Description
A level-\(\nu\) quantum divisible code is a generalized quantum divisible code whose coefficient vector \(t\) has entries in \(\{\pm 1\}\) [3; Def. V.1]. Each qubit is rotated about \(Z\) by \(\pi/2^{\nu-1}\), in a direction set by the sign of the corresponding entry of \(t\). This transversal rotation implements a gate at the \(\nu\)th level of the Clifford hierarchy on every logical qubit [3; Lemma V.3].
The coefficient signs partition the qubits into sets \(M^+\) and \(M^-\) that witness weak \(2^\nu\)-divisibility of the \(X\)-type stabilizer space [4; Defs. I.4 and I.5]. If all signs agree, this space is \(2^\nu\)-divisible in the ordinary sense. It is therefore doubly even at level two and triply even at level three, while mixed-sign codes need not have either property.
Transversal and Permutation-Based Gates
A level-\(\nu\) quantum divisible code admits a transversal product of \(Z\)-axis rotations by \(\pi/2^{\nu-1}\), with direction set by the coefficient vector. This product implements the same level-\(\nu\) rotation on every logical qubit [3; Lemma V.3].Cousins
- Divisible code— When all coefficient signs agree, the \(X\)-type stabilizers of a level-\(\nu\) quantum divisible code form a \(\nu\)-even linear binary code. Mixed signs instead give weak \(2^\nu\)-divisibility.
- Quasi-cyclic code— Certain double circulant codes can be used to construct doubly even \([[55,1,11]]\) and \([[87,1,15]]\) codes [5].
- Self-dual CSS code— A self-dual weakly doubly even \([[n,1,d]]\) CSS code admits a partitioned transversal physical \(S\) gate that realizes \(\overline{S}^m\), where \(m=|M^+|-|M^-| \pmod 4\); for odd \(m\), together with transversal Hadamard and CNOT, this yields the full logical Clifford group transversally [6][4; Lemma I.4].
- Quantum Reed-Muller (RM) code— Fault-tolerant universal computation can be achieved via code switching between the \([[127,1,15]]\) self-dual doubly even punctured quantum RM code and the \([[127,1,7]]\) triply even punctured quantum RM code [7].
- Doubled color code— Doubled color codes are subsystem codes constructed using a generalization of the doubling transformation [8] that combines doubly even linear binary codes to make triply even codes. The doubling transformation is a special case of level lifting (from two to three) [3; Sec. VI.D].
- Quantum quadratic-residue (QR) code— Qubit quantum QR codes are doubly even and admit transversal implementations of the single-qubit Clifford group [4]. They yield a family of high-distance triorthogonal and weak triply even codes via the doubling transformation [4]; such codes admit transversal implementations of the \(T\) gate.
Primary Hierarchy
Parents
Quantum divisible codes are generalized quantum divisible codes whose coefficient vector has entries in \(\{\pm1\}\) [3; Def. V.1].
The signs of the coefficient vector witness weak \(2^\nu\)-divisibility of the \(X\)-type stabilizer space [3; Def. V.1]. Quantum divisible codes additionally constrain the logical \(X\) representatives and all their joint products with stabilizers.
Quantum divisible code
Children
The \(X\)-type stabilizer space of the \([[17,1,5]]\) 4.8.8 color code is doubly even [4; Table II]. Taking the all-ones logical generator and \(t=(1,\ldots,1)\) makes the joint matrix satisfy the level-two quantum divisible conditions. The logical \(X\) operator is strongly transversal [4; Table I]. The two logical cosets therefore have weights congruent to zero and one modulo four.
The family satisfies the level-three quantum divisible conditions for a coefficient vector with \(\pm1\) entries [3; Sec. VI.C]. Its \(X\)-type stabilizer weight enumerator is \(1+x^8+6x^{2k+4}\), so the stabilizer space is triply even exactly when \(k\equiv2\pmod 4\) [9; Sec. VII]. Uniform coefficients \(t=(-1,\ldots,-1)\) work in that case, while the \(k\equiv0\pmod 4\) members require mixed signs.
The \(X\)-type stabilizer space of the \([[49,1,5]]\) triorthogonal code is triply even, and its logical generator is the all-ones vector [9; Appx. B]. The choice \(t=(1,\ldots,1)\) makes the joint matrix satisfy the level-three quantum divisible conditions [3; Sec. VI.D].
The \(X\)-type stabilizer space of each \([[2^r-1,1,3]]\) simplex code is \(2^{r-1}\)-divisible [10,11]. Taking the all-ones logical generator and \(t=(-1,\ldots,-1)\) makes the joint matrix satisfy the level-\((r-1)\) quantum divisible conditions [3; Sec. VI.B]. The two logical cosets therefore have weights congruent to \(0\) and \(2^{r-1}-1\) modulo \(2^{r-1}\), respectively.
References
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- K. Betsumiya and A. Munemasa, “On triply even binary codes”, Journal of the London Mathematical Society 86, 1 (2012) arXiv:1012.4134 DOI
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Page edit log
- Victor V. Albert (2026-09-06) — most recent
- Victor V. Albert (2026-09-01)
- Victor V. Albert (2026-06-08)
- Jingzhen Hu (2022-05-04)
- Victor V. Albert (2022-05-04)
Cite as:
“Quantum divisible code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/quantum_divisible, arXiv:2606.11484