In this paper the Clifford groups of general multipartite quantum systems with configuration space $\mathbb{Z}_{n_1}\oplus\cdots\oplus\mathbb{Z}_{n_k}$, $k\geq 1$ are studied. It is known that the Clifford group is a natural semidirect product provided the dimension $N=n_1\cdots n_k$ of the corresponding Hilbert space is an odd number.
For even $N$ 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 $\mathrm{Sp}_{[n_1,\dots,n_k]}$ we prove that for even $N=n_1\cdots n_k$ both Clifford group and the projective Clifford group are natural semidirect products if and only if $N$ is not divisible by four.
In this paper the Clifford groups of general multipartite quantum systems with configuration space Zn1⊕⋯⊕Znk, k≥1 are studied. It is known that the Clifford group is a natural semidirect product provided the dimension N=n1⋯nk of the corresponding Hilbert space is an odd number.
For even N 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] we prove that for even N=n1⋯nk both Clifford group and the projective Clifford group are natural semidirect products if and only if N is not divisible by four.
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.
Symmetries of finite Heisenberg groups represent an important tool for the study of deeper structure of finite-dimensional quantum mechanics. This short contribution presents extension of previous investigations to composite quantum systems comprised of k subsystems which are described with position and momentum variables in n i , i = 1,..., κ . Their Hilbert spaces are given by κ -fold tensor products of Hilbert spaces of dimensions n 1 ,..., n κ . Symmetry group of the corresponding finite Heisenberg group is given by the quotient group of a certain normalizer. We provide the description of the symmetry groups for arbitrary multipartite cases. The new class of symmetry groups represents very specific generalization of finite symplectic groups over modular rings.
A composite quantum system comprising a finite number k of subsystems which are described with position and momentum variables in , i = 1, ..., k , is considered. Its Hilbert space is given by a k -fold tensor product of Hilbert spaces of dimensions n 1 , ..., n k . The symmetry group of the respective finite Heisenberg group is given by the quotient group of certain normalizer. This paper extends our previous investigation of bipartite quantum systems to arbitrary multipartite systems of the above type. It provides detailed description of the normalizers and the corresponding symmetry groups. The new class of symmetry groups represents a very specific generalization of symplectic groups over modular rings. As an application, a new proof of existence of the maximal set of mutually unbiased bases in Hilbert spaces of prime power dimensions is provided.
The Clifford group is the quotient of the normalizer of the Weyl-Heisenberg group in dimension d by its centre. We prove that when d is not prime the Clifford group is not a group unitary 2-design. Furthermore, we prove that the multipartite Clifford group is not a group unitary 2-design except for the known cases wherein the local Hilbert space dimensions are a constant prime number. We also clarify the structure of projective group unitary 2-designs. We show that the adjoint action induced by a group unitary 2-design decomposes into exactly two irreducible components; moreover, a group is a unitary 2-design if and only if the character of its so-called UU representation is 2.
The Clifford hierarchy C1⊂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 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 ∪kCk 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∈C3 on the span of maximal stabilizer subgroups (arXiv:2006.14040). By extending their fixed-point arguments to the group Γ1(U) generated by UPU† and beyond, we prove the conjecture for k≤4 and any prime dimension d.
Our proof centers on conjugation groups Γ1(U),Γ2(U),… of U∈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.
Based on a presentation of Cn and the help of [The GAP Group 2022, https://www.gap-system.org] we construct the character table of the Clifford group Cn for n = 1, 2, 3. As an application, we can efficiently decompose the (higher power of) tensor product of the matrix representation in those cases. Our results recover some known results in Helsen et al. [J. Math. Phys. 59, 072201 (2018)] and Wallman and Flammia [New J. Phys. 16(10), 103032 (2014)] and reveal some new phenomena. We prove that when n ≥ 3, (1) the trivial character is the only linear character for Cn and hence Cn equals to its commutator subgroup, (2) the n-qubit Pauli group Pn is the only proper non-trivial normal subgroup of Cn, (3) the matrix representation M2n is a faithful representation for Cn. As a byproduct, we give a presentation of the finite symplectic group Sp(2n, 2) in terms of generators and relations.