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arXiv:1601.07423 · IEEE International Symposium on Information Theory (ISIT)

Concatenated codes for amplitude damping

Tyler Jackson, Markus Grassl, Bei Zeng

Published Jul 1, 2016• 8 citations
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Abstract

We discuss a method to construct quantum code correcting amplitude-damping errors via code concatenation. The inner codes are chosen as asymmetric Calderbank-Shor-Steane (CSS) codes. By concatenating with outer code-correcting symmetric errors, many new codes with good parameters are found, which outperform amplitude damping codes obtained by any previously known construction.

DOI
10.1109/isit.2016.7541703

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Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Mar 4, 2009•144 citations
We discuss a method to construct quantum code correcting amplitude-damping errors via code concatenation. The inner codes are chosen as asymmetric Calderbank-Shor-Steane (CSS) codes. By concatenating with outer code-correcting symmetric errors, many new codes with good parameters are found, which outperform amplitude damping codes obtained by any previously known construction.
We construct new families of multi-error-correcting quantum codes for the amplitude damping channel. Our key observation is that, with proper encoding, two uses of the amplitude damping channel simulate a quantum erasure channel. This allows us to use concatenated codes with quantum erasure-correcting codes as outer codes for correcting multiple amplitude damping errors. Our new codes are degenerate stabilizer codes and have parameters which are better than the amplitude damping codes obtained by any previously known construction.
A method for concatenating quantum error-correcting codes is presented. The method is applicable to a wide class of quantum error-correcting codes known as Calderbank-Shor-Steane (CSS) codes. As a result, codes that achieve a high rate in the Shannon-theoretic sense and that are decodable in polynomial time are presented. The rate is the highest among those known to be achievable by CSS codes. Moreover, the best known lower bound on the greatest minimum distance of codes constructible in polynomial time is improved for a wide range.
A method for concatenating quantum error-correcting codes is presented. The method is applicable to a wide class of quantum error-correcting codes known as Calderbank-Shor-Steane (CSS) codes. As a result, codes that achieve a high rate in the Shannon theoretic sense and that are decodable in polynomial time are presented. The rate is the highest among those known to be achievable by CSS codes. Moreover, the best known lower bound on the greatest minimum distance of codes constructible in polynomial time is improved for a wide range.
In this paper, we utilize a concatenation scheme to construct new families of quantum error correction codes achieving the quantum Gilbert-Varshamov (GV) bound asymptotically. Weconcatenate alternant codes with any linear code achievingthe classical GV bound to construct Calderbank-Shor-Steane (CSS) codes. We show that the concatenated code can achieve the quantum GV bound asymptotically and can approach the Hashing bound for asymmetric Pauli channels. By combing Steane’s enlargement construction of CSS codes, we derive a family of enlarged stabilizer codes achieving the quantum GV bound for enlarged CSS codes asymptotically. Asapplications, we derive two families of fast encodable and decodable CSS codes with parametersQ1=[[N,Ω(N),Ω(N)]]\mathscr {Q}_{1}=[[N,\Omega (\sqrt {N}),\Omega (\sqrt {N})]]Q1​=[[N,Ω(N​),Ω(N​)]], andQ2=[[N,Ω(N/log⁡N),Ω(N/log⁡N)/Ω(log⁡N)]]\mathscr {Q}_{2}=[[N,\Omega (N/\log N),\Omega (N/\log N)/\Omega (\log N)]]Q2​=[[N,Ω(N/logN),Ω(N/logN)/Ω(logN)]]. We show thatQ1\mathscr {Q}_{1}Q1​can be encoded very efficiently by circuits of sizeO(N)O(N)O(N)and depthO(N)O(\sqrt {N})O(N​). For an input error syndrome,Q1\mathscr {Q}_{1}Q1​can correct any adversarial error of weight up to half the minimum distance bound inO(N)O(N)O(N)time.Q1\mathscr {Q}_{1}Q1​can also be decoded in parallel inO(N)O(\sqrt {N})O(N​)time by usingO(N)O(\sqrt {N})O(N​)classical processors. For an input error syndrome, we proved thatQ2\mathscr {Q}_{2}Q2​can correct a linear number ofX{X}X-errors with high probability and an almost linear number ofZ{Z}Z-errors inO(N)O(N)O(N)time. Moreover,Q2\mathscr {Q}_{2}Q2​can be decoded in parallel inO(log⁡(N))O(\log (N))O(log(N))time by usingO(N)O(N)O(N)classical processors.
Recently, quantum error-correcting codes have been proposed that capitalize on the fact that many physical error models lead to a significant asymmetry between the probabilities for bit- and phase-flip errors. An example for a channel that exhibits such asymmetry is the combined amplitude damping and dephasing channel, where the probabilities of bit and phase flips can be related to relaxation and dephasing time, respectively. We study asymmetric quantum codes that are obtained from the Calderbank–Shor–Steane (CSS) construction. For such codes, we derive upper bounds on the code parameters using linear programming. A central result of this paper is the explicit construction of some new families of asymmetric quantum stabilizer codes from pairs of nested classical codes. For instance, we derive asymmetric codes using a combination of Bose–Chaudhuri–Hocquenghem (BCH) and finite geometry low-density parity-check (LDPC) codes. We show that the asymmetric quantum codes offer two advantages, namely to allow a higher rate without sacrificing performance when compared with symmetric codes and vice versa to allow a higher performance when compared with symmetric codes of comparable rates. Our approach is based on a CSS construction that combines BCH and finite geometry LDPC codes.