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arXiv:2305.13178 · Journal of Physics A Mathematical and Theoretical

Clifford group is not a semidirect product in dimensions N divisible by four

Miroslav Korbelář, J. Tolar

Published May 24, 2023• 3 citations
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Abstract

Abstract The paper is devoted to projective Clifford groups of quantum N-dimensional systems (with configuration space Z N ). Clearly, Clifford gates allow only the simplest quantum computations which can be simulated on a classical computer (Gottesmann–Knill theorem). However, it may serve as a cornerstone of full quantum computation. As to its group structure it is well-known that—in N-dimensional quantum mechanics—the Clifford group is a natural semidirect product provided the dimension N is an odd number. For even N special results on the Clifford groups are scattered in the mathematical literature, but they mostly do not concern the semidirect structure. Using appropriate group presentation of S L ( 2 , Z N ) it is proved that for even N the projective Clifford groups are not natural semidirect products if and only if N is divisible by four.

DOI
10.1088/1751-8121/acd891

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The Gottesman–Knill theorem asserts that quantum circuits composed solely of Clifford gates can be efficiently simulated classically. This theorem hinges on the fact that Clifford gates map Pauli strings to other Pauli strings, thereby allowing for a structured simulation process using classical computations. In this work, we break down the step-by-step procedure of the Gottesman–Knill theorem in a beginner-friendly manner, leveraging concepts such as matrix products, tensor products, commutation, anti-commutation, eigenvalues, and eigenvectors of quantum mechanical operators. Through detailed examples illustrating superposition and entanglement phenomena, we aim to provide a clear understanding of the classical simulation of Clifford gate-based quantum circuits. While we do not provide a formal proof of the theorem, we offer intuitive physical insights at each stage where necessary, empowering readers to grasp the fundamental principles underpinning this intriguing aspect of quantum computation. Quanta 2024; 13: 20–27.

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Generalised Clifford groups and simulation of associated quantum circuits

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Quantum computations that involve only Clifford operations are classically simulable despite the fact that they generate highly entangled states; this is the content of the Gottesman-Knill theorem. Here we isolate the ingredients of the theorem and provide generalisations of some of them with the aim of identifying new classes of simulable quantum computations. In the usual construction, Clifford operations arise as projective normalisers of the first and second tensor powers of the Pauli group. We consider replacing the Pauli group by an arbitrary finite subgroup G of U(d). In particular we seek G such that G tensor G has an entangling normaliser. Via a generalisation of the Gottesman-Knill theorem the resulting normalisers lead to classes of quantum circuits that can be classically efficiently simulated. For the qubit case d=2 we exhaustively treat all finite subgroups of U(2) and find that the only ones (up to unitary equivalence and trivial phase extensions) with entangling normalisers are the groups G_n generated by X and the n^th root of Z.

arXiv:quant-ph/0701103•Jan 15, 2007•4 citations
Abstract The paper is devoted to projective Clifford groups of quantum N-dimensional systems (with configuration space Z N ). Clearly, Clifford gates allow only the simplest quantum computations which can be simulated on a classical computer (Gottesmann–Knill theorem). However, it may serve as a cornerstone of full quantum computation. As to its group structure it is well-known that—in N-dimensional quantum mechanics—the Clifford group is a natural semidirect product provided the dimension N is an odd number. For even N special results on the Clifford groups are scattered in the mathematical literature, but they mostly do not concern the semidirect structure. Using appropriate group presentation of S L ( 2 , Z N ) it is proved that for even N the projective Clifford groups are not natural semidirect products if and only if N is divisible by four.
In this paper the Clifford groups of general multipartite quantum systems with configuration space Zn1⊕⋯⊕Znk\mathbb{Z}_{n_1}\oplus\cdots\oplus\mathbb{Z}_{n_k}Zn1​​⊕⋯⊕Znk​​, k≥1k\geq 1k≥1 are studied. It is known that the Clifford group is a natural semidirect product provided the dimension N=n1⋯nkN=n_1\cdots n_kN=n1​⋯nk​ of the corresponding Hilbert space is an odd number. For even NNN special results on the Clifford groups are scattered in the mathematical literature, but they do not concern the semidirect product structure. Using the relation of generators of the associated symplectic group Sp[n1,…,nk]\mathrm{Sp}_{[n_1,\dots,n_k]}Sp[n1​,…,nk​]​ we prove that for even N=n1⋯nkN=n_1\cdots n_kN=n1​⋯nk​ both Clifford group and the projective Clifford group are natural semidirect products if and only if NNN is not divisible by four.
The term Clifford group was introduced in 1998 by D. Gottesmann in his investigation of quantum error-correcting codes. The simplest Clifford group in multiqubit quantum computation is generated by a restricted set of unitary Clifford gates - the Hadamard, π /4-phase and controlled-X gates. Because of this restriction the Clifford model of quantum computation can be efficiently simulated on a classical computer (the Gottesmann-Knill theorem). However, this fact does not diminish the importance of the Clifford model, since it may serve as a suitable starting point for a full-fledged quantum computation. In the general case of a single or composite quantum system with finite-dimensional Hilbert space the finite Weyl-Heisenberg group of unitary operators defines the quantum kinematics and the states of the quantum register. Then the corresponding Clifford group is defined as the group of unitary operators leaving the Weyl-Heisenberg group invariant. The aim of this contribution is to show that our comprehensive results on symmetries of the Pauli gradings of quantum operator algebras - covering any single as well as composite finite quantum systems - directly correspond to Clifford groups defined as quotients with respect to U(1).
Quantum computations starting with computational basis states and involving only Clifford operations, are classically simulable despite the fact that they generate highly entangled states; this is the content of the Gottesman-Knill theorem. Here we isolate the ingredients of the theorem and provide generalisations of some of them with the aim of identifying new classes of simulable quantum computations. In the usual construction, Clifford operations arise as projective normalisers of the first and second tensor powers of the Pauli group. We consider replacing the Pauli group by an arbitrary finite subgroup GGG of U(d)U(d)U(d). In particular we seek GGG such that G⊗GG\otimes GG⊗G has an entangling normaliser. Via a generalisation of the Gottesman-Knill theorem the resulting normalisers lead to classes of quantum circuits that can be classically efficiently simulated. For the qubit case d=2d = 2d=2 we exhaustively treat all finite irreducible subgroups of U(2)U(2)U(2) and find that the only ones (up to unitary equivalence and trivial phase extensions) with entangling normalisers are the groups generated by X and the nthn^{\rm th}nth root of ZZZ for n∈Nn \in \Nn∈N.
The Gottesman–Knill theorem asserts that quantum circuits composed solely of Clifford gates can be efficiently simulated classically. This theorem hinges on the fact that Clifford gates map Pauli strings to other Pauli strings, thereby allowing for a structured simulation process using classical computations. In this work, we break down the step-by-step procedure of the Gottesman–Knill theorem in a beginner-friendly manner, leveraging concepts such as matrix products, tensor products, commutation, anti-commutation, eigenvalues, and eigenvectors of quantum mechanical operators. Through detailed examples illustrating superposition and entanglement phenomena, we aim to provide a clear understanding of the classical simulation of Clifford gate-based quantum circuits. While we do not provide a formal proof of the theorem, we offer intuitive physical insights at each stage where necessary, empowering readers to grasp the fundamental principles underpinning this intriguing aspect of quantum computation. Quanta 2024; 13: 20–27.
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.
Quantum computations that involve only Clifford operations are classically simulable despite the fact that they generate highly entangled states; this is the content of the Gottesman-Knill theorem. Here we isolate the ingredients of the theorem and provide generalisations of some of them with the aim of identifying new classes of simulable quantum computations. In the usual construction, Clifford operations arise as projective normalisers of the first and second tensor powers of the Pauli group. We consider replacing the Pauli group by an arbitrary finite subgroup G of U(d). In particular we seek G such that G tensor G has an entangling normaliser. Via a generalisation of the Gottesman-Knill theorem the resulting normalisers lead to classes of quantum circuits that can be classically efficiently simulated. For the qubit case d=2 we exhaustively treat all finite subgroups of U(2) and find that the only ones (up to unitary equivalence and trivial phase extensions) with entangling normalisers are the groups G_n generated by X and the n^th root of Z.