Michel Planat, Raymond Aschheim, Marcelo M. Amaral, Klee Irwin (2018)
It has been shown that non-stabilizer eigenstates of permutation gates are appropriate for allowing d-dimensional universal quantum computing (uqc) based on minimal informationally complete POVMs. The relevant quantum gates may be built from subgroups of finite index of the modular group Γ = PSL(2,Z) [M. Planat, Entropy 20, 16 (2018)] or more generally from subgroups of fundamental groups of 3- manifolds [M. Planat, R. Aschheim, M. M. Amaral and K. Irwin, arXiv 1802.04196(quant-ph)]. In this paper, previous work is encompassed by the use of torsion-free subgroups of Bianchi groups for deriving the quantum gate generators of uqc. A special role is played by a chain of Bianchi congruence n-cusped links starting with Thurston’s link.
arXiv:1808.06831v1 [math.GT] 21 Aug 2018QUANTUM COMPUTING WITH BIANCHI GROUPS
MICHEL PLANAT† , RAYMOND ASCHHEIM‡,
MARCELO M. AMARAL‡ AND KLEE IRWIN‡
Abstract. It has been shown that non-stabilizer eigenstates of permu-
tation gates are appropriate for allowing d-dimensional universal quan-
tum computing (uqc) based on minimal informationally complete POVMs.
The relevant quantum gates may be built from subgroups of finite in-
dex of the modular group Γ = PSL(2,Z) [M. Planat, Entropy 20, 16
(2018)] or more generally from subgroups of fundamental groups of 3-
manifolds [M. Planat, R. Aschheim, M. M. Amaral and K. Irwin, arXiv
1802.04196(quant-ph)].
In this paper, previous work is encompassed
by the use of torsion-free subgroups of Bianchi groups for deriving the
quantum gate generators of uqc. A special role is played by a chain of
Bianchi congruence n-cusped links starting with Thurston’s link.
PACS: 03.67.Lx, 03.65.Wj, 03.65.Aa, 02.20.-a, 02.10.Kn, 02.40.Pc, 02.40.Sf
MSC codes: 81P68, 57M25, 57M27, 20K15, 57R65, 14H30, 20E05, 57M12
Keywords: quantum computation, Bianchi groups, MIC-POVMs, knot and link theory,
three-manifolds, branch coverings, Dehn surgeries.
1. Introduction
A Bianchi group Γk = PSL(2,Ok) < PSL(2,C) acts as a subset of
orientation-preserving isometries of 3-dimensional hyperbolic space H3, with
Ok the ring of integers of the imaginary quadratic field I = Q(
√
−k). The
quotient space 3-orbifold PSL(2,Ok) \ H3 has a set of cusps in bijection
with the class group I [1, 2, 3]. A torsion-free subgroup Γk(l) of index l
of Γk is the fundamental group π1 of a 3-manifold defined by a link such
as the figure-eight knot [with Γ−3(12)], the Whitehead link [with Γ−1(12)]
or the Borromean rings [withΓ−1(24)]. The fundamental group of a knot
or link complement (such as the complement the figure-eight knot K4a1,
the Whitehead link L5a1 or Borromean rings L6a4) was used to construct
appropriate d-dimensional fiducial states for universal quantum compuring
(uqc) [4]. The latter states come from the permutation s