Existence of a ((7,3,3)) qubit code

Problem

Does a qubit quantum error-correcting code with parameters \(((7,3,3))_2\) exist, that is, a three-dimensional subspace of seven qubits with minimum distance at least three? Let \(\mathcal{P}_7:=\{I,X,Y,Z\}^{\otimes7}\), where \(I\) is the single-qubit identity and \(X,Y,Z\) are the Pauli matrices, and for \(E\in\mathcal{P}_7\) let \(\operatorname{wt}(E)\) be the number of non-identity tensor factors. A \(((7,3,3))_2\) code is the range of an operator \(P\) on \((\mathbb{C}^{2})^{\otimes7}\) with

\begin{equation} P=P^{\dagger}=P^{2}, \qquad \operatorname{Tr}P=3, \tag{1} \end{equation}

satisfying the Knill–Laflamme conditions

\begin{equation} PEP=\frac{\operatorname{Tr}(PE)}{3}\,P \qquad\text{for every }E\in\mathcal{P}_7\text{ with } 1\leq\operatorname{wt}(E)\leq2. \tag{2} \end{equation}

Equation (2) says that the code detects every Pauli error of weight at most two and hence, by linearity, exactly corrects an arbitrary error on any single qubit. No stabilizer structure, entanglement assistance, or other side resource is assumed; the codewords may be arbitrary vectors of \((\mathbb{C}^{2})^{\otimes7}\). Writing \(K_{\max}(7,3)\) for the largest dimension of a seven-qubit code of minimum distance at least three, the question is whether

\begin{equation} K_{\max}(7,3)\geq3. \tag{3} \end{equation}

Such a code would encode one logical qutrit. A complete answer is either an explicit rank-three projector satisfying Eqs. (1) and (2), or a proof that none exists.

Source

Cao, Zhang, Wu, Grassl, and Zeng state explicitly that the existence of a \(((7,3,3))_2\) code is an open question, Section 5.1 of [CZWGZ22]. Chuang, Cross, Smith, Smolin, and Zeng had earlier noted that the linear programming bound does not exclude these parameters [CCSSZ09].

Progress

The Steane code \([[7,1,3]]_2\) is a two-dimensional seven-qubit code of distance three [Ste96], so

\begin{equation} K_{\max}(7,3)\geq2. \tag{4} \end{equation}

A \(((7,3,3))_2\) code would strictly improve this dimension.
Chuang, Cross, Smith, Smolin, and Zeng prove that every \(((n,3,d))_2\) codeword-stabilized code, a subspace spanned by Pauli translates of a single stabilizer state, is a subcode of some \(((n,4,d))_2\) stabilizer code, Theorem 7 of [CCSSZ09]. Because the additive \(((7,2,3))_2\) code is optimal among stabilizer codes, they conclude that no \(((7,3,3))_2\) codeword-stabilized code exists even though the linear programming bound allows these parameters, Corollaries 2 and 3 there. This excludes only a structured class of codes.
Cao, Zhang, Wu, Grassl, and Zeng search for a \(((7,3,3))_2\) code with their variational algorithm VarQEC, using an overparameterized encoding circuit with \(L\) layers and \(N_{\mathrm{start}}\) optimization starting points, where

\begin{equation} L=31, \qquad N_{\mathrm{start}}=20\,000, \tag{5} \end{equation}

and find no such code; they take this as strong numerical evidence of nonexistence, Section 5.1 of [CZWGZ22]. A failed variational search does not certify that every rank-three projector violates Eq. (2).
Anglès Munné and Huber derive semidefinite-programming bounds on the largest dimension of an \(n\)-qubit code of given distance with exact rational infeasibility certificates. Their Table 1 nevertheless retains the gap

\begin{equation} 2\leq K_{\max}(7,3)\leq3, \tag{6} \end{equation}

so the rigorous upper bound still permits dimension three [AMH26].

Comment

The unresolved alternative is exactly \(K_{\max}(7,3)=2\) versus \(K_{\max}(7,3)=3\). The codeword-stabilized obstruction excludes only a structured class, and the numerical search provides evidence rather than a certificate of nonexistence; a rigorous exclusion would need, for example, an infeasibility certificate for Eqs. (1) and (2) over all rank-three projectors. Either an explicit construction or such a certificate would close this finite-length coding bound. The catalog’s record on the asymptotic Rains distance bound concerns how the distance of qubit codes scales with the block length; the present question is a single finite-length instance and is not implied by the asymptotic one.

References

[Ste96]: A. M. Steane, “Multiple-particle interference and quantum error correction,” Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 452, 2551–2577 (1996).DOI arXiv
[CCSSZ09]: I. L. Chuang, A. W. Cross, G. Smith, J. A. Smolin, and B. Zeng, “Codeword stabilized quantum codes: Algorithm and structure,” Journal of Mathematical Physics 50, 042109 (2009).DOI arXiv
[CZWGZ22]: C. Cao, C. Zhang, Z. Wu, M. Grassl, and B. Zeng, “Quantum variational learning for quantum error-correcting codes,” Quantum 6, 828 (2022).DOI arXiv
[AMH26]: G. Anglès Munné and F. Huber, “SDP bounds on quantum codes: rational certificates,” arXiv preprint (2026). exact certificates.arXiv link

Existence of a ((7,3,3)) qubit code

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