Qubit code distance bound
- Field
- Topic
Problem
Does there exist a sequence of qubit quantum codes \(Q_j\subseteq(\mathbb C^2)^{\otimes n_j}\), with \(n_j\to\infty\), \(\dim Q_j\geq2\), and minimum distances \(d_j\), satisfying
Here \(d_j\) is the least weight of a Pauli operator \(E\) for which \(P_jEP_j\) is not a scalar multiple of the projector \(P_j\) onto \(Q_j\); weight counts nonidentity tensor factors. Equation (1) allows nonadditive and degenerate codes, and the coding rate \(\log_2(\dim Q_j)/n_j\) may tend to zero.
Source
Contributor-supplied attainability question for Rains’s asymptotic upper bound, Theorem 5.6 and Eq. (5.45) of the arXiv version [Rai03]. The source proves the bound; the equality question is the supplied formulation.
Progress
Reports do not certify correctness or automatically change the problem's status. Progress policy.
Rains’s shadow-enumerator bound applies to arbitrary binary quantum-code families, including nonadditive and degenerate codes. The paragraph after Theorem 5.6 distinguishes a stronger Aaltonen-type bound valid only in a range of rates bounded away from zero. Section 6 discusses lower-order improvements, which do not exclude equality in Eq. (1) [Rai03].
The quantum Gilbert–Varshamov tradeoff gives stabilizer families with positive rate for every \(0<\delta<\delta_{\mathrm{GV}}\simeq0.1893\), defined by \(H_2(\delta_{\mathrm{GV}})+\delta_{\mathrm{GV}}\log_2 3=1\), where \(H_2(x)=-x\log_2x-(1-x)\log_2(1-x)\). Anand, Gorokhovsky, Hritz, and Sun realize this tradeoff with random Clifford encoders of depth \(O(\log n)\) (Theorem 1.1); the distance guarantee remains below the target [AGHS26].
Anglès Munné and Huber give exact rational semidefinite infeasibility certificates improving upper bounds for code sizes at block lengths \(6\leq n\leq19\) (Section 4, Table 4.1). These finite-length exclusions do not yield a smaller asymptotic constant. Their certificate files are public [AMH26].
Comment
The question concerns exact distance without a positive-rate, purity, or stabilizer restriction. Literature audit: 8 September 2026; Rains’s primary source and the cited 2026 construction and coding-bound preprints were checked, alongside searches for later improvements. No attaining construction or strictly smaller universal asymptotic constant was verified.
References
- [Rai03]
- E. M. Rains, “New asymptotic bounds for self-dual codes and lattices,” IEEE Transactions on Information Theory 49(5), 1261–1274 (2003).DOIarXiv
- [AGHS26]
- E. Anand, E. Gorokhovsky, J. Hritz, and J. Sun, “Good Stabilizer Codes from Shallow Clifford Circuits with Random Matchings,” arXiv preprint (2026).arXiv
- [AMH26]
- G. Anglès Munné and F. Huber, “SDP bounds on quantum codes: rational certificates,” arXiv preprint (2026). exact certificates.arXivlink