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One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. In past years, many good quantum error-correcting codes had been derived as binary stabilizer codes. Fault-tolerant quantum computation prompted the study of nonbinary quantum codes, but(More)
Two orthonormal bases B and B ′ of a d-dimensional complex inner-product space are called mutually unbiased if and only if |b|b ′ | 2 = 1/d holds for all b ∈ B and b ′ ∈ B ′. The size of any set containing pairwise mutually unbiased bases of C d cannot exceed d + 1. If d is a power of a prime, then extremal sets containing d + 1 mutually unbiased bases are(More)
Classical BCH codes that contain their (Euclidean or Hermitian) dual codes can be used to construct quantum stabilizer codes; this correspondence studies the properties of such codes. It is shown that a BCH code of length n can contain its dual code only if its designed distance δ = O(√ n), and the converse is proved in the case of narrow-sense codes.(More)
— Convolutional stabilizer codes promise to make quantum communication more reliable with attractive online encoding and decoding algorithms. This paper introduces a new approach to convolutional stabilizer codes based on direct limit constructions. Two families of quantum convolutional codes are derived from generalized Reed-Solomon codes and from(More)
— Recently, it has been shown that the max flow capacity can be achieved in a multicast network using network coding. In this paper, we propose and analyze a more realistic model for wireless random networks. We prove that the capacity of network coding for this model is concentrated around the expected value of its minimum cut. Furthermore, we establish(More)
The controlled-not gate and the single qubit gates are considered elementary gates in quantum computing. It is natural to ask how many such elementary gates are needed to implement more elaborate gates or circuits. Recall that a controlled-U gate can be realized with two controlled-not gates and four single qubit gates. We prove that this implementation is(More)
— Unitary error bases generalize the Pauli matrices to higher dimensional systems. Two basic constructions of unitary error bases are known: An algebraic construction by Knill, which yields nice error bases, and a combinato-rial construction by Werner, which yields shift-and-multiply bases. An open problem posed by Schlingemann and Werner relates these two(More)
— Quantum convolutional codes can be used to protect a sequence of qubits of arbitrary length against decoherence. We introduce two new families of quantum convolutional codes. Our construction is based on an algebraic method which allows to construct classical convolutional codes from block codes, in particular BCH codes. These codes have the property that(More)