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Nonadditive Quantum Codes from $\Z_4$-Codes

San Ling, Patrick Solé

Published Jan 1, 2008• 1 citation
OpenAlex

Abstract

Non additive binary quantum codes are constructed by using classical $\Z_4$-linear binary codes.

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Nonadditive Quantum Codes from Z4\Z_4Z4​-Codes
Non additive binary quantum codes are constructed by using classical Z4\Z_4Z4​-linear binary codes.
Binary Quantum Codes from Cyclic Codes over Rs1,s2\mathfrak {R}_{s_{1},s_{2}}Rs1​,s2​​
We first present a useful characterization of additive (stabilizer) quantum error-correcting codes. Then we present several examples of We first present a useful characterization of additive (stabilizer) quantum error--correcting codes. Then we present several examples of nonadditive codes. We show that there exist infinitely many non-trivial nonadditive codes with different minimum distances, and high rates. In fact, we show that nonadditive codes that correct t errors can reach the asymptotic rate R=1-2H(2t/n), where H(x) is the binary entropy function. Finally, we introduce the notion of strongly nonadditive codes (i.e., quantum codes with the following property: the trivial code consisting of the entire Hilbert space is the only additive code that is equivalent to any code containing the given code), and provide a construction for an ((11,2,3)) strongly nonadditive code.
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 ZZZ 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.
Quantum Codes Over an Extension of Z4{\mathbb {Z}_4}Z4​