Optimal dimension of nine-qubit distance-three codes
- Field
- Topic
Problem
Is the known non-additive \(((9,12,3))_2\) code of largest possible dimension, or does a nine-qubit code of dimension thirteen and distance three exist? Let \(\mathcal P_9:=\{I,X,Y,Z\}^{\otimes9}\) be the Pauli basis, where the weight \(\operatorname{wt}(E)\) counts nonidentity tensor factors, and define
where each \(c_E\) is a scalar depending only on \(E\). The error condition is imposed for the full tensor-product Pauli basis, without an additivity, codeword-stabilized, or purity assumption. Since a \(((9,12,3))_2\) code is known and no code of dimension fourteen or more exists, \(K_{\max}^{(2)}(9,3)\in\{12,13\}\). The question is whether some projector with \(\operatorname{Tr}P=13\) satisfies the constraints in Eq. (1), which would give a \(((9,13,3))_2\) code correcting an arbitrary single-qubit error.
Source
Derived from the gap between the lower and upper bounds for nine qubits and distance three recorded in Table III of [ROJ19] and Table 1 of [AMH26], whose lower bound is the non-additive code of [YCLO08]. These sources record the gap but do not pose the existence of a \(((9,13,3))_2\) code as a named question or conjecture.
Progress
Yu, Chen, Lai, and Oh constructed a twelve-dimensional nine-qubit code, spanned by twelve \(Z\)-type translates of the nine-vertex cycle graph state, that corrects an arbitrary single-qubit error. The best nine-qubit distance-three stabilizer code, \([[9,3,3]]\), has dimension eight, so non-additivity already increases the dimension at these parameters; the question is whether this advantage is the largest possible [YCLO08].
The linear-programming bound gives dimension at most \(13\) (Table III of [ROJ19]). The table of Anglès Munné and Huber, whose upper bounds all carry exact rational semidefinite-programming certificates, still lists \(12\)–\(13\) for block length \(9\) and distance \(3\) (Table 1, called Table 4.1 in the text of Section 4.2). Ruling out dimension thirteen would therefore determine \(K_{\max}^{(2)}(9,3)\) completely [AMH26].
Rigby, Olivier, and Jarvis report an exhaustive search over nine-vertex graphs up to local complementation and isomorphism: eight classes yield twelve-dimensional distance-three codeword-stabilized (CWS) codes, and none yields a thirteen-dimensional one (Section III-B). A \(((9,13,3))_2\) code would therefore lie outside the CWS framework, and since local unitaries preserve the code parameters, it could not be local-unitary equivalent to a CWS code either [ROJ19].
The neighboring eight-qubit case is settled. Anglès Munné and Huber give an exact rational certificate that no \(((8,9,3))_2\) code exists, CWS or not, so the largest dimension for eight qubits and distance three is \(8\) (Section 4.2 and Table 1). This does not by itself settle the nine-qubit question: transferring it would require a puncturing or shortening argument that preserves the required dimension and distance [AMH26].
Comment
Literature checked through 15 September 2026: no construction and no general exclusion of a \(((9,13,3))_2\) code was located, so the alternative \(K_{\max}^{(2)}(9,3)\in\{12,13\}\) remains open. This is an optimality problem for an existing non-additive code, not a named conjecture. Both outcomes are informative: a thirteen-dimensional code would improve a longstanding finite-length benchmark and necessarily lie outside the CWS framework, while a nonexistence proof would certify that the code of [YCLO08] has optimal dimension. The exhaustive CWS search implies neither outcome, since it is conclusive only within the class it enumerates. Likewise, a feasible weight-enumerator vector or a feasible finite-level semidefinite relaxation is only a necessary condition, not a code. The analogous question for seven qubits and distance two is the optimal dimension of seven-qubit distance-two codes.