Theoretical and Mathematical Physics

Groups of diagonal gates in the Clifford hierarchy

Lingxuan Feng, Shunlong Luo

Published Dec 1, 2024• 5 citations

Abstract

The Clifford hierarchy plays a crucial role in the stabilizer formalism of quantum error correction and quantum computation. Apart form the zeroth level (the discrete Heisenberg–Weyl group) and the first level (the Clifford group), all other levels of the Clifford hierarchy are not groups. However, the diagonal gates at all levels do form groups, and it is desirable to characterize their generators and structures. In this paper, we study the diagonal gates at the second level of the Clifford hierarchy. For this, we introduce the notion of a $$T$$ -gate in an arbitrary dimension, generalizing the corresponding notion in prime dimensions. By the use of the $$T$$ -gate, we are able to completely characterize the group structures of the diagonal gates at the second level of the Clifford hierarchy in any (not necessarily prime) dimension. It turns out that the classification depends crucially on the number-theoretic nature of the dimension. The results highlight the special role of the first two primes, $$2$$ and $$3$$ , in the prime factorization of the dimension. The $$T$$ -gate in an arbitrary dimension, apart from its key role as a generator of the diagonal gates, may have independent interest and further applications in quantum theory.

DOI: 10.1134/s0040577924120018

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Theoretical and Mathematical Physics

Groups of diagonal gates in the Clifford hierarchy

Lingxuan Feng, Shunlong Luo

Published Dec 1, 2024• 5 citations

Publisher

Abstract

T

-gate in an arbitrary dimension, generalizing the corresponding notion in prime dimensions. By the use of the

T

-gate, we are able to completely characterize the group structures of the diagonal gates at the second level of the Clifford hierarchy in any (not necessarily prime) dimension. It turns out that the classification depends crucially on the number-theoretic nature of the dimension. The results highlight the special role of the first two primes,

2

and

3

, in the prime factorization of the dimension. The

T

-gate in an arbitrary dimension, apart from its key role as a generator of the diagonal gates, may have independent interest and further applications in quantum theory.

DOI: 10.1134/s0040577924120018

Related papers

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.

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

Yifei Qi, Rahul Sarkar

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.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.

quant-pharXiv:2610.06836Oct 6, 20260 citations

T

-gate in an arbitrary dimension, generalizing the corresponding notion in prime dimensions. By the use of the

T

2

and

3

, in the prime factorization of the dimension. The

T

-gate in an arbitrary dimension, apart from its key role as a generator of the diagonal gates, may have independent interest and further applications in quantum theory.

{p}^{m}\mathrm{th}

roots of unity raised to polynomial functions of the basis state to which they are applied, and we determine which level of the Clifford hierarchy a given gate sits in based on

m

and the degree of the polynomial.

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.

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.