Generalized semi-Clifford conjecture

For every number of qubits $n$ , every level $k\geq 1$ of the Clifford hierarchy, and every $n$ -qubit unitary $U\in\mathcal{C}_k$ , there exist Clifford unitaries $C_1,C_2$ , a computational-basis permutation unitary $\Pi$ , and a diagonal unitary $D$ such that $U=C_1\Pi D C_2$ . Here $\mathcal{C}_1$ is the Clifford group and, recursively, $\mathcal{C}_{k}=\{U:UPU^\dagger\in\mathcal{C}_{k-1}\text{ for every Pauli operator }P\}$ ; equivalently, every gate in the Clifford hierarchy is generalised semi-Clifford.

References

Primary source

arXiv

Additional references

The generalised semi-Clifford conjecture is false — arXiv — Nadish de Silva, Oscar Lautsch

quantum-computation quantum-information

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A September 2026 preprint claims a counterexample that disproves the conjecture at higher levels, but the result has not been independently confirmed.

The conjecture, attributed to Zeng and collaborators, asserts that every gate in each level of the Clifford hierarchy has a generalized semi-Clifford structure. Earlier work settled the third-level case, while the higher-level question remained open.

Known results

Gottesman and Mochon disproved the ordinary semi-Clifford conjecture at the third level.
Beigi and Shor proved the generalized conjecture at the third level.
de Silva and Lautsch proved a stronger structural third-level statement.
Chen and de Silva (2023; revised 2024) proved third-level semi-Cliffordness for up to two qudits of prime dimension.

September 2026 claimed counterexample

A preprint by Nadish de Silva and Oscar Lautsch claims that the generalized conjecture is false beyond the third level. The example is also claimed to show that the Clifford hierarchy is not closed under inverses. The claim is unrefereed and independently unconfirmed.

Current status (as of September 2026): The third-level case is settled, while a higher-level counterexample and the claimed failure of inverse closure remain unverified.

Sources

No solutions have been posted yet.