Semi-Clifford structure of two-qudit hierarchy gates above the third level

Problem

Is every two-qudit gate in every level \(k\geq4\) of the Clifford hierarchy semi-Clifford when the local dimension is an odd prime? Let \(p\) be an odd prime, let \(\omega:=e^{2\pi i/p}\), and define on \(\mathbb{C}^p\) the operators \(X|j\rangle=|j+1\bmod p\rangle\) and \(Z|j\rangle=\omega^j|j\rangle\) for \(j\in\{0,\ldots,p-1\}\). For \(n\) qudits, let \(\mathcal{P}_{n,p}\) be the group generated by arbitrary global phases and the single-site operators \(X_i,Z_i\) (\(1\leq i\leq n\)), each acting as the identity on the other sites, and define

\begin{equation} \mathcal{C}_1(n,p):=\mathcal{P}_{n,p}, \qquad \mathcal{C}_{\ell+1}(n,p):= \bigl\{U\in U(p^n):UPU^\dagger\in\mathcal{C}_\ell(n,p) \ \text{for every}\ P\in\mathcal{P}_{n,p}\bigr\}, \qquad \ell\geq1. \tag{1} \end{equation}

A unitary is semi-Clifford when it equals \(C_LDC_R\) with \(C_L,C_R\in\mathcal{C}_2(n,p)\) and \(D\) diagonal in the computational basis; write \(\mathrm{SC}(n,p)\) for this set. The question asks whether, with the levels of Eq. (1),

\begin{equation} \mathcal{C}_k(2,p)\subseteq\mathrm{SC}(2,p) \qquad\text{for every odd prime }p\text{ and every integer }k\geq4. \tag{2} \end{equation}

A complete answer either proves Eq. (2) or exhibits an odd prime \(p\), a level \(k\geq4\), and a gate in \(\mathcal{C}_k(2,p)\) that is not semi-Clifford.

Source

De Silva conjectured that every \(k\)th-level gate of one or two qudits of any prime dimension is semi-Clifford (Conjecture 3 in Section V.C of the arXiv version), and in the concluding Section VI proposed the two-qudit, all-level statement again by analogy with the two-qubit theorem of Zeng, Chen, and Chuang [dS21]. Chen and de Silva, having proved the third-level case, name its generalization to higher levels as a natural follow-up question in their concluding Section 4 [CdS24]. Eq. (2) is the part of the two-qudit conjecture not settled by the results below: qubits and levels \(k\leq3\) are excluded.

Progress

For qubits, Zeng, Chen, and Chuang proved by induction on the level, using exhaustive computations, that every two-qubit hierarchy gate is semi-Clifford (Theorem 1), and showed that the property fails for three qubits at every level \(k\geq4\) (Theorem 3):

\begin{equation} \mathcal{C}_k(2,2)\subseteq\mathrm{SC}(2,2)\quad(k\geq1), \qquad \mathcal{C}_k(3,2)\not\subseteq\mathrm{SC}(3,2)\quad(k\geq4), \tag{3} \end{equation}

where \(\mathcal{C}_k(n,2)\) and \(\mathrm{SC}(n,2)\) denote the sets defined as in Eq. (1) with \(p=2\). The two-qubit case motivates Eq. (2), but its computational proof does not extend to odd prime dimensions [ZCC08].
De Silva proved that every third-level gate of one qudit of any prime dimension (Theorem 8) and every third-level gate of two qutrits (Theorem 9, by exhaustive computation) is semi-Clifford, and reported numerical evidence that the property extends to further cases. These results do not reach levels \(k\geq4\) for two qudits [dS21].
Chen and de Silva proved that, for every odd prime \(p\), every two-qudit third-level gate is semi-Clifford (Theorem 1):

\begin{equation} \mathcal{C}_3(2,p)\subseteq\mathrm{SC}(2,p) \qquad(p\ \text{an odd prime}). \tag{4} \end{equation}

Their argument describes a simplified two-qudit third-level gate by the Clifford gates to which it conjugates the basic Pauli operators, turns the third-level and semi-Clifford conditions into polynomial systems over \(\mathbb{Z}_p\), and compares the rational points of the resulting algebraic sets uniformly in \(p\). At levels \(k\geq4\) these conjugates are no longer Clifford gates, and Eq. (4) does not establish Eq. (2) [CdS24].
De Silva and Lautsch proved that, for every odd prime \(p\), every one-qudit hierarchy gate is semi-Clifford (Theorem 6), and derived a unique normal form \(MDC\) for non-Clifford one-qudit hierarchy gates, with \(M\) from a fixed finite set of Clifford gates, \(D\) diagonal, and \(C\) Clifford (Theorem 7):

\begin{equation} \mathcal{C}_k(1,p)\subseteq\mathrm{SC}(1,p) \qquad(p\ \text{an odd prime},\ k\geq1). \tag{5} \end{equation}

In Section 5 (Conjecture 1 of the arXiv version) they conjecture that every two-qudit hierarchy gate is either Clifford or uniquely of the form \(M_1M_2DC\) with Clifford gates \(M_1,M_2\) from explicit finite families, \(D\) diagonal, and \(C\) Clifford. Such a gate is semi-Clifford, so that unproved normal form would imply Eq. (2); the one-qudit theorem in Eq. (5) does not [dSL25].

Comment

What remains open is Eq. (2) at every level \(k\geq4\) for every odd prime \(p\). The established results cover two qubits at every level, Eq. (3); two odd-prime-dimensional qudits at level three, Eq. (4); and one qudit at every level, Eq. (5). None covers two odd-prime-dimensional qudits at levels four and above. A counterexample must be a gate in \(\mathcal{C}_k(2,p)\) for the two-qudit Pauli group of Eq. (1); reproducing a multiqubit counterexample inside an encoded subspace does not qualify. For qubits, the weaker generalized semi-Clifford property, which allows an additional computational-basis permutation between the Clifford factors, fails for a five-qubit gate in the fifth level, and the same authors announce forthcoming qudit counterexamples without specifying the number of qudits or the level [dSL26]; a two-qudit gate at some level \(k\geq4\) that is not generalized semi-Clifford would in particular answer the question negatively. Literature checked through 15 September 2026; no proof of Eq. (2) and no two-qudit counterexample was found. The zoo’s record “Smallest qubit number for a non-semi-Clifford third-level gate” concerns the least number of qubits at which the third level stops being semi-Clifford.

References

[ZCC08]: B. Zeng, X. Chen, and I. L. Chuang, “Semi-Clifford operations, structure of \(\mathcal{C}_k\) hierarchy, and gate complexity for fault-tolerant quantum computation,” Physical Review A 77, 042313 (2008).DOI arXiv
[dS21]: N. de Silva, “Efficient quantum gate teleportation in higher dimensions,” Proceedings of the Royal Society A 477, 20200865 (2021).DOI arXiv
[CdS24]: I. Chen and N. de Silva, “Characterising semi-Clifford gates using algebraic sets,” Communications in Mathematical Physics 405, 201 (2024).DOI arXiv
[dSL25]: N. de Silva and O. Lautsch, “The Clifford hierarchy for one qubit or qudit,” Proceedings of the Royal Society A 481, 20250035 (2025).DOI arXiv
[dSL26]: N. de Silva and O. Lautsch, “The generalised semi-Clifford conjecture is false,” arXiv preprint, preliminary draft (2026).DOI arXiv

Semi-Clifford structure of two-qudit hierarchy gates above the third level

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