The recent proposal (M Planat and M Kibler, Preprint 0807.3650 [quantph]) of representing Clifford quantum gates in terms of unitary reflections is revisited. In this essay, the geometry of a Clifford group G is expressed as a BN-pair, i.e. a pair of subgroups B and N that generate G, is such that intersection H = B \cap N is normal in G, the group W = N/H is a Coxeter group and two extra axioms are satisfied by the double cosets acting on B. The BN-pair used in this decomposition relies on the swap and match gates already introduced for classically simulating quantum circuits (R Jozsa and A Miyake, Preprint arXiv:0804.4050 [quant-ph]). The two- and three-qubit steps are related to the configuration with 27 lines on a smooth cubic surface.
Peres/Mermin arguments about no-hidden variables in quantum mechanics are used for displaying a pair (R, S) of entangling Clifford quantum gates, acting on two qubits. From them, a natural unitary representation of Coxeter/Weyl groups W(D5) and W(F4) emerges, which is also reflected into the splitting of the n-qubit Clifford group Cn into dipoles C
±n . The union of the three-qubit real Clifford group C+ 3 and the Toffoli gate ensures a orthogonal representation of the Weyl/Coxeter group W(E8), and of its relatives. Other concepts involved are complex reflection groups, BN pairs, unitary group designs and entangled states of the GHZ family.
This paper explores the representation of quantum computing in terms of unitary reflections (unitary transformations that leave invariant a hyperplane of a vector space). The symmetries of qubit systems are found to be supported by Euclidean real reflections (i.e., Coxeter groups) or by specific imprimitive reflection groups, introduced (but not named) in a recent paper [Planat M and Jorrand Ph 2008, {\it J Phys A: Math Theor} {\bf 41}, 182001]. The automorphisms of multiple qubit systems are found to relate to some Clifford operations once the corresponding group of reflections is identified. For a short list, one may point out the Coxeter systems of type
B3 and
G2 (for single qubits),
D5 and
A4 (for two qubits),
E7 and
E6 (for three qubits), the complex reflection groups
G(2l,2,5) and groups No 9 and 31 in the Shephard-Todd list. The relevant fault tolerant subsets of the Clifford groups (the Bell groups) are generated by the Hadamard gate, the
π/4 phase gate and an entangling (braid) gate [Kauffman L H and Lomonaco S J 2004 {\it New J. of Phys.} {\bf 6}, 134]. Links to the topological view of quantum computing, the lattice approach and the geometry of smooth cubic surfaces are discussed.
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.
One learned from Gottesman-Knill theorem that the Clifford model of quantum computing \cite{Clark07} may be generated from a few quantum gates, the Hadamard, Phase and Controlled-Z gates, and efficiently simulated on a classical computer. We employ the group theoretical package GAP\cite{GAP} for simulating the two qubit Clifford group
C2. We already found that the symmetric group S(6), aka the automorphism group of the generalized quadrangle W(2), controls the geometry of the two-qubit Pauli graph \cite{Pauligraphs}. Now we find that the {\it inner} group
Inn(C2)=C2/Center(C2) exactly contains two normal subgroups, one isomorphic to
Z2×4 (of order 16), and the second isomorphic to the parent
A′(6) (of order 5760) of the alternating group A(6). The group
A′(6) stabilizes an {\it hexad} in the Steiner system
S(3,6,22) attached to the Mathieu group M(22). Both groups A(6) and
A′(6) have an {\it outer} automorphism group
Z2×Z2, a feature we associate to two-qubit quantum entanglement.
Peres/Mermin arguments about no-hidden variables in quantum mechanics are used for displaying a pair (R;S) of entangling Clifiord quantum gates, acting on two qubits. From them, a unitary realization of Coxeter/Weyl groups W(D5) and W(F4) emerges, which is also re∞ected into the splitting of the nqubit Clifiord group Cn into subgroup dipoles C § . Generators of the three-qubit real Clifiord group C + 3 with the Tofioli gate ensures a orthogonal realization of the Weyl/Coxeter group W(E8), and of its relatives. Other concepts involved are complex re∞ection groups, BN pairs, unitary group designs and entangled states of the GHZ, W and chain families.