Qutrit magic-distillation thresholds beyond the ternary Golay code

Problem

Does any finite qutrit stabilizer-code protocol distill depolarized Strange states above the ternary Golay code’s threshold?

Let

\begin{equation} |S\rangle=\frac{|1\rangle-|2\rangle}{\sqrt2}, \qquad \rho_S(\varepsilon)=(1-\varepsilon)|S\rangle\langle S|+\varepsilon\frac{I_3}{3},\qquad 0\leq\varepsilon\leq1. \tag{1} \end{equation}

For a finite \(n\)-to-\(1\) qutrit stabilizer-code protocol \(\mathcal P\), let \(F_{\mathcal P}(\varepsilon)\) be the output mixing parameter after accepted syndrome projection, Clifford decoding and correction, and twirling back to the family in Eq. (1). Define

\begin{equation} \varepsilon_*(\mathcal P)=\sup\left\{a\in[0,1]: \lim_{r\to\infty}F_{\mathcal P}^{\circ r}(\varepsilon)=0 \text{ for every }0<\varepsilon<a\right\}, \tag{2} \end{equation}

requiring nonzero acceptance probability at every finite iteration. If \(\varepsilon_G\approx0.38715\) is the threshold of the \([[11,1,5]]_3\) ternary Golay protocol, does some finite protocol satisfy \(\varepsilon_*(\mathcal P)>\varepsilon_G\) as defined in Eq. (2)?

Source

The question is explicitly posed or retained as open in the cited primary literature [Prakash20]. The statement is rewritten here to make its hypotheses and success criterion self-contained.

Progress

This question directly concerns whether the quantum ternary Golay code is a fundamental threshold benchmark or only the best construction currently known. It is an explicit literature question, not an assertion that Golay optimality has been conjectured or proved.
Original formulation and benchmark. Prakash’s 2020 paper introduces ternary Golay distillation of the Strange state with cubic suppression of small depolarizing noise and threshold approximately \(0.38715\). Section 7 explicitly asks: “Do other codes exist with better thresholds?” The paper uses more than one noise convention; the definition above fixes the convention in which the benchmark is \(0.38715\) [Prakash20], Sections 3, 5, and 7. The comparison in the question is with the exact threshold \(\epsilon_G\), not with its rounded decimal representation.
Substantial search progress, but no improvement of the threshold. Prakash and Singhal found \(646\) inequivalent indecomposable \(23\)-qutrit CSS Strange-state distillers with cubic suppression, none exceeding Golay. Their exhaustive searches impose full transversal Clifford symmetry for \(n\le23\), or transversal \(H^2\) for \(n\le9\), where \(H\) is the qutrit Fourier transform. This does not exclude unrestricted stabilizer codes. Their discussion explicitly retains the better-threshold question [Prakash26], Sections IV–V.
A subsequent family-specific investigation. Zurel, Jana, and de Silva studied quantum quadratic-residue codes in March 2026 and evaluated Strange-state distillation for the qutrit members through blocklength \(41\). Their Table 3 reports [Zurel26]:
Number of physical qutrits: 11, the ternary Golay case; Strange-state depolarizing threshold: \(0.38715\). Number of physical qutrits: 17; Strange-state depolarizing threshold: \(0.34394\). Number of physical qutrits: 23; Strange-state depolarizing threshold: \(0.16636\). Number of physical qutrits: 41; Strange-state depolarizing threshold: \(0.31877\).
Thus this later search also fails to beat Golay. The table above uses quantum blocklengths only: the classical \(\mathbb F_9\) code parameters appearing in the source must not be mistaken for quantum code distances. This negative result concerns the examined quadratic-residue family, not arbitrary stabilizer codes [Zurel26], Section 4.3.2 and Table 3.
Apparent counterexamples. Sharma and Garani’s 2024 \(13\)- and \(29\)-qutrit proposals were followed by an October 15, 2024 erratum [Sharma24]. Independent calculations show that neither construction distills the Strange state [Prakash26], Appendix B. These are not counterexamples. The literature check examined the publisher’s erratum record and the independent calculations; the full erratum text was not accessible.
Important distinctions. The existence of non-Golay Strange-state distillers is therefore settled [Prakash26]. Simply concatenating a known successful protocol does not improve its threshold. If \(F^{\circ r}(\epsilon)\) denotes its iterates, grouping iterations into blocks replaces \(F\) by \(F^{\circ m}\) without creating convergence outside the original basin, assuming continuity at the target fixed point. Therefore, concatenation alone does not answer the present question, even though it changes the effective small-error exponent.
Research significance. A higher threshold would enlarge the interval of noisy Strange states handled by an explicit code-based distillation construction. A negative resolution would require a genuine bound across the entire protocol class, not the failure of a finite search. Either result would clarify what makes the ternary Golay construction exceptional. Threshold optimality is also distinct from optimizing acceptance probability, yield, or physical implementation overhead.

Comment

The surviving question is whether any finite qutrit stabilizer-code protocol can strictly exceed the ternary Golay Strange-state threshold, not whether other distillation codes exist. The 2026 searches provide substantial restricted-family evidence but neither a better construction nor a universal Golay-optimality theorem. This remains unresolved in the publicly accessible literature located through September 15, 2026; an unrestricted optimality claim would go beyond that evidence.

References

[Prakash20]: Shiroman Prakash, “Magic state distillation with the ternary Golay code,” Proceedings of the Royal Society A 476, 20200187 (2020). DOI:10.1098/rspa.2020.0187; See Sections 3 and 5 for conventions and performance, and Section 7 for the explicit open question.DOI arXiv
[Prakash26]: Shiroman Prakash and Rishabh Singhal, “Search for high-threshold qutrit magic-state distillation routines,” Physical Review A 113, 042404 (April 1, 2026). DOI:10.1103/61q5-f754; whose preprint title includes an initial “A.” Original preprint August 1, 2024; version 2 dated February 3, 2026. See Sections IV–V and Appendix B.DOI arXiv
[Zurel26]: Michael Zurel, Santanil Jana, and Nadish de Silva, “High-threshold magic state distillation with quantum quadratic residue codes,” arXiv:2603.18560, March 19, 2026. arXiv record. See Section 4.3.2 and Table 3 for the qutrit Strange-state thresholds.arXiv
[Sharma24]: Abhi Kumar Sharma and Shayan Srinivasa Garani, “Erratum: Near-threshold qudit stabilizer codes with efficient encoding circuits for magic-state distillation [Phys. Rev. A 109, 062426 (2024)],” Physical Review A 110, 049902 (October 15, 2024). DOI:10.1103/PhysRevA.110.049902. Read together with the independent calculations in reference [Prakash26], Appendix B.DOI

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