Partially Concatenated Calderbank-Shor-Steane Codes Achieving the Quantum Gilbert-Varshamov Bound Asymptotically
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Abstract
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 parameters <inline-formula> <tex-math notation="LaTeX">$\mathscr {Q}_{1}=[[N,\Omega (\sqrt {N}),\Omega (\sqrt {N})]]$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$\mathscr {Q}_{2}=[[N,\Omega (N/\log N),\Omega (N/\log N)/\Omega (\log N)]]$ </tex-math></inline-formula>. We show that <inline-formula> <tex-math notation="LaTeX">$\mathscr {Q}_{1}$ </tex-math></inline-formula> can be encoded very efficiently by circuits of size <inline-formula> <tex-math notation="LaTeX">$O(N)$ </tex-math></inline-formula> and depth <inline-formula> <tex-math notation="LaTeX">$O(\sqrt {N})$ </tex-math></inline-formula>. For an input error syndrome, <inline-formula> <tex-math notation="LaTeX">$\mathscr {Q}_{1}$ </tex-math></inline-formula> can correct any adversarial error of weight up to half the minimum distance bound in <inline-formula> <tex-math notation="LaTeX">$O(N)$ </tex-math></inline-formula> time. <inline-formula> <tex-math notation="LaTeX">$\mathscr {Q}_{1}$ </tex-math></inline-formula> can also be decoded in parallel in <inline-formula> <tex-math notation="LaTeX">$O(\sqrt {N})$ </tex-math></inline-formula> time by using <inline-formula> <tex-math notation="LaTeX">$O(\sqrt {N})$ </tex-math></inline-formula> classical processors. For an input error syndrome, we proved that <inline-formula> <tex-math notation="LaTeX">$\mathscr {Q}_{2}$ </tex-math></inline-formula> can correct a linear number of <inline-formula> <tex-math notation="LaTeX">${X}$ </tex-math></inline-formula>-errors with high probability and an almost linear number of <inline-formula> <tex-math notation="LaTeX">${Z}$ </tex-math></inline-formula>-errors in <inline-formula> <tex-math notation="LaTeX">$O(N)$ </tex-math></inline-formula> time. Moreover, <inline-formula> <tex-math notation="LaTeX">$\mathscr {Q}_{2}$ </tex-math></inline-formula> can be decoded in parallel in <inline-formula> <tex-math notation="LaTeX">$O(\log (N))$ </tex-math></inline-formula> time by using <inline-formula> <tex-math notation="LaTeX">$O(N)$ </tex-math></inline-formula> classical processors.