isotropically distributed unitary matrices are used for information transmission) is closely approximated by these packings. Index Terms—Chordal distance, Grassmannian packings, noncoherent multiple-input–multiple-output (MIMO) channel, Reed–Muller codes, space-time codes. I.
Codeword Stabilized Quantum Codes: Algorithm & Structure
, 2008
"... The codeword stabilized (“CWS”) quantum codes formalism presents a unifying approach to both additive and nonadditive quantum error-correcting codes (arXiv:quantph/0708.1021). This formalism reduces the problem of constructing such quantum codes to finding a binary classical code correcting an erro ..."
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Cited by 5 (2 self)
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The codeword stabilized (“CWS”) quantum codes formalism presents a unifying approach to both additive and nonadditive quantum error-correcting codes (arXiv:quantph/0708.1021). This formalism reduces the problem of constructing such quantum codes to finding a binary classical code correcting an error pattern induced by a graph state. Finding such a classical code can be very difficult. Here, we consider an algorithm which maps the search for CWS codes to a problem of identifying maximum cliques in a graph. While solving this problem is in general very hard, we provide three structure theorems which reduce the search space, specifying certain admissible and optimal ((n, K, d)) additive codes. In particular, we find there does not exist any ((7,3,3)) CWS code though the linear programing bound does not rule it out. The complexity of the CWS-search algorithm is compared with the contrasting method introduced by Aggarwal & Calderbank (arXiv:cs/0610159).
Non-Additive Quantum Codes from Goethals and Preparata Codes
, 2008
"... We extend the stabilizer formalism to a class of nonadditive quantum codes which are constructed from non-linear classical codes. As an example, we present infinite families of nonadditive codes which are derived from Goethals and Preparata codes. ..."
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Cited by 5 (3 self)
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We extend the stabilizer formalism to a class of nonadditive quantum codes which are constructed from non-linear classical codes. As an example, we present infinite families of nonadditive codes which are derived from Goethals and Preparata codes.
( Polar Quantum Channel Coding for Symmetric Capacity Achieving) 양 재 승, 박 주 용, 이 문 호
"... We demonstrate a fashion of quantum channel combining and splitting, called polar quantum channel coding, to generate a quantum bit (qubit) sequence that achieves the symmetric capacity for any given binary input discrete quantum channels. The present capacity is achievable subject to input of arbit ..."
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We demonstrate a fashion of quantum channel combining and splitting, called polar quantum channel coding, to generate a quantum bit (qubit) sequence that achieves the symmetric capacity for any given binary input discrete quantum channels. The present capacity is achievable subject to input of arbitrary qubits with equal probability. The polarizing quantum channels can be well-conditioned for quantum error-correction coding, which transmits partially quantum data
Fast Constructions of Quantum Codes Based on Residues Pauli Block Matrices
"... We demonstrate how to fast construct quantum error-correction codes based on quadratic residues Pauli block transforms. The present quantum codes have an advantage of being fast designed from Abelian groups on the basis of Pauli block matrices that can be yielded from quadratic residues with much e ..."
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We demonstrate how to fast construct quantum error-correction codes based on quadratic residues Pauli block transforms. The present quantum codes have an advantage of being fast designed from Abelian groups on the basis of Pauli block matrices that can be yielded from quadratic residues with much efficiency.
Logic Functions and Quantum Error Correcting Codes
, 712
"... Abstract. In this paper, based on the relationship between logic functions and quantum error correcting codes(QECCs), we unify the construction of QECCs via graphs, projectors and logic functions. A construction of QECCs over a prime field Fp is given, and one of the results given by Ref[8] can be v ..."
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Abstract. In this paper, based on the relationship between logic functions and quantum error correcting codes(QECCs), we unify the construction of QECCs via graphs, projectors and logic functions. A construction of QECCs over a prime field Fp is given, and one of the results given by Ref[8] can be viewed as a corollary of one theorem in this paper. With the help of Boolean functions, we give a clear proof of the existence of a graphical QECC in mathematical view, and find that the existence of an [[n, k, d]] QECC over Fp requires similar conditions with that depicted in Ref[9]. The result that under the correspondence defined in Ref[17], every [[n, 0, d]] QECC over F2 corresponding to a simple undirected graph has a Boolean basis state, which is closely related to the adjacency matrix of the graph, is given. After a modification of the definition of operators, we find that some QECCs constructed via projectors depicted in Ref[11] can have Boolean basis states. A necessary condition for a Boolean function being used in the construction via projectors is given. We also give some examples to illustrate our results. 1
