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arXiv:2006.14040 · Quantum

Un-Weyl-ing the Clifford Hierarchy

Tefjol Pllaha, Narayanan Rengaswamy, Olav Tirkkonen, Robert Calderbank

Published Dec 11, 2020• 16 citations
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Abstract

The teleportation model of quantum computation introduced by Gottesman and Chuang (1999) motivated the development of the Clifford hierarchy. Despite its intrinsic value for quantum computing, the widespread use of magic state distillation, which is closely related to this model, emphasizes the importance of comprehending the hierarchy. There is currently a limited understanding of the structure of this hierarchy, apart from the case of diagonal unitaries (Cui et al., 2017; Rengaswamy et al. 2019). We explore the structure of the second and third levels of the hierarchy, the first level being the ubiquitous Pauli group, via the Weyl (i.e., Pauli) expansion of unitaries at these levels. In particular, we characterize the support of the standard Clifford operations on the Pauli group. Since conjugation of a Pauli by a third level unitary produces traceless Hermitian Cliffords, we characterize their Pauli support as well. Semi-Clifford unitaries are known to have ancilla savings in the teleportation model, and we explore their Pauli support via symplectic transvections. Finally, we show that, up to multiplication by a Clifford, every third level unitary commutes with at least one Pauli matrix. This can be used inductively to show that, up to a multiplication by a Clifford, every third level unitary is supported on a maximal commutative subgroup of the Pauli group. Additionally, it can be easily seen that the latter implies the generalized semi-Clifford conjecture, proven by Beigi and Shor (2010). We discuss potential applications in quantum error correction and the design of flag gadgets.

DOI
10.22331/q-2020-12-11-370

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The teleportation model of quantum computation introduced by Gottesman and Chuang (1999) motivated the development of the Clifford hierarchy. Despite its intrinsic value for quantum computing, the widespread use of magic state distillation, which is closely related to this model, emphasizes the importance of comprehending the hierarchy. There is currently a limited understanding of the structure of this hierarchy, apart from the case of diagonal unitaries (Cui et al., 2017; Rengaswamy et al. 2019). We explore the structure of the second and third levels of the hierarchy, the first level being the ubiquitous Pauli group, via the Weyl (i.e., Pauli) expansion of unitaries at these levels. In particular, we characterize the support of the standard Clifford operations on the Pauli group. Since conjugation of a Pauli by a third level unitary produces traceless Hermitian Cliffords, we characterize their Pauli support as well. Semi-Clifford unitaries are known to have ancilla savings in the teleportation model, and we explore their Pauli support via symplectic transvections. Finally, we show that, up to multiplication by a Clifford, every third level unitary commutes with at least one Pauli matrix. This can be used inductively to show that, up to a multiplication by a Clifford, every third level unitary is supported on a maximal commutative subgroup of the Pauli group. Additionally, it can be easily seen that the latter implies the generalized semi-Clifford conjecture, proven by Beigi and Shor (2010). We discuss potential applications in quantum error correction and the design of flag gadgets.
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.
The Clifford hierarchy of unitary operators is a foundational concept for universal quantum computation. It was introduced to show that universal quantum computation can be realized via quantum teleportation, given access to certain standard resources. While the full structure of the hierarchy is still not understood, Cui et al. [S. X. Cui et al., Phys. Rev. A 95, 012329 (2017)] recently described the structure of diagonal unitaries in the hierarchy. They considered diagonal unitaries whose action on a computational basis qudit state is described by a 2kth{2}^{k}\mathrm{th}2kth root of unity raised to some polynomial function of the state, and they established the level of such unitaries in the hierarchy as a function of kkk and the degree of the polynomial. For qubit systems, we consider kthk\mathrm{th}kth-level diagonal unitaries that can be described just by quadratic forms of the state over the ring Z2k{\mathbb{Z}}_{{2}^{k}}Z2k​ of integers modulo 2k{2}^{k}2k. The quadratic forms involve symmetric matrices over Z2k{\mathbb{Z}}_{{2}^{k}}Z2k​ that can be used to efficiently describe all two-local and certain higher locality diagonal gates in the hierarchy. We also provide explicit algebraic descriptions of their action on Pauli matrices, which establishes a natural recursion to diagonal unitaries from lower levels. The result involves symplectic matrices over Z2k{\mathbb{Z}}_{{2}^{k}}Z2k​ and hence our perspective unifies a subgroup of diagonal gates in the Clifford hierarchy with the binary symplectic framework for gates in the Clifford group. We augment our description with simple examples for certain standard gates. In addition to demonstrating structure, these formulas might prove useful in applications such as (i) classical simulation of quantum circuits, especially via the stabilizer rank approach, (ii) synthesis of logical non-Clifford unitaries, specifically alternatives to expensive magic state distillation, and (iii) decomposition of arbitrary unitaries beyond the Clifford+T\mathrm{Clifford}+TClifford+T set of gates, perhaps leading to shorter depth circuits. Our results suggest that some nondiagonal gates in the hierarchy might also be understood by generalizing other binary symplectic matrices to integer rings.
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.
The Clifford hierarchy is a nested sequence of sets of quantum gates critical to achieving fault-tolerant quantum computation. Diagonal gates of the Clifford hierarchy and ‘nearly diagonal’ semi-Clifford gates are particularly important: they admit efficient gate teleportation protocols that implement these gates with fewer ancillary quantum resources such as magic states. Despite the practical importance of these sets of gates, many questions about their structure remain open; this is especially true in the higher-dimensional qudit setting. Our contribution is to leverage the discrete Stone–von Neumann theorem and the symplectic formalism of qudit stabilizer theory towards extending the results of Zeng et al . (2008) and Beigi & Shor (2010) to higher dimensions in a uniform manner. We further give a simple algorithm for recursively enumerating all gates of the Clifford hierarchy, a simple algorithm for recognizing and diagonalizing semi-Clifford gates, and a concise proof of the classification of the diagonal Clifford hierarchy gates due to Cui et al . (2016) for the single-qudit case. We generalize the efficient gate teleportation protocols of semi-Clifford gates to the qudit setting and prove that every third-level gate of one qudit (of any prime dimension) and of two qutrits can be implemented efficiently. Numerical evidence gathered via the aforementioned algorithms supports the conjecture that higher-level gates can be implemented efficiently.
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.
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.