arXiv:2410.11818

Permutation gates in the third level of the Clifford hierarchy

Zhiyang He, Luke Robitaille, Xinyu Tan

Published Oct 15, 2024• 1 citation

Abstract

The Clifford hierarchy is a fundamental structure in quantum computation, classifying unitary operators based on their commutation relations with the Pauli group. Despite its significance, the mathematical structure of the hierarchy is not well understood at the third level and higher. In this work, we study permutations in the hierarchy: gates which permute the $2^n$ basis states. We fully characterize all the semi-Clifford permutation gates. Moreover, we prove that any permutation gate in the third level, not necessarily semi-Clifford, must be a product of Toffoli gates in what we define as staircase form, up to left and right multiplications of Clifford permutations. Finally, we show that the smallest number of qubits for which there exists a non-semi-Clifford permutation in the third level is $7$.

DOI: 10.48550/arxiv.2410.11818

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arXiv:2410.11818

Permutation gates in the third level of the Clifford hierarchy

Zhiyang He, Luke Robitaille, Xinyu Tan

Published Oct 15, 2024• 1 citation

arXiv HTML PDF

Abstract

2^n

basis states. We fully characterize all the semi-Clifford permutation gates. Moreover, we prove that any permutation gate in the third level, not necessarily semi-Clifford, must be a product of Toffoli gates in what we define as staircase form, up to left and right multiplications of Clifford permutations. Finally, we show that the smallest number of qubits for which there exists a non-semi-Clifford permutation in the third level is

7

DOI: 10.48550/arxiv.2410.11818

Related papers

Characterization of Permutation Gates in the Third Level of the Clifford Hierarchy

Zhiyang He, Luke Robitaille, Xinyu Tan

The Clifford hierarchy is a fundamental structure in quantum computation whose mathematical properties are not fully understood. In this work, we characterize permutation gates -- unitaries which permute the $2^n$ basis states -- in the third level of the hierarchy. We prove that any permutation gate in the third level must be a product of Toffoli gates in what we define as \emph{staircase form}, up to left and right multiplications by Clifford permutations. We then present necessary and sufficient conditions for a staircase form permutation gate to be in the third level of the Clifford hierarchy. As a corollary, we construct a family of non-semi-Clifford permutation gates $\{U_k\}_{k\geq 3}$ in staircase form such that each $U_k$ is in the third level but its inverse is not in the $k$ -th level.The Clifford hierarchy is a fundamental structure in quantum computation whose mathematical properties are not fully understood. In this work, we characterize permutation gates -- unitaries which permute the $2^n$ basis states -- in the third level of the hierarchy. We prove that any permutation gate in the third level must be a product of Toffoli gates in what we define as \emph{staircase form}, up to left and right multiplications by Clifford permutations. We then present necessary and sufficient conditions for a staircase form permutation gate to be in the third level of the Clifford hierarchy. As a corollary, we construct a family of non-semi-Clifford permutation gates $\{U_k\}_{k\geq 3}$ in staircase form such that each $U_k$ is in the third level but its inverse is not in the $k$-th level.

arXiv:2510.04993Leibniz international proceedings in informaticsOct 6, 20250 citations

When is the Clifford hierarchy generalized semi-Clifford?

Rongbiao Thomas Wang, Bobby Zixuan Zhang

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 $k\le4$ or $n\le3$ 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 $k\le4$ or $n\le4$ . 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.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 $k\le4$ or $n\le3$ 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 $k\le4$ or $n\le4$. 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.

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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.

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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.

quant-pharXiv:2609.11903Sep 11, 20260 citations

The Generalized Semi-Clifford Conjecture Holds at Level 4

Maxwell Marcus, Sathyawageeswar Subramanian, Marcel Dall'Agnol

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 $Γ_1(U), Γ_2(U), \ldots$ of $U \in \mathsf{C}_k$ , 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 $\mathsf{C}_1 \subset \mathsf{C}_2 \subset \cdots$ 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 $Γ_1(U), Γ_2(U), \ldots$ of $U \in \mathsf{C}_k$, 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.

quant-pharXiv:2609.38751Oct 1, 20260 citations

2^n

7

2^n

basis states -- in the third level of the hierarchy. We prove that any permutation gate in the third level must be a product of Toffoli gates in what we define as \emph{staircase form}, up to left and right multiplications by Clifford permutations. We then present necessary and sufficient conditions for a staircase form permutation gate to be in the third level of the Clifford hierarchy. As a corollary, we construct a family of non-semi-Clifford permutation gates

\{U_k\}_{k\geq 3}

in staircase form such that each

U_k

is in the third level but its inverse is not in the

k

-th level.

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.

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.

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

\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.