\([[47,3,3]]\) \(CCZ\) distillation code[1; Sec. V.C]
Description
A \([[47,3,3]]\) generalized triorthogonal CSS code that distills a \(CCZ\) magic state from 47 noisy \(T\) states at distance three [1; Sec. V.C]. It is the smallest known CSS code with three logical qubits, \(d_Z\geq3\), and a quasi-transversal \(CCZ\) gate [2].
Its columns come from a nine-qubit circuit of 47 \(Z\)-type Pauli-product \(\pi/8\) rotations that implements \(CCZ\) [1; Sec. V.C]. One of the three \(CCZ\) qubits is encoded in a seven-qubit phase-flip repetition code [1; Secs. III.D and V.C]. Its generalized triorthogonal matrix has three logical rows and six \(X\)-type stabilizer rows with distinct nonzero stabilizer columns, and it satisfies the nine Campbell-Howard conditions [1; App. D][2; Ex. 3.7].
The same construction yields a 48-qubit code with three logical qubits and seven \(X\)-type stabilizer generators whose \(CCZ\)-state distillation protocol has distance four and output error \(2816p^4\) [1; Sec. V.C and App. D]. The \([[47,3,3]]\) code is a shortening of this \([[48,3,4]]\) code.
Protection
Distance three, with \(d_X=16\) and \(d_Z=3\) [2; Ex. 3.7]. No CSS code with three logical qubits, \(d_Z\geq3\), and a quasi-transversal \(CCZ\) gate exists for \(n\leq38\) [2; Thm. 4.5]. Its \(T\)-count 47 and nine matrix rows form the only Pareto-optimal point for distance-three \(CCZ\)-state distillation among generalized triorthogonal codes with \(n\leq54\) [3; Table II].Transversal and Permutation-Based Gates
Logical \(CCZ\) gate from physical \(T\) gates followed by a diagonal Clifford correction [2; Ex. 3.7]. No pattern of single-qubit \(T\) and \(T^\dagger\) gates implements the logical \(CCZ\) gate, so the correction must contain entangling \(CZ\) gates [2; Ex. 3.7].Gates
Output error \(236p^3\) for input error \(p\) [1; Sec. V.C].Primary Hierarchy
References
- [1]
- H. Jacinto, X. Valcarce, V. Barizien, É. Gouzien, and N. Sangouard, “Exploring the landscape of compact magic-state distillation factories”, (2026) arXiv:2606.07734
- [2]
- B. Lu and K. R. Brown, “Building codes with transversal CCZ using projective geometry and SAT solvers”, (2026) arXiv:2610.09341
- [3]
- A. Wills, S. P. Jain, and S. Singh, “Classification of Generalised Triorthogonal Codes through Length 54”, (2026) arXiv:2609.30860
Page edit log
- Victor V. Albert (2026-10-13) — most recent
- Victor V. Albert (2026-10-08)
Cite as:
“\([[47,3,3]]\) \(CCZ\) distillation code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_47_3_3, arXiv:2606.11484