Seven qubits protecting a qutrit: does a ((7,3,3)) code exist?

Steane's famous code uses seven qubits to protect one qubit against an arbitrary error on any single qubit. Can the same seven qubits protect a three-level system, a qutrit, instead? This is the smallest open existence question for qubit codes of distance three, and either answer can be certified by a finite computation.

Problem

Let $P$ be the orthogonal projector onto a subspace $\mathcal{C}\subseteq(\mathbb{C}^2)^{\otimes 7}$ with $\dim\mathcal{C}=3$. The code $\mathcal{C}$ has minimum distance at least $3$ if it satisfies the Knill–Laflamme conditions

$$PEP=c(E)\,P\qquad\text{for every Pauli operator } E \text{ of weight } 1 \text{ or } 2,$$

where $c(E)$ is a complex number and the weight of $E$ is the number of qubits on which it acts nontrivially. Such a code corrects an arbitrary error on any one qubit.

Question. Does a $((7,3,3))_2$ code exist? Equivalently, writing $K_{\max}(7,3)$ for the largest dimension of a seven-qubit code of distance three, is $K_{\max}(7,3)=3$ rather than $2$?

No stabilizer structure is assumed: the code may be any subspace.

Background

Stabilizer codes always have a dimension that is a power of two, so a $((7,3,3))_2$ code would have to be non-additive. In 2009 Chuang, Cross, Smith, Smolin and Zeng showed that no such code exists within the large codeword-stabilized family, even though the linear programming bound does not rule these parameters out [2]. In 2022 Cao, Zhang, Wu, Grassl and Zeng described the existence of a $((7,3,3))_2$ code as an open question [3]. The problem sits at the edge of what bounds on general quantum codes can decide, which makes it a test case for new bounding techniques.

What is known

• Steane's $[[7,1,3]]$ code gives $K_{\max}(7,3)\ge2$ [1].
• There is no codeword-stabilized $((7,3,3))_2$ code (Corollary 3 of [2]). Codeword-stabilized codes do not cover everything: the same paper notes $((7,2,3))$ and $((9,2,3))$ codes that lie outside the construction [2].
• A variational search with an overparameterized encoding circuit of 31 layers and 20,000 optimization starting points found no $((7,3,3))_2$ code, which the authors take as strong evidence that none exists [3].
• Semidefinite-programming bounds with exact rational infeasibility certificates, which improve 18 upper bounds for $6\le n\le 19$, still leave $2\le K_{\max}(7,3)\le3$ [4].

What would count as progress

• An explicit $((7,3,3))_2$ code: three orthonormal vectors in $\mathbb{C}^{128}$ whose span satisfies the 210 conditions above (21 Pauli errors of weight one and 189 of weight two). This can be verified in milliseconds, and exactly if the entries are algebraic.
• A proof that none exists, for example a rational semidefinite infeasibility certificate of the kind used in [4], checkable in exact arithmetic.
• Partial exclusions beyond the codeword-stabilized class, such as ruling out codes with a given symmetry group or with a nontrivial stabilizer subgroup.

References

1. A. M. Steane, "Multiple-particle interference and quantum error correction", Proceedings of the Royal Society A 452, 2551–2577 (1996). doi:10.1098/rspa.1996.0136
2. 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:10.1063/1.3086833
3. C. Cao, C. Zhang, Z. Wu, M. Grassl and B. Zeng, "Quantum variational learning for quantum error-correcting codes", Quantum 6, 828 (2022). doi:10.22331/q-2022-10-06-828
4. G. Anglès Munné and F. Huber, "SDP bounds on quantum codes: rational certificates", preprint (2026). arXiv:2603.19901

Source

Imported from QIQCOP Zoo op_4765bc6f2ac91d46, curated by Bikun Li, Qicheng Tang, Changhao Li, Chengkai Zhu, Minbo Gao, Zhong-Xia Shang, Bin Cheng and Naixu Guo. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.