Weight-four generators for an [[11,3,3]] stabilizer code
- Field
- Topic
Problem
Does there exist a qubit stabilizer code with parameters \([[11,3,d]]\) and \(d\geq3\) whose stabilizer group is generated by eight independent commuting Pauli operators of weight at most four?
Write an \(11\)-qubit Pauli operator, up to a phase, as \(X^{a}Z^{b}=\bigotimes_{j=1}^{11}X^{a_j}Z^{b_j}\) with \(v=(a,b)\in\mathbb F_2^{22}\). Its weight is \(\operatorname{wt}(v)=|\{j:(a_j,b_j)\neq(0,0)\}|\), and two Pauli operators commute exactly when the symplectic form
vanishes. Commuting Hermitian Pauli operators whose eight binary vectors are linearly independent generate a group not containing \(-I\), and this group defines a code with \(k=11-8=3\) logical qubits. Its distance is the least weight of a Pauli operator that commutes with all generators but is not a scalar multiple of an element of the stabilizer group; it does not depend on the signs of the generators. With the form in Eq. (1), the question therefore asks whether there are \(g_1,\dots,g_8\in\mathbb F_2^{22}\) such that
The second line of Eq. (2) is the condition \(d\geq3\). Degenerate codes are allowed, and no CSS structure is required.
Source
Derived from Table II of Wei, Han, He, Li, and Liu (Section VII of arXiv version 2, 13 September 2026). The table leaves the optimal maximum generator weight of \([[11,3,3]]\) stabilizer codes between four and five, and Section VIII names the unmatched table entries as targets for stronger lower bounds and improved constructions [WHH+26]. The statement asks whether the value four is attained.
Progress
Wei et al. bracket the optimal generator weight \(W_{\mathrm{opt}}(11,3,3)\) between four and five (Table II). Weight five is realized by their \([[9,3,3]]\) code with generators of weight five, extended by two qubits carrying single-qubit stabilizers; Appendix J lists this \([[11,3,3]]\) code as not known to be optimal. Weight four is the general lower bound for \(d\geq3\). Their analytic bound \(W_{\mathrm{opt}}(n,k,d)\geq\max\{4,\lceil(11n+k+1)/(4(n-k))\rceil\}\) for \(d\geq3\) gives only \(\lceil125/32\rceil=4\) here (Theorem 4). Their weight-constrained linear program (Section V, with Algorithm 1 in Appendix L) supplies only necessary conditions for existence, and it also leaves four. The same table records that no \([[11,3,4]]\) or \([[11,3,5]]\) stabilizer code exists [WHH+26]; larger distances violate the quantum Singleton bound \(n-k\geq2(d-1)\), so \(d\geq3\) is equivalent to \(d=3\).
The CSS case is excluded. Every CSS \([[n,k,d]]\) code with \(k\geq1\) and \(d\geq3\) satisfies
\begin{equation} W(\mathcal S)\geq\left\lceil\frac{3n+k}{n-k}\right\rceil, \qquad \left\lceil\frac{3\cdot11+3}{11-3}\right\rceil=\left\lceil\frac{36}{8}\right\rceil=5, \tag{3} \end{equation}where \(W(\mathcal S)\) is the least maximum weight over generating sets of the stabilizer group \(\mathcal S\) (Theorem 4, proved as Theorem 25 in Appendix F). By Lemma 24, the bound in Eq. (3) applies to all generating sets of a CSS stabilizer group, including mixed ones [WHH+26]. Since single-qubit Clifford unitaries preserve weight, commutation, and distance, a positive answer requires a stabilizer group that is not equivalent to a CSS group under such unitaries.
Nearby lengths delimit the case \(n=11\). For \((k,d)=(3,3)\), Wei et al. obtain \(W_{\mathrm{opt}}(10,3,3)=5\) (Table II) and \(W_{\mathrm{opt}}(n,3,3)=4\) for all \(n\geq15\) (Eq. (4)); three copies of the \([[5,1,3]]\) code give weight four at \(n=15\). Since \(W_{\mathrm{opt}}(n,k,d)\) is nonincreasing in \(n\) (Theorem 2), the least length admitting an \([[n,3,3]]\) stabilizer code with generators of weight at most four lies between \(11\) and \(15\). Table II also leaves \(W_{\mathrm{opt}}(12,3,3)\) between four and five [WHH+26].
Comment
Weight five is achieved, so the question is exactly whether generators satisfying Eq. (2) exist; either an explicit generating set or a proof that none exists would settle it. Equation (2) involves only finitely many candidate vectors, so the question is decidable by exhaustive search. Neither the linear-programming bounds nor the constructions of Wei et al. decide it. A negative answer would leave the corresponding questions for \(12\leq n\leq14\) open. This finite question arises from the same theory of weight-constrained codes as the square-root distance conjecture for weight-four stabilizer codes. Literature checked through 15 September 2026.