Computer Engineering and Applications Journal

Existence of quantum codes [[7,1,4]] p(p > 3) via graphs

Cheng Stella Qian

Published Jan 1, 2012• 0 citations

Abstract

The existence of quantum stabilizer with parameters [[n,k,d]] p = [[7,1,4]] p is showed for all primes(p 3) by using graph machinery,given by Schingemann and Werner,with a little number theory and combinatorics.

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Computer Engineering and Applications Journal

Existence of quantum codes [[7,1,4]] p(p > 3) via graphs

Cheng Stella Qian

Published Jan 1, 2012• 0 citations

OpenAlex

Abstract

Related papers

Quantum codes [[6, 2, 3]]/sub p/ and [[7, 3, 3]]/sub p/ (p ≥ 3) exist

Keqin Feng

IEEE Transactions on Information TheoryAug 1, 200251 citations

Existence of Quantum Codes [[n, k, d]]_p(p > 3, n + k = 8) via Graphs

Cheng Qia

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.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.

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Lifting an $[n,k,d]$ Binary Linear Code to an $[[n,k,d]]$ Quantum Stabilizer CodeLifting an $[n,k,d]$ Binary Linear Code to an $[[n,k,d]]$ Quantum Stabilizer Code

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The codeword stabilized (CWS) framework provides a systematic method for constructing quantum error correcting codes from a simple undirected graph and a classical code. When the underlying classical code is linear, the resulting quantum code is a stabilizer code. In this work, we investigate the conditions under which an arbitrary $[n,k,d]$ classical binary linear code can be lifted to an $[[n,k,d]]$ quantum stabilizer code using the CWS framework. We analyze the random CWS code constructed from a given $[n,k,d]$ binary linear code $\mathcal{C}$ and an Erdős-Rényi random graph $\mathrm{G}(n, \frac12)$ . We establish a lower bound on the probability that the resulting quantum code preserves the minimum distance $d$ of the classical code. This probabilistic analysis yields a Gilbert-Varshamov-type condition on the $n,k,d$ parameters of the classical code that guarantees the existence of a graph resulting in a quantum code with the same $n,k,d$ parameters.The codeword stabilized (CWS) framework provides a systematic method for constructing quantum error correcting codes from a simple undirected graph and a classical code. When the underlying classical code is linear, the resulting quantum code is a stabilizer code. In this work, we investigate the conditions under which an arbitrary $[n,k,d]$ classical binary linear code can be lifted to an $[[n,k,d]]$ quantum stabilizer code using the CWS framework. We analyze the random CWS code constructed from a given $[n,k,d]$ binary linear code $\mathcal{C}$ and an Erdős-Rényi random graph $\mathrm{G}(n, \frac12)$. We establish a lower bound on the probability that the resulting quantum code preserves the minimum distance $d$ of the classical code. This probabilistic analysis yields a Gilbert-Varshamov-type condition on the $n,k,d$ parameters of the classical code that guarantees the existence of a graph resulting in a quantum code with the same $n,k,d$ parameters.

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Scheme for constructing graphs associated with stabilizer quantum codes

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arXiv:1407.2777Jul 10, 20144 citations

Lifting an

[n,k,d]

Binary Linear Code to an

[[n,k,d]]

Quantum Stabilizer Code

[n,k,d]

classical binary linear code can be lifted to an

[[n,k,d]]

quantum stabilizer code using the CWS framework. We analyze the random CWS code constructed from a given

[n,k,d]

binary linear code

\mathcal{C}

and an Erdős-Rényi random graph

\mathrm{G}(n, \frac12)

. We establish a lower bound on the probability that the resulting quantum code preserves the minimum distance

d

of the classical code. This probabilistic analysis yields a Gilbert-Varshamov-type condition on the

n,k,d

parameters of the classical code that guarantees the existence of a graph resulting in a quantum code with the same

n,k,d

parameters.