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Lijuan Xing - One of the best experts on this subject based on the ideXlab platform.
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On a Problem Concerning the Quantum Hamming Bound for Impure Quantum Codes
IEEE Transactions on Information Theory, 2010Co-Authors: Lijuan XingAbstract:A famous open problem in the theory of Quantum error-correcting Codes is whether or not the parameters of an impure Quantum Code can violate the Quantum Hamming bound for pure Quantum Codes. We partially solve this problem. We demonstrate that there exists a threshold N(d,m) such that an arbitrary ((n,K,d))m Quantum Code must obey the Quantum Hamming bound whenever n ≥ N(d,m). We list some values of N(d,m) for small d and binary Quantum Codes.
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Progress on Problem about Quantum Hamming Bound for Impure Quantum Codes
arXiv: Quantum Physics, 2009Co-Authors: Lijuan XingAbstract:A famous open problem in the theory of Quantum error-correcting Codes is whether or not the parameters of an impure Quantum Code can violate the Quantum Hamming bound for pure Quantum Codes. We partially solve this problem. We demonstrate that there exists a threshold such that an arbitrary Quantum Code must obey the Quantum Hamming bound whenever . We list some values of for small d and binary Quantum Codes.
Xinglijuan - One of the best experts on this subject based on the ideXlab platform.
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On a problem concerning the Quantum hamming bound for impure Quantum Codes
IEEE Transactions on Information Theory, 2010Co-Authors: Lizhuo, XinglijuanAbstract:A famous open problem in the theory of Quantum error-correcting Codes is whether or not the parameters of an impure Quantum Code can violate the Quantum Hamming bound for pure Quantum Codes. We par...
Mark M. Wilde - One of the best experts on this subject based on the ideXlab platform.
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Quantum convolutional coding with shared entanglement: general structure
Quantum Information Processing, 2010Co-Authors: Mark M. Wilde, Todd A. BrunAbstract:We present a general theory of entanglement-assisted Quantum convolutional coding. The Codes have a convolutional or memory structure, they assume that the sender and receiver share noiseless entanglement prior to Quantum communication, and they are not restricted to possess the Calderbank–Shor–Steane structure as in previous work. We provide two significant advances for Quantum convolutional coding theory. We first show how to “expand” a given set of Quantum convolutional generators. This expansion step acts as a preprocessor for a polynomial symplectic Gram–Schmidt orthogonalization procedure that simplifies the commutation relations of the expanded generators to be the same as those of entangled Bell states (ebits) and ancilla qubits. The above two steps produce a set of generators with equivalent error-correcting properties to those of the original generators. We then demonstrate how to perform online encoding and decoding for a stream of information qubits, halves of ebits, and ancilla qubits. The upshot of our theory is that the Quantum Code designer can engineer Quantum convolutional Codes with desirable error-correcting properties without having to worry about the commutation relations of these generators.
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Nonlocal Quantum information in bipartite Quantum error correction
Quantum Information Processing, 2010Co-Authors: Mark M. Wilde, David FattalAbstract:We show how to convert an arbitrary stabilizer Code into a bipartite Quantum Code. A bipartite Quantum Code is one that involves two senders and one receiver. The two senders exploit both nonlocal and local Quantum resources to enCode Quantum information with local encoding circuits. They transmit their enCoded Quantum data to a single receiver who then deCodes the transmitted Quantum information. The nonlocal resources in a bipartite Code are ebits and nonlocal information qubits and the local resources are ancillas and local information qubits. The technique of bipartite Quantum error correction is useful in both the Quantum communication scenario described above and in fault-tolerant Quantum computation. It has application in fault-tolerant Quantum computation because we can prepare nonlocal resources offline and exploit local encoding circuits. In particular, we derive an encoding circuit for a bipartite version of the Steane Code that is local and additionally requires only nearest-neighbor interactions. We have simulated this encoding in the CNOT extended rectangle with a publicly available fault-tolerant simulation software. The result is that there is an improvement in the "pseudothreshold" with respect to the baseline Steane Code, under the assumption that Quantum memory errors occur less frequently than Quantum gate errors.
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Logical operators of Quantum Codes
Physical Review A, 2009Co-Authors: Mark M. WildeAbstract:(Dated: March 30, 2009)I show how applying a symplectic Gram-Schmidt orthogonalization to the normalizer of a QuantumCode gives a different way of determining the Code’s logical operators. This approach may be morenatural in the setting where we produce a Quantum Code from classical Codes because the generatormatrices of the classical Codes form the normalizer of the resulting Quantum Code. This technique isparticularly useful in determining the logical operators of an entanglement-assisted Code producedfrom two classical binary Codes or from one classical quaternary Code. Finally, this approach givesadditional formulas for computing the amount of entanglement that an entanglement-assisted Coderequires.
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Optimal entanglement formulas for entanglement-assisted Quantum coding
Physical Review A, 2008Co-Authors: Mark M. Wilde, Todd A. BrunAbstract:We provide several formulas that determine the optimal number of entangled bits (ebits) that a general entanglement-assisted Quantum Code requires. Our first theorem gives a formula that applies to an arbitrary entanglement-assisted block Code. Corollaries of this theorem give formulas that apply to a Code imported from two classical binary block Codes, to a Code imported from a classical quaternary block Code, and to a continuous-variable entanglement-assisted Quantum block Code. Finally, we conjecture two formulas that apply to entanglement-assisted Quantum convolutional Codes.
Todd A. Brun - One of the best experts on this subject based on the ideXlab platform.
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Codeword-stabilized Quantum Codes on subsystems
Physical Review A, 2012Co-Authors: Jeonghwan Shin, Jun Heo, Todd A. BrunAbstract:Codeword stabilized Quantum Codes provide a unified approach to constructing Quantum error-correcting Codes, including both additive and non-additive Quantum Codes. Standard Codeword stabilized Quantum Codes enCode Quantum information into subspaces. The more general notion of encoding Quantum information into a subsystem is known as an operator (or subsystem) Quantum error correcting Code. Most operator Codes studied to date are based in the usual stabilizer formalism. We introduce operator Quantum Codes based on the Codeword stabilized Quantum Code framework. Based on the necessary and sufficient conditions for operator Quantum error correction, we derive a error correction condition for operator Codeword stabilized Quantum Codes. Based on this condition, the word operators of a operator Codeword stabilized Quantum Code are constructed from a set of classical binary errors induced by generators of the gauge group. We use this scheme to construct examples of both additive and non-additive Codes that enCode Quantum information into a subsystem.
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Entanglement-assisted Codeword stabilized Quantum Codes
Physical Review A, 2011Co-Authors: Jeonghwan Shin, Jun Heo, Todd A. BrunAbstract:Entangled qubits can increase the capacity of Quantum error-correcting Codes based on stabilizer Codes. In addition, by using entanglement Quantum stabilizer Codes can be construct from classical linear Codes that do not satisfy the dual-containing constraint. We show that it is possible to construct both additive and nonadditive Quantum Codes using the Codeword stabilized Quantum Code framework. Nonadditive Codes may offer improved performance over the more common stabilizer Codes. Like other entanglement-assisted Codes, the encoding procedure acts only on the qubits on Alice's side, and only these qubits are assumed to pass through the channel. However, errors in the Codeword stabilized Quantum Code framework give rise to effective Z errors on Bob's side. We use this scheme to construct entanglement-assisted nonadditive Quantum Codes, in particular, ((5,16,2;1)) and ((7,4,5;4)) Codes.
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Quantum convolutional coding with shared entanglement: general structure
Quantum Information Processing, 2010Co-Authors: Mark M. Wilde, Todd A. BrunAbstract:We present a general theory of entanglement-assisted Quantum convolutional coding. The Codes have a convolutional or memory structure, they assume that the sender and receiver share noiseless entanglement prior to Quantum communication, and they are not restricted to possess the Calderbank–Shor–Steane structure as in previous work. We provide two significant advances for Quantum convolutional coding theory. We first show how to “expand” a given set of Quantum convolutional generators. This expansion step acts as a preprocessor for a polynomial symplectic Gram–Schmidt orthogonalization procedure that simplifies the commutation relations of the expanded generators to be the same as those of entangled Bell states (ebits) and ancilla qubits. The above two steps produce a set of generators with equivalent error-correcting properties to those of the original generators. We then demonstrate how to perform online encoding and decoding for a stream of information qubits, halves of ebits, and ancilla qubits. The upshot of our theory is that the Quantum Code designer can engineer Quantum convolutional Codes with desirable error-correcting properties without having to worry about the commutation relations of these generators.
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Optimal entanglement formulas for entanglement-assisted Quantum coding
Physical Review A, 2008Co-Authors: Mark M. Wilde, Todd A. BrunAbstract:We provide several formulas that determine the optimal number of entangled bits (ebits) that a general entanglement-assisted Quantum Code requires. Our first theorem gives a formula that applies to an arbitrary entanglement-assisted block Code. Corollaries of this theorem give formulas that apply to a Code imported from two classical binary block Codes, to a Code imported from a classical quaternary block Code, and to a continuous-variable entanglement-assisted Quantum block Code. Finally, we conjecture two formulas that apply to entanglement-assisted Quantum convolutional Codes.
Lizhuo - One of the best experts on this subject based on the ideXlab platform.
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On a problem concerning the Quantum hamming bound for impure Quantum Codes
IEEE Transactions on Information Theory, 2010Co-Authors: Lizhuo, XinglijuanAbstract:A famous open problem in the theory of Quantum error-correcting Codes is whether or not the parameters of an impure Quantum Code can violate the Quantum Hamming bound for pure Quantum Codes. We par...
