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Science China Information Sciences

Construction of nonbinary quantum cyclic codes by using graph method

Liu ., Tailin, Wen Wen, Qiaoyan, Zihui

Published Jan 1, 2005• 0 citations
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Abstract

Using the graph method proposed by Schlingemann and Werner, this paper introduces a technique to construct nonbinary quantum cyclic codes and provides a specific example. We also construct the quantum codes [[8, 2, 4]]p and [[n, n - 2, 2]]p for all odd primes p by the graph method.

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In this paper, we present a systematic way based on the nonbinary graph state of constructing good nonbinary quantum codes, both additive and nonadditive, for systems with integer dimensions. With a computer search, which results in many interesting codes including some nonadditive codes meeting the Singleton bounds, we are able to construct explicitly four families of optimal codes, namely, [[6,2,3]]p{[[6,2,3]]}_{p}[[6,2,3]]p​, [[7,3,3]]p{[[7,3,3]]}_{p}[[7,3,3]]p​, [[8,2,4]]p{[[8,2,4]]}_{p}[[8,2,4]]p​, and [[8,4,3]]p{[[8,4,3]]}_{p}[[8,4,3]]p​ for any odd dimension p\mathit{p}p and a family of nonadditive codes ((5,p,3))p{((5,p,3))}_{p}((5,p,3))p​ for arbitrary p>3p>3p>3. In the case of composite numbers as dimensions, we also construct a family of stabilizer codes ((6,2p2,3))2p{((6,2{p}^{2},3))}_{2p}((6,2p2,3))2p​ for odd p\mathit{p}p, whose coding subspace is not of a dimension that is a power of the dimension of the physical subsystem.
We propose a systematic scheme for the construction of graphs associated with binary stabilizer codes. The scheme is characterized by three main steps: first, the stabilizer code is realized as a codeword-stabilized (CWS) quantum code; second, the canonical form of the CWS code is uncovered; third, the input vertices are attached to the graphs. To check the effectiveness of the scheme, we discuss several graphical constructions of various useful stabilizer codes characterized by single and multi-qubit encoding operators. In particular, the error-correcting capabilities of such quantum codes are verified in graph-theoretic terms as originally advocated by Schlingemann and Werner. Finally, possible generalizations of our scheme for the graphical construction of both (stabilizer and nonadditive) nonbinary and continuous-variable quantum codes are briefly addressed.
We propose a systematic scheme for the construction of graphs associated with binary stabilizer codes. The scheme is characterized by three main steps: first, the stabilizer code is realized as a codeword-stabilized (CWS) quantum code; second, the canonical form of the CWS code is uncovered; third, the input vertices are attached to the graphs. To check the effectiveness of the scheme, we discuss several graphical constructions of various useful stabilizer codes characterized by single and multi-qubit encoding operators. In particular, the error-correcting capabilities of such quantum codes are verified in graph-theoretic terms as originally advocated by Schlingemann and Werner. Finally, possible generalizations of our scheme for the graphical construction of both (stabilizer and nonadditive) nonbinary and continuous-variable quantum codes are briefly addressed.
Quantum error correction plays a crucial role in quantum information theory.Schlingemann and Werner presented a new way to construct quantum stabilizer codes by finding certain graphs(or matrices) with specific properties, and they constructed several new non-binary quantum codes, in particular, they gave a new proof on the existence of quantum codes [[5, 1, 3]] for all odd primes. In this paper, using the same method, we prove the existence of MDS quantum codes with the sum n and k being 8 for all primes exceeding three.