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Good quantum LDPC codes with a non-Clifford transversal CCZ gate

Unsolved ID op_15c6d5adf1796d86 Last edited 15 September 2026
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Do there exist triples of qubit CSS LDPC codes with linear combined rate and linear distance on which the transversal CCZ gate induces a non-Clifford logical operation? Precisely, do there exist constants \(r,\delta>0\) and \(w\in\mathbb N\), and triples, with \(n\to\infty\), of qubit CSS codes \(\mathcal Q_{a,n}\) with parameters \([[n,k_{a,n},d_{a,n}]]\), indexed by \(a\in\{1,2,3\}\), such that each code has \(k_{a,n}\ge1\) and \(X\)- and \(Z\)-check matrices whose row weights (check weights) and column weights (qubit degrees) are at most \(w\),

\begin{equation} \sum_{a=1}^{3}k_{a,n}\ge rn,\qquad \min_{a\in\{1,2,3\}}d_{a,n}\ge\delta n, \tag{1} \end{equation}

and the transversal gate

\begin{equation} U_n:=\prod_{j=1}^{n}\operatorname{CCZ}_{(1,j),(2,j),(3,j)}, \qquad \operatorname{CCZ}|x,y,z\rangle=(-1)^{xyz}|x,y,z\rangle \quad\text{for }x,y,z\in\{0,1\}, \tag{2} \end{equation}

preserves \(\mathcal Q_{1,n}\otimes\mathcal Q_{2,n}\otimes\mathcal Q_{3,n}\) and induces a logical unitary that does not normalize its encoded Pauli group? Here \((a,j)\) denotes physical qubit \(j\) of block \(a\), so Eq. (2) applies one CCZ gate to each aligned triple of physical qubits.

Source

Golowich and Guruswami pose the construction of LDPC codes with asymptotically optimal dimension and distance and transversal non-Clifford gates as an open problem [GG25a], Section 1.1. The formulation here specializes it to qubit CSS code triples with linear combined rate, linear distance, bounded check weights and qubit degrees, and a non-Clifford logical action of the full transversal CCZ product in Eq. (2).

Progress

  • Li, Li, and Liu construct quantum LDPC codes supporting nontrivial transversal logical CCZ gates with parameters

    \begin{equation} [[N,\Theta(N),\Theta(N/(\log N)^2)]] \tag{3} \end{equation}

    [LLL26], Theorem 1.1 with \(r=3\) and Corollary 5.7. A constant-depth logical CCZ circuit is made transversal at a constant-factor loss in parameters (Lemma 5.6). At least one of the three blocks is only known to have a constant number of logical qubits, so linear rate is established for the combined blocks rather than for each block (Section 5.3 and Section 6); the combined-rate condition in Eq. (1) accommodates this. The distance in Eq. (3) is a factor \((\log N)^2\) short of linear. The revision of 2 July 2026 replaces the argument of the first version, whose cap product did not satisfy the Leibniz rule required for the gate argument, by one based on cup products and two-way product-expanding punctured Reed–Solomon local codes; the results cited here are those of the revised version.

  • Golowich and Guruswami construct, for every fixed prime power \(q\) including \(q=2\), CSS codes with dimension and distance linear in the block length that support a transversal CCZ gate [GG25a], Theorem 1.1. Their stabilizers have high weight, so these codes are not LDPC (Section 1.1).

  • An intermediate locality regime is reached by subsystem codes supporting transversal CCZ with dimension and distance \(\Theta(N)\) and locality \(O(\sqrt N)\) over alphabets of size \(O(\sqrt N)\); reducing to qubits gives \([[N,\tilde\Theta(N),\tilde\Theta(N)]]_2\) codes of locality \(\tilde O(\sqrt N)\), where the tilde hides polylogarithmic factors [GG25b], Theorem 1.1. Neither the locality nor, over qubits, the distance meets Eq. (1).

  • Golowich, Tamo, and Zhu construct, for every \(0<\varepsilon<1\) and every prime power alphabet size including qubits, subsystem codes with constant-weight stabilizers whose depth-one transversal CCZ circuit induces logical CCZ gates on at least \(N^{1-\varepsilon}\) disjoint triples of logical qubits, with distance at least \(N^{(1-\varepsilon)/3}\) [GTZ26], Theorem 1.1 with \(r=3\) (informal statement of Corollary 5.1). Appendix A extends this to addressable gates. These results increase the number and addressability of logical CCZ gates at constant stabilizer weight, but the distance is polynomially sublinear.

Comment

Literature checked through 15 September 2026: no family meeting Eq. (1) with bounded check weights and qubit degrees and a non-Clifford logical action of Eq. (2) was found. Distance \(\Theta(N/(\log N)^2)\) as in Eq. (3) is not a resolution.

The combined code has \(N=3n\) physical qubits. Only the total number of logical qubits must be linear, which makes the rate accounting explicit without requiring linearly many logical qubits in every block; each block must still encode at least one qubit and have linear distance. The question asks only for a non-Clifford logical action, not for linearly many independent or addressable logical CCZ gates. A physical gate that preserves the code space but acts trivially or as a logical Clifford operation does not qualify. The statement fixes the full product \(U_n\); some cited constructions use more general transversal circuits, such as constant-depth circuits or gates on selected physical triples, which would need an additional argument to meet Eq. (2).

The locally testable analogue for good quantum LDPC codes is asymptotically good quantum locally testable codes; the construction of [LLL26] also gives almost-good quantum locally testable codes with transversal non-Clifford gates.

References

[LLL26]
Y. Li, Z. Li, and Z.-W. Liu, “Transversal Non-Clifford Gates on Almost-Good Quantum LDPC and Quantum Locally Testable Codes,” arXiv:2604.01874 (2026).DOIarXiv
[GG25a]
L. Golowich and V. Guruswami, “Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates,” in Proceedings of the 57th Annual ACM Symposium on Theory of Computing (STOC 2025), 707–717 (2025).DOIarXiv
[GG25b]
L. Golowich and V. Guruswami, “Near-Asymptotically-Good Quantum Codes with Transversal CCZ Gates and Sublinear-Weight Parity-Checks,” in 2025 IEEE 66th Annual Symposium on Foundations of Computer Science (FOCS), 1561–1569 (2025).DOIarXiv
[GTZ26]
L. Golowich, I. Tamo, and G. Zhu, “Improved Transversal Non-Clifford Gates from Cup Products,” Electronic Colloquium on Computational Complexity (2026). ECCC TR26-160.link

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“Good quantum LDPC codes with a non-Clifford transversal CCZ gate,” Quantum Information and Quantum Computation Open Problem Zoo (QIQCOP Zoo), ID op_15c6d5adf1796d86, accessed 2026-09-18.

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@incollection{qiqcop_op_15c6d5adf1796d86,
  title = {Good quantum LDPC codes with a non-Clifford transversal CCZ gate},
  booktitle = {Quantum Information and Quantum Computation Open Problem Zoo (QIQCOP Zoo)},
  year = {2026},
  howpublished = {\url{https://qiqc-op.com/problem/op_15c6d5adf1796d86/}},
  note = {Stable ID op_15c6d5adf1796d86; status: Unsolved; accessed 2026-09-18}
}

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“Good quantum LDPC codes with a non-Clifford transversal CCZ gate,” Quantum Information and Quantum Computation Open Problem Zoo (QIQCOP Zoo), https://qiqc-op.com/problem/op_15c6d5adf1796d86/, ID op_15c6d5adf1796d86, accessed 2026-09-18.

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op_15c6d5adf1796d86
01M2JD0QDEDZ78QV24WQ3VS9BN