Quantum Logic Code (QLC)[1]
Description
Family of qubit self-dual CSS stabilizer codes designed to support a complete transversal logical Clifford basis instruction set made of codespace-preserving \(W\leq 2\)-fold transversal gates and permutations [1].
Starting from a self-dual \([[n_0,2,d_0]]\) core code with \((n_0,d_0)\in\{(4,2),(18,5),(20,6)\}\), the construction takes \(r\) disjoint copies and \(\ell\) levels of concatenation with the \([[7,1,3]]\) Steane code to yield \begin{align} [[n_0 r 7^{\ell},2r,d_0 3^{\ell}]] \tag*{(1)}\end{align} codes with rate \(2/(n_0 7^{\ell})\) and transversally generated logical Clifford group \(\mathrm{Sp}(4r,\mathbb{F}_2)\). The three core codes are group-algebra CSS codes: the \([[4,2,2]]\) core comes from \(G=\mathbb{Z}_4\) and \(c=1+x+x^2+x^3\); the \([[18,2,5]]\) core uses \(G=D_9\) and \(c=\{0,2,5,7,11,12,13,17\}\); and the \([[20,2,6]]\) core uses \(G=AGL(1,5)\cong\mathbb{Z}_5\rtimes\mathbb{Z}_4\) and \(c=\{0,1,2,4,6,7,10,16\}\) [1].
Protection
The core distances are \(d_0\in\{2,5,6\}\). Tiling preserves distance, while each Steane concatenation level multiplies distance by three; the family therefore has exact distance \(d_0 3^{\ell}\) [1]. For \(r=7^{\ell}\) and the \([[4,2,2]]\) core, one obtains asymptotic parameters \([[n,\sqrt{n},\Theta(n^\beta)]]\) with \(\beta=\log_{49}3\approx0.2823\) [1].Rate
Family members have rate \(2/(n_0 7^{\ell})\). For fixed \(\ell\), tiling increases both \(n\) and \(k\) linearly while preserving distance [1].Transversal and Permutation-Based Gates
Each family member carries a complete transversal logical Clifford basis ISA generated by individually targeted \(\overline{S}_i\), \(\overline{SHS}_i=\sqrt{\overline{X}_i}\), and \(\overline{CZ}_{i,j}\) gates, together with logical Hadamards and logical-qubit permutations [1].For the \(\mathbb{Z}_4\) and \(AGL(1,5)\) cores, all diagonal/CZ generators are single codespace-preserving \(W\leq2\) layers, with global \(\overline{H}\) available in depth two. For the \(D_9\) core, \(\overline{CZ}_{0,1}\) is depth one while \(\overline{S}_i\) and \(\overline{SHS}_i\) use two layers, and a SAT search finds no depth-one solution for those generators [1].The Steane-concatenation axis preserves the per-generator layer counts, while \(r\)-fold tiling supplies inter-copy aggregate \(\overline{CZ}\) gates in depth one. Individually addressable inter-copy \(\overline{CZ}_{i^{(a)},j^{(b)}}\) gates have depth two for the \(\mathbb{Z}_4\) and \(AGL(1,5)\) core families and depth at most \(18\) for the \(D_9\) core family [1].Cousins
- Concatenated qubit code— The \(\ell>0\) members are obtained by concatenating the QLC cores with the Steane code, and the construction also permits any self-dual doubly-even \([[n_i,1,d_i]]\) inner code with suitable transversal Clifford gates [1].
- Concatenated Steane code— The main QLC concatenation axis uses the \([[7,1,3]]\) Steane code as the inner code, multiplying blocklength and distance by \(7\) and \(3\), respectively [1].
Primary Hierarchy
References
- [1]
- A. Holmes, “Quantum Logic Codes: Complete Transversal Logical Clifford Instruction Sets for High-Rate Stabilizer Quantum Error Correcting Codes”, (2026) arXiv:2606.13521
Page edit log
- Victor V. Albert (2026-06-29) — most recent
Cite as:
“Quantum Logic Code (QLC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/quantum_logic, arXiv:2606.11484