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arXiv:0712.2084 · Physical Review A

Semi-Clifford operations, structure of C k hierarchy, and gate complexity for fault-tolerant quantum computation

Bei Zeng, Xie Chen, Isaac L. Chuang

Published Apr 16, 2008• 43 citations
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Abstract

Teleportation is a crucial element in fault-tolerant quantum computation and a complete understanding of its capacity is very important for the practical implementation of optimal fault-tolerant architectures. It is known that stabilizer codes support a natural set of gates that can be more easily implemented by teleportation than any other gates. These gates belong to the so-called ${\mathcal{C}}_{k}$ hierarchy introduced by Gottesman and Chuang [Nature (London) 402, 390 (1999)]. Moreover, a subset of ${\mathcal{C}}_{k}$ gates, called semi-Clifford operations, can be implemented by an even simpler architecture than the traditional teleportation setup [X. Zhou, D. W. Leung, and I. L. Chuang, Phys. Rev. A 62, 052316 (2000)]. However, the precise set of gates in ${\mathcal{C}}_{k}$ remains unknown, even for a fixed number of qubits $n$, which prevents us from knowing exactly what teleportation is capable of. In this paper we study the structure of ${\mathcal{C}}_{k}$ in terms of semi-Clifford operations, which send by conjugation at least one maximal Abelian subgroup of the $n$-qubit Pauli group into another one. We show that for $n=1,2$, all the ${\mathcal{C}}_{k}$ gates are semi-Clifford, which is also true for ${n=3,k=3}$. However, this is no longer true for ${n>2,k>3}$. To measure the capability of this teleportation primitive, we introduce a quantity called ``teleportation depth,'' which characterizes how many teleportation steps are necessary, on average, to implement a given gate. We calculate upper bounds for teleportation depth by decomposing gates into both semi-Clifford ${\mathcal{C}}_{k}$ gates and those ${\mathcal{C}}_{k}$ gates beyond semi-Clifford operations, and compare their efficiency.

DOI
10.1103/physreva.77.042313

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Teleportation is a crucial element in fault-tolerant quantum computation and a complete understanding of its capacity is very important for the practical implementation of optimal fault-tolerant architectures. It is known that stabilizer codes support a natural set of gates that can be more easily implemented by teleportation than any other gates. These gates belong to the so-called Ck{\mathcal{C}}_{k}Ck​ hierarchy introduced by Gottesman and Chuang [Nature (London) 402, 390 (1999)]. Moreover, a subset of Ck{\mathcal{C}}_{k}Ck​ gates, called semi-Clifford operations, can be implemented by an even simpler architecture than the traditional teleportation setup [X. Zhou, D. W. Leung, and I. L. Chuang, Phys. Rev. A 62, 052316 (2000)]. However, the precise set of gates in Ck{\mathcal{C}}_{k}Ck​ remains unknown, even for a fixed number of qubits nnn, which prevents us from knowing exactly what teleportation is capable of. In this paper we study the structure of Ck{\mathcal{C}}_{k}Ck​ in terms of semi-Clifford operations, which send by conjugation at least one maximal Abelian subgroup of the nnn-qubit Pauli group into another one. We show that for n=1,2n=1,2n=1,2, all the Ck{\mathcal{C}}_{k}Ck​ gates are semi-Clifford, which is also true for n=3,k=3{n=3,k=3}n=3,k=3. However, this is no longer true for n>2,k>3{n>2,k>3}n>2,k>3. To measure the capability of this teleportation primitive, we introduce a quantity called ``teleportation depth,'' which characterizes how many teleportation steps are necessary, on average, to implement a given gate. We calculate upper bounds for teleportation depth by decomposing gates into both semi-Clifford Ck{\mathcal{C}}_{k}Ck​ gates and those Ck{\mathcal{C}}_{k}Ck​ gates beyond semi-Clifford operations, and compare their efficiency.
Fault-tolerant quantum computation is a basic problem in quantum computation, and teleportation is one of the main techniques in this theory. Using teleportation on stabilizer codes, the most well-known quantum codes, Pauli gates and Clifford operators can be applied fault-tolerantly. Indeed, this technique can be generalized for an extended set of gates, the so called Ck{\mathcal{C}}_kCk​ hierarchy gates, introduced by Gottesman and Chuang (Nature, 402, 390-392). Ck{\mathcal{C}}_kCk​ gates are a generalization of Clifford operators, but our knowledge of these sets is not as rich as our knowledge of Clifford gates. Zeng et al. in (Phys. Rev. A 77, 042313) raise the question of the relation between Ck{\mathcal{C}}_kCk​ hierarchy and the set of semi-Clifford and generalized semi-Clifford operators. They conjecture that any Ck{\mathcal{C}}_kCk​ gate is a generalized semi-Clifford operator. In this paper, we prove this conjecture for k=3k=3k=3. Using the techniques that we develop, we obtain more insight on how to characterize C3{\mathcal{C}}_3C3​ gates. Indeed, the more we understand C3{\mathcal{C}}_3C3​, the more intuition we have on Ck{\mathcal{C}}_kCk​, k≥4k\geq 4k≥4, and then we have a way of attacking the conjecture for larger kkk.
The Clifford hierarchy C1⊂C2⊂⋯\mathsf{C}_1 \subset \mathsf{C}_2 \subset \cdotsC1​⊂C2​⊂⋯ 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 Ck\mathsf{C}_kCk​ for k>3k > 3k>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 ddd or with few qudits. The generalized semi-Clifford conjecture, proposed by Zeng et al. (arXiv:0712.2084), states that every gate in ∪kCk\cup_k \mathsf{C}_k∪k​Ck​ is, up to multiplication by Cliffords, the product of a permutation and a diagonal matrix. Beigi and Shor proved the case d=2d=2d=2, k=3k=3k=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∈C3U \in \mathsf{C}_3U∈C3​ on the span of maximal stabilizer subgroups (arXiv:2006.14040). By extending their fixed-point arguments to the group Γ1(U)Γ_1(U)Γ1​(U) generated by UPU†U \mathsf{P} U^\daggerUPU† and beyond, we prove the conjecture for k≤4k \leq 4k≤4 and any prime dimension ddd. Our proof centers on conjugation groups Γ1(U),Γ2(U),…Γ_1(U), Γ_2(U), \ldotsΓ1​(U),Γ2​(U),… of U∈CkU \in \mathsf{C}_kU∈Ck​, 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.
Fault-tolerant quantum computation is a basic problem in quantum computation, and teleportation is one of the main techniques in this theory. Using teleportation on stabilizer codes, the most well-known quantum codes, Pauli gates and Clifford operators can be applied fault-tolerantly. Indeed, this technique can be generalized for an extended set of gates, the so called Ck{\mathcal{C}}_kCk​ hierarchy gates, introduced by Gottesman and Chuang (Nature, 402, 390-392). Ck{\mathcal{C}}_kCk​ gates are a generalization of Clifford operators, but our knowledge of these sets is not as rich as our knowledge of Clifford gates. Zeng et al. in (Phys. Rev. A 77, 042313) raise the question of the relation between Ck{\mathcal{C}}_kCk​ hierarchy and the set of semi-Clifford and generalized semi-Clifford operators. They conjecture that any Ck{\mathcal{C}}_kCk​ gate is a generalized semi-Clifford operator. In this paper, we prove this conjecture for k=3k=3k=3. Using the techniques that we develop, we obtain more insight on how to characterize C3{\mathcal{C}}_3C3​ gates. Indeed, the more we understand C3{\mathcal{C}}_3C3​, the more intuition we have on Ck{\mathcal{C}}_kCk​, k≥4k\geq 4k≥4, and then we have a way of attacking the conjecture for larger kkk.
Fault-tolerant quantum computation is a basic problem in quantum computation, andteleportation is one of the main techniques in this theory. Using teleportation on stabi-lizer codes, the most well-known quantum codes, Pauli gates and Clifford operators canbe applied fault-tolerantly. Indeed, this technique can be generalized for an extended setof gates, the so called Ck hierarchy gates, introduced by Gottesman and Chuang (Nature,402, 390-392). Ck gates are a generalization of Clifford operators, but our knowledge ofthese sets is not as rich as our knowledge of Clifford gates. Zeng et al. in (Phys. Rev.A 77, 042313) raise the question of the relation between Ck hierarchy and the set ofsemi-Clifford and generalized semi-Clifford operators. They conjecture that any Ck gateis a generalized semi-Clifford operator. In this paper, we prove this conjecture for k = 3.Using the techniques that we develop, we obtain more insight on how to characterize Ckgates. Indeed, the more we understand Ck, the more intuition we have on Ck, k ≥ 4, and then we have a way of attacking the conjecture for larger k.
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 C1ΠDC2C_1 ΠD C_2C1​ΠDC2​ for Clifford gates C1,C2C_1, C_2C1​,C2​, a permutation gate ΠΠΠ, and a diagonal gate DDD; 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.
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 ddd is semi-Clifford, if and only if ddd is square-free. In composite dimensions, Zd2\mathbb{Z}_d^2Zd2​ is a symplectic module rather than a vector space, motivating a distinction between four types of gates. Semi-Clifford gates (SC\mathcal{SC}SC) are the ones that become diagonal under left and right multiplication by Clifford gates, while a broader class N\mathcal{N}N admits a permutation diagonal form, under left and right multiplication by Cliffords. Lagrangian semi-Clifford gates (LSC\mathcal{LSC}LSC) conjugate some maximal abelian Pauli subgroup to another, whereas generalized semi-Clifford gates (GSC\mathcal{GSC}GSC) map some maximal abelian Pauli subalgebra to another under conjugation. In square-free dimensions, we have SC=LSC\mathcal{SC}=\mathcal{LSC}SC=LSC and N=GSC\mathcal{N}=\mathcal{GSC}N=GSC, but when ddd 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.