Weakly divisible CSS code

Weakly divisible CSS code [1][2; Defs. I.4 and I.5]

Description

A weakly $\Delta$-divisible CSS code for $\Delta>1$ is a CSS code whose $X$-type stabilizer space obeys a signed divisibility condition on two disjoint sets of qubits. The condition allows coordinates outside those sets and does not constrain the logical $X$ representatives. Weakly four-divisible and weakly eight-divisible CSS codes are also called $\mathrm{DE}^*$ and $\mathrm{TE}^*$ codes, respectively.

More explicitly, the binary space $C_2$ of $X$-type stabilizers is weakly $\Delta$-divisible if there are disjoint subsets $M^+,M^-\subseteq[n]$ such that \begin{align} |u\cap M^+|-|u\cap M^-|\equiv0\pmod\Delta\tag*{$(1)$} \end{align}for every $u\in C_2$ [2; Defs. I.4 and I.5]. At least one of $M^+$ and $M^-$ is required to be nonempty [1]. Coordinates outside $M^+\cup M^-$ do not contribute to the congruence. Ordinary $\Delta$-divisibility is recovered by taking $M^+=[n]$ and $M^-=\emptyset$. Weak $\Delta$-divisibility implies weak $\Delta^{\prime}$-divisibility for every divisor $\Delta^{\prime}$ of $\Delta$.

Coset divisibility

A stronger condition applies to a CSS code defined by binary linear codes $C_2\subseteq C_1$ and a character vector $y$ fixing the signs of the $Z$-type stabilizers. Coset divisibility requires the codeword weights to be constant modulo $\Delta$ within each coset of $C_2$ in $C_1+y$ [3; Thm. 7][4; Cor. 7].

More explicitly, coset divisibility requires \begin{align} \operatorname{wt}(u+w)\equiv\operatorname{wt}(w)\pmod\Delta\tag*{$(2)$} \end{align}for every $u\in C_2$ and $w\in C_1+y$. Taking $w=y$ shows that every coset-divisible code is weakly divisible with $M^-=\operatorname{supp}(y)$ and $M^+=[n]\setminus M^-$. The converse need not hold because weak divisibility tests only this one element of $C_1+y$, rather than every logical coset representative. Positive signs correspond to $y=0$, in which case the trivial coset shows that $C_2$ is a $\Delta$-divisible classical code. A $\Delta$-coset-divisible code is also $\Delta^{\prime}$-coset-divisible for every divisor $\Delta^{\prime}$ of $\Delta$.

For $\Delta=2^\nu$ and $y=0$, the above is equivalent to $2^\nu$ dividing $\operatorname{wt}(u)$ for every $u\in C_2$, together with $2^{\nu-1}$ dividing $\operatorname{wt}(u*w)$ for every $u\in C_2$ and $w\in C_1$ [3; Rem. 13]. Here $*$ denotes the entrywise product.

Transversal and Permutation-Based Gates

A weakly triply even $[[n,1,d]]$ CSS code with positive $Z$-type stabilizer signs and a strongly transversal logical $X$ gate admits a partitioned transversal physical $T$ gate that realizes $\overline{T}^m$, where $m=|M^+|-|M^-| \pmod 8$ [1][2; Lemma I.3]. The logical gate is non-Clifford when $m$ is odd. For the coset-divisible subclass, the uniform transversal gate $\operatorname{diag}(1,e^{2\pi i/\Delta})$ preserves the code space and induces a logical diagonal gate determined by the coset residues [4; Cor. 7]. For $\Delta=2^\nu$, this gate corresponds to the all-ones exponent vector in the group of diagonal transversal gates preserving the code [5; Thm. 3.2 and Rem. 3.16]. Algorithms determine this group for a given CSS code [6,7]. Let $C$ be a $2^m$-divisible $[n,k,d]$ code with $m\geq2$ and dual distance $d^{\perp}$, and let $t<\min\{k,d,d^{\perp}\}$. Puncturing $C$ on $t$ coordinates yields $C_1$, and shortening $C$ on the same coordinates yields $C_2\subseteq C_1$ [8; Sec. III.A]. Repeating both codes $2^p$ times yields a $2^{m+p}$-coset-divisible $[[2^p(n-t),t,\geq\min\{d,d^{\perp}\}-t]]$ CSS code with positive signs [8; Sec. III and Prop. 22]. Its transversal $R_Z(\pi/2^{l})$ gate realizes the logical transversal $R_Z(\pi/2^{l-p})^{\dagger}$ gate for $p\leq l\leq m+p-1$ and the logical identity for $l\leq p-1$ [8; Lemma 7]. Consider a CSS code with positive $Z$-type stabilizer signs whose $X$-type stabilizers form a $\nu$-even space, and for which the entrywise product of every $X$-type logical operator and $X$-type stabilizer has weight divisible by $2^{\nu-1}$. Such a code admits a diagonal transversal gate at the $\nu$th level of the Clifford hierarchy [9; Prop. 8]. Transversal $T^\dagger$ preserves the quadratic-form family and induces, up to global phase, $\exp(-i\pi Z^{\otimes k}/8)$ on its $k$ logical qubits [4; Thm. 10]. This logical gate decomposes into $T$ on every logical qubit, controlled-Phase$^\dagger$ on every pair, and $CCZ$ on every triple.

Fault Tolerance

The $[[31,5,3]]$ and $[[63,7,3]]$ members can serve as outer codes for the five-qubit and Steane codes, respectively. The resulting layered constructions implement a fault-tolerant logical $T$ gate on the inner code without teleporting magic states [4; Sec. V]. Encoding and decoding are required to pass between the inner and outer codes.

Realizations

Triply even codes can yield secure multi-party quantum computation [10].

Cousins

Generalized quantum divisible code— Weak divisibility constrains signed weights in the $X$-type stabilizer space, while generalized quantum divisibility constrains the joint matrix of logical $X$ representatives and stabilizers using odd coefficients on every qubit.
$[2^m-1,m,2^{m-1}]$ simplex code— One coset-divisible family takes $C_2$ to be the length-$2^m-1$ simplex code and obtains $C_1$ by adjoining evaluation vectors of bounded-rank quadratic forms [4; Lemma 9]. This yields $[[2^m-1,k,3]]$ codes for $m\geq4$ with $k=1+\sum_{i=1}^{m-4}(m-i)$ [4; Thm. 10]. Deleting quadratic-form rows from the logical generator matrix gives the members with smaller $k$ [4; Rem. 12].
Triorthogonal code— The $[[31,5,3]]$ member together with the five-qubit code can be viewed as a factorization of a $[[31,1,3]]$ triorthogonal code [4; Sec. V].
Concatenated qubit code— Particular coset-divisible codes can be used as outer codes in layered constructions that implement a fault-tolerant $T$ gate on the five-qubit or Steane code [4; Sec. V].
Divisible code— Let $t$ be less than the dimension, distance, and dual distance of a $2^m$-divisible binary linear code with $m\geq2$. Puncturing the code on $t$ coordinates and shortening it on the same coordinates yields a pair $C_2\subseteq C_1$ defining a $2^m$-coset-divisible CSS code with positive signs [8; Prop. 22]. Its transversal $R_Z(\pi/2^{l})$ gate realizes the logical transversal $R_Z(\pi/2^{l})^{\dagger}$ gate for $l\leq m-1$ [8; Lemma 7].
$[[5,1,3]]$ Five-qubit perfect code— A fault-tolerant logical $T$ gate can be obtained by encoding the five-qubit code’s five physical qubits into the five logical qubits of a $[[31,5,3]]$ outer coset-divisible code preserved by transversal $T^\dagger$ [4; Sec. V].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeQubit Stabilizer Hamiltonian-based QECC Quantum

Coherent-parity-check (CPC) code

Qubit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum

Parents

Qubit CSS code

Weakly divisible CSS codes are CSS codes whose $X$-type stabilizer spaces satisfy a signed divisibility condition [2; Def. I.5].

Weakly divisible CSS code

Children

Quantum divisible code

The signs of the coefficient vector witness weak $2^\nu$-divisibility of the $X$-type stabilizer space [11; Def. V.1]. Quantum divisible codes additionally constrain the logical $X$ representatives and all their joint products with stabilizers.

$[[47,3,3]]$ $CCZ$ distillation code

$[[48,3,3]]$ transversal $CCZ$ code

The $X$-type stabilizer code of the $[[48,3,3]]$ transversal $CCZ$ code is the triply-even code FG48 [12; Prop. 2.3]. Its codewords also have signed weights divisible by eight with respect to the 22 qubits carrying $T^\dagger$ [12; Prop. 3.9].

CSS-T code

CSS-T codes are weakly eight-divisible CSS codes that satisfy the stronger coset-divisibility condition at $\Delta=8$. A strongly transversal $T$ gate preserves the code space if and only if $\operatorname{wt}(x)-2\operatorname{wt}(x*z)\equiv0\pmod 8$ for every $x\in C_2$ and every $z\in C_1+y$ [2; Thm. I.1]. Here $*$ is the entrywise product and $y$ is the character vector. The identity $\operatorname{wt}(x+z)-\operatorname{wt}(z)=\operatorname{wt}(x)-2\operatorname{wt}(x*z)$ makes this coset divisibility at $\Delta=8$.

References

[1]
S. Bravyi and A. Cross, “Doubled Color Codes”, (2015) arXiv:1509.03239
[2]
S. P. Jain and V. V. Albert, “Transversal Clifford and T -Gate Codes of Short Length and High Distance”, IEEE Journal on Selected Areas in Information Theory 6, 127 (2025) arXiv:2408.12752 DOI
[3]
J. Hu, Q. Liang, and R. Calderbank, “Designing the Quantum Channels Induced by Diagonal Gates”, Quantum 6, 802 (2022) arXiv:2109.13481 DOI
[4]
J. Hu, Q. Liang, and R. Calderbank, “Divisible Codes for Quantum Computation”, (2022) arXiv:2204.13176
[5]
E. Camps-Moreno, H. H. López, G. L. Matthews, N. Rengaswamy, and R. San-José, “Transversal gates for quantum CSS codes”, (2026) arXiv:2601.21514
[6]
M. A. Webster, A. O. Quintavalle, and S. D. Bartlett, “Transversal diagonal logical operators for stabiliser codes”, New Journal of Physics 25, 103018 (2023) arXiv:2303.15615 DOI
[7]
A. Bauer, “Finding diagonal logical gates in CSS codes and circuits”, (2026) arXiv:2607.26477
[8]
K. S. M. Reddy and N. Kashyap, “Asymptotically good CSS codes that realize the logical transversal Clifford group fault-tolerantly”, (2026) arXiv:2601.08568
[9]
C. Vuillot and N. P. Breuckmann, “Quantum Pin Codes”, IEEE Transactions on Information Theory 68, 5955 (2022) arXiv:1906.11394 DOI
[10]
P. A. Mishchenko and K. Xagawa, “Secure multiparty quantum computation protocol for quantum circuits: The exploitation of triply even quantum error-correcting codes”, Physical Review A 110, (2024) arXiv:2206.04871 DOI
[11]
J. Haah, “Towers of generalized divisible quantum codes”, Physical Review A 97, (2018) arXiv:1709.08658 DOI
[12]
B. Lu and K. R. Brown, “Building codes with transversal CCZ using projective geometry and SAT solvers”, (2026) arXiv:2610.09341

Page edit log

Victor V. Albert (2026-10-26) — most recent
Victor V. Albert (2026-09-06)
Victor V. Albert (2026-09-01)

Cite as:

“Weakly divisible CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/weakly_divisible_css, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/magic/weakly_divisible_css/weakly_divisible_css.yml.