arXiv:2610.06836

Semi-Cliffordness of the Clifford hierarchy for a single qudit in composite dimensions

Yifei Qi, Rahul Sarkar

quant-phSubmitted Oct 5, 2026• Announced Oct 6, 2026• 0 citations

Abstract

An important question in quantum information theory asks whether every gate in the Clifford hierarchy is semi-Clifford or not, as such gates admit resource-efficient implementations via gate teleportation. Extending recent results for prime dimensions~\cite{silva_clifford_2025}, we prove that every hierarchy gate on a single qudit of dimension $d$ is semi-Clifford, if and only if $d$ is square-free. In composite dimensions, $\mathbb{Z}_d^2$ is a symplectic module rather than a vector space, motivating a distinction between four types of gates. Semi-Clifford gates ($\mathcal{SC}$) are the ones that become diagonal under left and right multiplication by Clifford gates, while a broader class $\mathcal{N}$ admits a permutation diagonal form, under left and right multiplication by Cliffords. Lagrangian semi-Clifford gates ($\mathcal{LSC}$) conjugate some maximal abelian Pauli subgroup to another, whereas generalized semi-Clifford gates ($\mathcal{GSC}$) map some maximal abelian Pauli subalgebra to another under conjugation. In square-free dimensions, we have $\mathcal{SC}=\mathcal{LSC}$ and $\mathcal{N}=\mathcal{GSC}$, but when $d$ is not square-free, these equivalences can fail, as Lagrangian submodules can become non-free. We also show that every third-level gate of the one-qudit Clifford hierarchy is a generalized semi-Clifford in arbitrary dimensions.

Comments: 25 pages, 2 figures

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arXiv:2610.06836

Semi-Cliffordness of the Clifford hierarchy for a single qudit in composite dimensions

Yifei Qi, Rahul Sarkar

quant-phSubmitted Oct 5, 2026• Announced Oct 6, 2026• 0 citations

arXiv HTML PDF

Abstract

\mathcal{SC}

) are the ones that become diagonal under left and right multiplication by Clifford gates, while a broader class

\mathcal{N}

admits a permutation diagonal form, under left and right multiplication by Cliffords. Lagrangian semi-Clifford gates (

\mathcal{LSC}

) conjugate some maximal abelian Pauli subgroup to another, whereas generalized semi-Clifford gates (

\mathcal{GSC}

) map some maximal abelian Pauli subalgebra to another under conjugation. In square-free dimensions, we have (\mathcal{SC}=\mathcal{LSC}) and (\mathcal{N}=\mathcal{GSC}), but when (d) is not square-free, these equivalences can fail, as Lagrangian submodules can become non-free. We also show that every third-level gate of the one-qudit Clifford hierarchy is a generalized semi-Clifford in arbitrary dimensions.

Comments: 25 pages, 2 figures

Related papers

The Generalized Semi-Clifford Conjecture Holds at Level 4

Maxwell Marcus, Sathyawageeswar Subramanian, Marcel Dall'Agnol

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arXiv:2011.00127Proceedings of the Royal Society A Mathematical Physical and Engineering SciencesJul 1, 202116 citations

The generalised semi-Clifford conjecture is false

Nadish de Silva, Oscar Lautsch

The Clifford hierarchy is a nested sequence of sets of quantum gates that can be fault-tolerantly performed using gate teleportation within standard quantum error correction schemes. The importance of these gates has motivated numerous studies of their structure. Zeng-Chen-Chuang conjectured in 2007 that all hierarchy gates are generalised semi-Clifford, i.e. take the form $C_1 ΠD C_2$ for Clifford gates $C_1, C_2$ , a permutation gate $Π$ , and a diagonal gate $D$ ; Beigi-Shor proved in 2008 that this holds for all third-level gates. We construct a five-qubit gate that is in the fifth level of the Clifford hierarchy but is not generalised semi-Clifford. Rather than simply present and verify our counterexample to the generalised semi-Clifford conjecture, we show how its form can be deduced. Our counterexample also demonstrates that the Clifford hierarchy is not closed under inverses.The Clifford hierarchy is a nested sequence of sets of quantum gates that can be fault-tolerantly performed using gate teleportation within standard quantum error correction schemes. The importance of these gates has motivated numerous studies of their structure. Zeng-Chen-Chuang conjectured in 2007 that all hierarchy gates are generalised semi-Clifford, i.e. take the form $C_1 ΠD C_2$ for Clifford gates $C_1, C_2$, a permutation gate $Π$, and a diagonal gate $D$; Beigi-Shor proved in 2008 that this holds for all third-level gates. We construct a five-qubit gate that is in the fifth level of the Clifford hierarchy but is not generalised semi-Clifford. Rather than simply present and verify our counterexample to the generalised semi-Clifford conjecture, we show how its form can be deduced. Our counterexample also demonstrates that the Clifford hierarchy is not closed under inverses.

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When is the Clifford hierarchy generalized semi-Clifford?

Rongbiao Thomas Wang, Bobby Zixuan Zhang

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quant-pharXiv:2610.06402Oct 6, 20260 citations

The Clifford hierarchy for one qubit or qudit

Nadish de Silva, Oscar Lautsch

The Clifford hierarchy is a nested sequence of sets of quantum gates that can be fault-tolerantly performed using gate teleportation within standard quantum error correction schemes. The groups of Pauli and Clifford gates constitute the first and second ‘levels’, respectively. Non-Clifford gates from the third level or higher, such as the T gate, are necessary for achieving fault-tolerant universal quantum computation. Since it was defined 26 years ago by Gottesman–Chuang, two questions have been studied by numerous researchers. First, precisely which gates constitute the Clifford hierarchy? Second, which subset of the hierarchy gates admit efficient gate teleportation protocols? We completely solve both questions in the practically relevant case of the Clifford hierarchy for gates of one qubit or one qudit of prime dimension. We express every such hierarchy gate uniquely as a product of three simple gates, yielding also a formula for the size of every level. These results are a consequence of our finding that all such hierarchy gates can be expressed in a certain form that guarantees efficient gate teleportation. Our decomposition of Clifford gates as a unique product of three elementary Clifford gates is of broad applicability.The Clifford hierarchy is a nested sequence of sets of quantum gates that can be fault-tolerantly performed using gate teleportation within standard quantum error correction schemes. The groups of Pauli and Clifford gates constitute the first and second ‘levels’, respectively. Non-Clifford gates from the third level or higher, such as the T gate, are necessary for achieving fault-tolerant universal quantum computation. Since it was defined 26 years ago by Gottesman–Chuang, two questions have been studied by numerous researchers. First, precisely which gates constitute the Clifford hierarchy? Second, which subset of the hierarchy gates admit efficient gate teleportation protocols? We completely solve both questions in the practically relevant case of the Clifford hierarchy for gates of one qubit or one qudit of prime dimension. We express every such hierarchy gate uniquely as a product of three simple gates, yielding also a formula for the size of every level. These results are a consequence of our finding that all such hierarchy gates can be expressed in a certain form that guarantees efficient gate teleportation. Our decomposition of Clifford gates as a unique product of three elementary Clifford gates is of broad applicability.

arXiv:2501.07939Proceedings of the Royal Society A Mathematical Physical and Engineering SciencesOct 1, 20251 citation

A Cohomological Characterization of the Clifford Hierarchy

Junaid Aftab

The Clifford hierarchy plays a central role in fault-tolerant quantum computation, but its higher levels remain only partially understood. Recent work on quantum higher order Fourier analysis characterizes membership in the Clifford hierarchy using quantum derivatives, but does not provide an explicit description of the individual levels. In this work, we address this problem by showing that the collection of quantum derivatives of a unitary operator forms a non-abelian $1$ -cocycle, and conversely that every such $1$ -cocycle arises from a unitary operator. This yields a recursive cohomological characterization of the Clifford hierarchy. We then specialize this framework to the third level and decompose the cocycle data into symplectic, affine, and phase components, each admitting a cohomological interpretation. As applications, we give a new proof that every two-qudit gate in the third level of the Clifford hierarchy is semi-Clifford, and prove for the first time that every three-qudit gate in the third level is semi-Clifford.The Clifford hierarchy plays a central role in fault-tolerant quantum computation, but its higher levels remain only partially understood. Recent work on quantum higher order Fourier analysis characterizes membership in the Clifford hierarchy using quantum derivatives, but does not provide an explicit description of the individual levels. In this work, we address this problem by showing that the collection of quantum derivatives of a unitary operator forms a non-abelian $1$-cocycle, and conversely that every such $1$-cocycle arises from a unitary operator. This yields a recursive cohomological characterization of the Clifford hierarchy. We then specialize this framework to the third level and decompose the cocycle data into symplectic, affine, and phase components, each admitting a cohomological interpretation. As applications, we give a new proof that every two-qudit gate in the third level of the Clifford hierarchy is semi-Clifford, and prove for the first time that every three-qudit gate in the third level is semi-Clifford.

math-phCross-listedarXiv:2610.06699Oct 6, 20260 citations

\mathcal{SC}

) are the ones that become diagonal under left and right multiplication by Clifford gates, while a broader class

\mathcal{N}

admits a permutation diagonal form, under left and right multiplication by Cliffords. Lagrangian semi-Clifford gates (

\mathcal{LSC}

) conjugate some maximal abelian Pauli subgroup to another, whereas generalized semi-Clifford gates (

\mathcal{GSC}

The Clifford hierarchy

\mathsf{C}_1 \subset \mathsf{C}_2 \subset \cdots

was introduced by Gottesman and Chuang (arXiv:quant-ph/9908010) to characterize gates that admit fault-tolerant implementation by gate teleportation. Yet, despite its rich mathematical structure and the attention it has received in recent years, little is known about

\mathsf{C}_k

for

k > 3

. Most progress has focused on identifying structural properties of restrictions of the hierarchy, such as diagonal gates and gates on systems of small dimension

d

or with few qudits. The generalized semi-Clifford conjecture, proposed by Zeng et al. (arXiv:0712.2084), states that every gate in

\cup_k \mathsf{C}_k

is, up to multiplication by Cliffords, the product of a permutation and a diagonal matrix. Beigi and Shor proved the case

d=2

k=3

(arXiv:0810.5108) and Pllaha et al. found an alternative proof by exploiting fixed points of the conjugation map induced by (a Clifford correction of)

U \in \mathsf{C}_3

on the span of maximal stabilizer subgroups (arXiv:2006.14040). By extending their fixed-point arguments to the group

Γ_1(U)

generated by

U \mathsf{P} U^\dagger

and beyond, we prove the conjecture for

k \leq 4

and any prime dimension

d

. Our proof centers on conjugation groups

Γ_1(U), Γ_2(U), \ldots

U \in \mathsf{C}_k

, which we expect to be a useful tool in the study of the Clifford hierarchy more generally. We also show a natural sufficient condition on such groups for gates in higher levels to be generalized semi-Clifford.

The Clifford hierarchy is a nested sequence of sets of quantum gates that can be fault-tolerantly performed using gate teleportation within standard quantum error correction schemes. The importance of these gates has motivated numerous studies of their structure. Zeng-Chen-Chuang conjectured in 2007 that all hierarchy gates are generalised semi-Clifford, i.e. take the form

C_1 ΠD C_2

for Clifford gates

C_1, C_2

, a permutation gate

Π

, and a diagonal gate

D

; Beigi-Shor proved in 2008 that this holds for all third-level gates. We construct a five-qubit gate that is in the fifth level of the Clifford hierarchy but is not generalised semi-Clifford. Rather than simply present and verify our counterexample to the generalised semi-Clifford conjecture, we show how its form can be deduced. Our counterexample also demonstrates that the Clifford hierarchy is not closed under inverses.

We study semi-Clifford and generalized semi-Clifford structures in the Clifford hierarchy as a function of hierarchy level

k

and qubit number

n

. We show that

\mathcal{C}_k(n)

is semi-Clifford if and only if

k\le2

n\le2

, or

k=3

and

n\le6

. We also establish that

\mathcal{C}_k(n)

is generalized semi-Clifford when

k\le4

n\le3

and construct counterexamples when

k,n \geq 5

. For

n=4

and

k\ge5

, we show that every gate defined over

\mathbb{Q}(i)

up to a scalar is generalized semi-Clifford, reduce the general case to a single level, and propose a strategy to prove our conjecture that

\mathcal{C}_k(n)

is generalized semi-Clifford if and only if

k\le4

n\le4

. Lastly, we go beyond this structure by introducing

n_c

-generalized semi-Clifford gates, which we show are exactly the gates of the form

\sum_x|π(x)\rangle\langle x|\otimes V_x

up to Clifford gates on both sides, where

π

permutes the computational basis states of

n_c

control qubits. We conjecture that for

k\ge3

every gate in

\mathcal{C}_k(n)

has this form for some

n_c\ge1

with every

V_x\in\mathcal{C}_{k-3}(n-n_c)

, refining the conjecture of Zeng, Chen, and Chuang.

1

-cocycle, and conversely that every such

1

-cocycle arises from a unitary operator. This yields a recursive cohomological characterization of the Clifford hierarchy. We then specialize this framework to the third level and decompose the cocycle data into symplectic, affine, and phase components, each admitting a cohomological interpretation. As applications, we give a new proof that every two-qudit gate in the third level of the Clifford hierarchy is semi-Clifford, and prove for the first time that every three-qudit gate in the third level is semi-Clifford.