Square-root distance bound for weight-four stabilizer codes
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Problem
Is there a universal constant \(C>0\) such that every qubit stabilizer code that encodes at least one logical qubit, and whose stabilizer group is generated by Pauli operators of weight at most four, has distance at most \(C\sqrt n\) on \(n\) physical qubits?
A stabilizer group \(\mathcal S\) on \(n\) qubits is an abelian subgroup of the \(n\)-qubit Pauli group with \(-I\notin\mathcal S\); if \(\mathcal S\) has \(n-k\) independent generators, its common \(+1\) eigenspace encodes \(k\) logical qubits. The weight \(\operatorname{wt}(P)\) of a Pauli operator \(P\) is the number of qubits on which \(P\) acts nontrivially. For \(k\geq1\), the distance \(d\) is the least weight of a Pauli operator that commutes with every element of \(\mathcal S\) but is not a scalar multiple of an element of \(\mathcal S\). The optimal generator weight of \(\mathcal S\) is
With \(W(\mathcal S)\) as in Eq. (1), the question asks whether
Equation (2) assumes no CSS structure, no bound on the number of generators acting on a qubit, and no geometric locality. A negative answer requires stabilizer codes with \(k\geq1\) and \(W(\mathcal S)\leq4\) whose ratios \(d/\sqrt n\) are unbounded.
Source
Wei, Han, He, Li, and Liu conjecture that every stabilizer code with \(W(\mathcal S)=4\) satisfies \(d=O(\sqrt n)\). They work with general, not necessarily CSS, stabilizer codes in the regime \(k\geq1\); the conjecture is in Section IV of arXiv version 2 (13 September 2026) and already appears in version 1 (27 January 2026) [WHH+26]. Equation (2) states it with a universal constant. Allowing \(W(\mathcal S)\leq4\) rather than \(W(\mathcal S)=4\) does not change the question, because generator weight at most three forces \(d\leq2\) when \(k\geq1\) [WHH+26], [WLL+26].
Progress
Weight three is settled. Wang, Liu, Li, Kubica, and Gu prove that a stabilizer code whose checks have weight at most three has \(d\leq2\) or \(k=0\), with no assumption on qubit degree (Theorem III.7) [WLL+26]. Wei et al. independently prove that \(k\geq1\), \(d\geq2\), and \(W(\mathcal S)=3\) imply \(d=2\) and \(k/n\leq1/4\) (Theorem 5), and that \(W(\mathcal S)\leq2\) forces \(d=1\) [WHH+26]. Four is therefore the smallest generator weight compatible with growing distance.
The exponent in Eq. (2) would be optimal. The rotated surface code on an \(L\times L\) lattice has parameters \([[L^2,1,L]]\) and \(W(\mathcal S)=4\) (proof of Theorem 2), and the toric code and other weight-four hypergraph-product codes have \(d=\Theta(\sqrt n)\). As evidence for the conjecture, Wei et al. observe that all established constructions exceeding the square-root barrier use checks of weight larger than four (Section IV) [WHH+26].
Wang et al. prove a stronger bound for a structured CSS class. Consider a CSS code with \(k\geq1\), given by a possibly linearly dependent set of \(X\)- and \(Z\)-type checks of weight at most four, in which every qubit lies in exactly two \(X\) checks and exactly two \(Z\) checks. Then
\begin{equation} kd^2\leq An \tag{3} \end{equation}for a universal constant \(A\) (Theorem III.9). After disentangled pairs of qubits are removed, such a code is a surface code on a closed cellulated surface with vertex and face degrees at most four, and Eq. (3) follows from Fetaya’s systolic bound. The authors expect an extension to qubits lying in at most two checks of each type, which would require an analogous bound for surfaces with boundary. Non-CSS codes and other incidence patterns are not covered [WLL+26].
Asymptotically good codes are available at higher weight. Yuan, Baspin, and Williamson convert every CSS \([[n,k,d]]\) code with maximum check weight \(w\) and qubit degree \(q\) into a CSS \([[O(w^4q^4n),k,\Omega(wq^2d)]]\) code with maximum check weight six and total qubit degree six (Theorem I.1); applied to a good CSS quantum LDPC family, this gives good codes of check weight six [YBW26]. Hsieh, Li, and Lin state a weight reduction from codes of weight at most \(w\) to \([[O(nw^2\log w),k,\Omega(dw)]]\) codes with check weight at most five and qubit weight at most six (Theorem 1.1), which would give good codes of check weight five [HLL25]. Yuan, Baspin, and Williamson write that parts of that proof require further clarification. They also show that an error in Lemma 8 of Hastings’s earlier weight-reduction procedure weakens its guaranteed weights \((w_X,w_Z,q_X,q_Z)\) from the claimed \((5,5,3,5)\) to at most \((42,36,4,3)\), calling both issues minor for asymptotic scaling (Introduction and Appendix B) [YBW26]. Wei et al. describe weight five as subtle (Section IV) [WHH+26]. None of these results concerns weight four.
Comment
The remaining gap is either a proof of Eq. (2) for every stabilizer group with \(W(\mathcal S)\leq4\) and \(k\geq1\), or a family of such codes with \(d/\sqrt n\) unbounded. A proof must cover non-CSS codes, codes with unbounded qubit degree, and CSS codes outside the incidence pattern of Eq. (3); that theorem is a special case, not the general statement. Restricting to CSS codes or to bounded-degree quantum LDPC families gives weaker versions of the question. The eleven-qubit weight-four stabilizer code question arises from the same theory of weight-constrained codes. Literature checked through 15 September 2026.