Unitary Reflection Groups for Quantum Fault Tolerance

Abstract

Abstract.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, J Phys A: Math Theor 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 B 3 and G 2 (for single qubits), D 5 and A 4 (for two qubits), E 7 and E 6 (for three qubits), the complex reflection groups G(2 l , 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 New J. of Phys. 6, 134]. Links to the topological view of quantum computing, the lattice approach and the geometry of smooth cubic surfaces are discussed.

Zoology of Atlas-Groups: Dessins D’enfants, Finite Geometries and Quantum Commutation

“…It is found to be of type G 4 . The representation of index 1120 and rank four of O 7 (3) found in the Atlas is associated with the dual of W 5 (3) that is the dense near hexagon DQ (6,3). See below Table 9 for further details.…”

“…The representation of index 560 of rank 17 and sub-degrees 1, 13 3 , 26 6 , 52 7 leads to a configuration of type [560 13 , 1820 4 ] (i.e., every point is on 13 lines, and there are 1820 lines of size four). The Atlas also provides a representation of index 1456 and rank 79 that leads to another geometry, of order (3,4), with again 1820 lines of size four (see also the relevant item in Table 10). The physical meaning of both representations, if any, has not been discovered.…”

“…The near hexagon O − 8 (2) on 765 points is described in the text. (12,92,12,3) [120 28 , 1120 3 ] (3) , srg, NO + (8, 2), pg (7,8,4) 0.817 (15,99,11,6) [135 64 , 960 9 ] (3) , srg, G 4 , pg (8,7,4), 3QB * 0.770 (96,624,72,85) [960 36 , 4320 8 ] (4) 0.923 (4,6). Since the permutation representation is a subgroup of the modular group Γ = PSL(2, Z), it is possible to see the dessin D as an hyperbolic polygon D H .…”

“…Over the last few years, it has been recognized that the detailed investigation of commutation between the elements of generalized Pauli groups-the qudits and arbitrary collections of them [1]-is useful for a better understanding of the concepts of quantum information, such as error correction [2,3], entanglement [4,5] and contextuality [6][7][8], that are cornerstones of quantum algorithms and quantum computation. Only recently, the first author observed that much of the information needed is encapsulated in permutation representations, of rank larger than two, available in the Atlas of finite group representations [9].…”

Zoology of Atlas-Groups: Dessins D’enfants, Finite Geometries and Quantum Commutation

“…Over the last years, it has been recognized that the detailed investigation of commutation between the elements of generalized Pauli groups -the qudits and arbitrary collections of them [1]-is useful for a better understanding of concepts of quantum information such as error correction [2,3], entanglement [4,5] and contextuality [6]- [8], that are cornerstones of quantum algorithms and quantum computation. Only recently the first author observed that much of the information needed is encapsulated in permutation representations, of rank larger than two, available in the Atlas of finite group representations [9].…”