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 structure of cosets
in some d-coverings of subgroups Γk(l) [alias the subgroups of index d of
subgroups Γk(l) given a selected pair (k, l)].
In this paper, one starts by upgrading these models of uqc by using other
torsion-free subgroups of Bianchi groups and the corresponding 3-manifold
such as the Bergé manifold that comes from the Bergé link L6a2 [with
Γ−3(24)] or the so-called magic manifold that comes from the link L6a5
[with Γ−7(6)]. The latter link is a small congruence link [5] and belongs
to a chain of eight links starting with Thurston’s eight-cusped congruence
link [with Γ(−3) and ideal
〈
(5 +
√
−3)/2
〉
] and ending with the Whitehead
link and the figure-eight knot. In Sec. 2, it is presented the permutation
based model of universal quantum computing developed by the authors and
its relationship to some minimal informationally complete POVMs or MICs
[6, 7, 8]. In Sec. 3, important small index torsion-free subgroups of Bianchi
1
2MICHEL PLANAT† , RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡ANDKLEE IRWIN‡
groups and the corresponding substructure of 3-manifolds are derived. In
Sec. 4, one specializes in the connection of the aforementioned Bianchi
subgroups to our version of uqc. One focuses on a remarkable chain of n-
cusped links, n = 8..1 obtained thanks to (±1, 1)-slope Dehn filling starting
with congruence Thurston’s link. Their possible role for uqc and the relevant
3-manifolds is discussed.
2. Minimal informationally complete POVMs with
permutations
In our approach [7, 8], minimal informationally complete POVMs (MICs)
are derived from appropriate fiducial states under the action of the (gen-
eralized) Pauli group. The fiducial states also allow to perform universal
quantum computation [6].
A POVM is a collection of positive semi-definite operators {E1, . . . , Em}
that sum to the identity. In the measurement of a state ρ, the i-th outcome
is obtained with a probability given by the Born rule p(i) = tr(ρEi). For a
minimal IC-POVM, one needs d2 one-dimensional projectors Πi = |ψi〉 〈ψi|,
with Πi = dEi, such that the rank of the Gram matrix with elements
tr(ΠiΠj), is precisely d
2. A SIC-POVM (the S means symmetric) obeys
the relation|〈ψi|ψj〉|2 = tr(ΠiΠj) = dδij+1
d+1 , that allows the explicit recovery
of the density matrix as in [9, eq. (29)].
New MICs (i.e. whose rank of the Gram matrix is d2) and with Hermitian
angles |〈ψi|ψj〉|i
6=j ∈ A = {a1, . . . , al} have been discovered [8]. A SIC is
equiangular with |A| = 1 and a1 =
1
√
d+1
. The states encountered are
considered to live in a cyclotomic field F = Q[exp(2iπ
n
)], with n = GCD(d, r),
the greatest common divisor of d and r, for some r. The Hermitian angle
is defined as |〈ψi|ψj〉|i
6=j = ‖(ψi, ψj)‖
1
deg , where ‖.‖ means the field norm
of the pair (ψi, ψj) in F and deg is the degree of the extension F over the
rational field Q [7].
The fiducial states for SIC-POVMs are quite difficult to derive and seem
to follow from algebraic number theory [10]. Except for d = 3, the MICs
derived from permutation groups are not symmetric and most of them can
be recovered thanks to subgroups of index d of the modular group Γ [8, Table
1]. For instance, for d = 3, the action of a magic state of type (0, 1,±1)
results into the Hesse SIC shown in Fig 1a, arising from the congruence
subgroup Γ0(2) of Γ. For d = 4, the action of the two-qubit Pauli group
on the magic/fiducial state of type (0, 1,−ω6, ω6 − 1) with ω6 = exp(2iπ
6 )
results into a MIC whose geometry of triple products of projectors Πi, arising
from the congruence subgroup Γ0(3) of Γ, turns out to correspond to the
commutation graph of Pauli operators, see Fig. 1b and [8, Fig. 2c]. For
d = 5, the congruence subgroup 5A0 of Γ is used to get a MIC whose
geometry consists of copies of the Petersen graph (see Fig. 1c and [8, Fig.
3c]. For d = 6, all five congruence subgroups Γ′, Γ(2), 3C0, Γ0(4) or Γ0(5)
point out the geometry of Borromean rings (see [8, Fig. 4c] and Table 1 of
[4]).
The modular group Γ first served as a motivation for investigating the
trefoil knot manifold 31 in relation to uqc and the corresponding MICs,
QUANTUM COMPUTING WITH BIANCHI GROUPS
3
Figure 1. Geometrical structure of low dimensional MICs:
(a) the qutrit Hesse SIC, (b) the two-qubit MIC that is the
generalized quadrangle of order two GQ(2, 2), (c) the basic
component of the 5-dit MIC that is the Petersen graph. The
coordinates on each diagram are the d-dimensional Pauli op-
erators that act on the fiducial state, as shown. For the 6-dit
case in (d), the coordinates are made explicit in the caption
of [8, Fig. 4].
then the uqc problem was put in the wider frame of Poincaré conjecture,
the Thurston’s geometrization conjecture and the related 3-manifolds. E.g.
MICs may follow from hyperbolic or Seifert 3-manifolds as shown in Tables
2 to 5 of [4]. A further step is obtained here by restricting our choice of
3-manifolds to low degree coverings of Bianchi subgroups. This is a quite
natural procedure to base our uqc models on Bianchi groups Γk defined over
4MICHEL PLANAT† , RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡ANDKLEE IRWIN‡
the ring Ok of imaginary quadratic integers [11] since they are a natural
generalization of the modular group Γ.
3. Torsion-free subgroups of Bianchi groups
k
index cusps
names
other name
note
-1
12
2
L5a1, ooct0100001 Whitehead link: WL
L6a5(1,1)
.
.
.
L13n5885, ooct0100000
sister of WL L10n113(1,1)(1,1)(1,1)
.
24
3
L6a4, ooct0200005
Borromean rings
.
.
.
L8n7, ooct0200001
L5a1(1,1)
.
.
.
L10n84, ooct0200002
-2
12
2
L9a32, 9240
.
.
.
L9a33, 9224
-3
12
2
K4a1, otet0200001
figure-eight knot
L5a1(1,1)
.
.
1
m003, otet0200000
sister of K4a1
.
24
2
L6a2, 622, otet0400001
Bergé manifold
.
.
1
m206, otet0400002
.
.
.
m207, otet0400001
-7
6
2
L6a1, 623
.
.
3
L6a5, 631
magic manifold
congruence
.
12
3
L10n81, 10315
cyc. 2-cover of L6a5
.
.
4
L12n2205
-15 6
6
〈
(1 +
√
−15)/2
〉
Baker&Reid [5]
congruence
Table 1. Low index torsion-free subgroups of Bianchi
groups Γk = PSL(2,Ok), k = −1,−2,−3,−7, their names
in SnapPy in column 4 and their popular name in column 5,
when known. The last column shows a connection to Dehn
filling of slope (1, 1) of the corresponding manifold. Bold
characters point out manifolds investigated in this paper for
their connection to MICs.
Table 1 provides a short list of low index torsion-free subgroups of Bianchi
groups Γk, k ∈ {−1,−2,−3,−7,−15} [11], obtained from the software
Magma [12] and their identification as fundamental groups of 3-manifolds
obtained from the software SnapPy [13].
In particular, one recovers the
Whitehead link and the Borromean rings (k = −1 with index l = 12 and 24,
respectively) as well as the figure-eight knot (k = −3 with index 12) whose
relationship to uqc was explored in [4]. Other links of importance in our
paper are the link L6a1, the Bergé link L6a2 and the magic link L6a5 whose
connections to uqc are summarized in Table 3.
According to [14], a d-fold covering of the fundamental group π1(M) of a
3-manifold M is uniquely determined by the conjugacy class of a subgroup
of index d of π1. To recognize the d-covering from the conjugacy class
and conversely, one makes use of the cardinality signature of subgroups of
π1(M), denoted ηd(π1(M)), d = 1..dmax and one identifies ηd(π1(M)) both
in a particular d-covering of M (with SnapPy) and in the representative of
a particular conjugacy class of subgroups of π1(M) (with Magma).
QUANTUM COMPUTING WITH BIANCHI GROUPS
5
Table 2 provides the identification of all 3-manifolds coming from d-
fold coverings (d ≤ 5) of the Bergé link and identifies the ones related
to MICs. Observe that it becomes a cumbersome task for manifolds with
many cusps. E.g., for the manifold otet1600025 that one finds corresponding
to the two-qubit MIC, one gets ηd(M) = [15, 70, 642, 2206, 30192, · · · ] when
d = [2, 3, 4, 5, 6, · · · ].
d homology
cusps
sym
names
note
2 cyc: 1 + 1
2
D4
otet0800002 , L10n46
.
cyc: 15 + 1 + 1
.
G16
otet0800010
3 cyc: 1 + 1 + 1 + 1
4
Z
2 ×O
otet1200009 , L12n2208
.
cyc: 13 +
1
3 + 1 + 1
.
D6
otet1200020
4 cyc: 1 + 1 + 1
2
D8
otet1600013 , L14n17878
.
cyc: 13
+2
+ 1+2
.
G16
otet1600091
.
cyc: 15 + 1 + 1
.
Z
2 ×D4
otet1600063
.
irr: 1 + 1 + 1 + 1
4
Z
2 +
Z
2
otet1600025 2QB MIC
.
reg: 15 + 1 + 1
2
G32
otet1600092
5 cyc: 1 + 1
2
D10
otet2000443 ,
.
cyc: 12
+4
+ 1 + 1
.
G20
otet2001343 ,
.
irr: 1+4
4
Z
2
otet2000543; otet2000041; otet2000549 5-dit MIC
.
irr: 12 + 1
+4
.
.
otet2000574 5-dit MIC
.
irr: 13 +
1
3 + 1 + 1
2
D4
otet2000665 5-dit MIC
.
irr: 1+6
6
Z
2 ×D4
otet2000573
Table 2. Table of 3-manifolds (in column 5) found from sub-
groups of finite index d of the fundamental group of Bergé
link (alias the d-fold coverings of L6a2). The terminology
is that of SnapPy [13] with a first homology group such as
Z denoted as 1 in column 2 or 1+s for Z + · · ·Z (s times).
Column 4 is the symmetry group of the manifold with O the
octahedral group and Ga a group of order a. The three mani-
folds separated by a semicolon are distinct although with the
same homology, symmetry and number of cusps.
4. A Bianchi factory for quantum computing
First one provides a reminder about the concept of Dehn filling that will
be useful in Sec. 4.2. Let us start with a Lens space L(p, q) that is 3-
manifold obtained by gluing the boundaries of two solid tori together, so
that the meridian of the first solid torus goes to a (p, q)-curve on the second
solid torus [where a (p, q)-curve wraps around the longitude p times and
around the meridian q times]. Then we generalize this concept to a knot
exterior, i.e. the complement of an open solid torus knotted like the knot.
One glues a solid torus so that its meridian curve goes to a (p, q)-curve on
the torus boundary of the knot exterior, an operation called Dehn surgery [2,
p. 275],[3, p 259], [15],[16]. According to Lickorish’s theorem, every closed,
6MICHEL PLANAT† , RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡ANDKLEE IRWIN‡
orientable, connected 3-manifold is obtained by performing Dehn surgery on
a link in the 3-sphere.
Another operation useful for creating new 3-manifolds from old ones is
that of drilling out along a closed geodesic. Let us quote Thurston [17] In
SnapPea there is a module for drilling out geodesic loops that are identified
geometrically. Given a hyperbolic manifold, the program will present a list of
loops up to a specified combinatorial complexity sorted in order of hyperbolic
length. SnapPea can remove any of these geodesics from the manifold, in
essence deforming its cone angle to 0 so as to obtain a complete hyperbolic
structure for its complement... Repeatedly drilling out a relatively short
geodesics always seems to start crystallizing a 3-manifold, so that its cusp
neighbourhoods can be adjusted in size to pack together in regular crystalline
patterns, most commonly in a square- looking pattern as for the Borromean
rings.
One shows below how the Bianchi subgroups are related to uqc and the
related MICs and how useful is Dehn filling and conversely drilling in our
context.
source
index cusps
homology
names or ηd(M)
uqc
L6a1
3
3
irr: 1+4
L12n2181 Hesse SIC
.
4
5
irr: 1+5
L14n63905 2QB MIC
Bergé L6a2
4
4
irr: 1+4
otet1600025 2QB MIC
.
5
4
irr: 1+4
otet2000543 ; otet2000041 ; otet2000549 5-dit MIC
.
5
2
irr: 12 + 1
+4
otet2000574
.
.
5
2
irr: 13
+2
+ 12
otet2000665
.
magic L6a5
3
5
irr: 1+5
L14n63788 Hesse SIC
.
4
4
irr: 12 + 1
+4
[31, 174, 4324, 82357, · · · ]
2QB MIC
Borr L6a4
3
4
irr: 1+4
ooct0600466 Hesse SIC
.
3
4
irr: 1+5
ooct0600398
.
.
4
4
irr: 12
+2
+ 1+4
[63, 300, 10747, · · · ]
2QB MIC
.
4
6
irr: 12 + 1
+6
[127, 2871, 478956, · · · ]
.
4-link L8n7
3
6
irr: 1+6
ooct0600340 Hesse SIC
5-link L10n113
3
8
irr: 1+8
ocube0603005;ocube0603322 Hesse SIC
6-link L12n2256 3
9
irr: 1+9
[511, 20122, · · · ] Hesse SIC
Table 3. A few 3-manifolds (in column 5) occuring as a
model of uqc (in column 6). When not identified in SnapPy
the cardinality structure ηd(M) of finite index subgroups of
the 3-manifold M is written explicitely. The fourth column
identifies the first homology type, with 1 meaning the ring of
integers Z. The 3-manifolds occuring as a model of uqc with
source the trefoil knot K3a= 31, the figure-eight knot K4a1
and the Whitehead link L5a1 have been investigated in [4].
4.1. Universal quantum computing from the Bianchi factory. In
this subsection, one specializes on uqc based on the qutrit Hesse SIC (shown
in Fig. 1a) and on the two-qubit geometry of the generalized quadrangle
QUANTUM COMPUTING WITH BIANCHI GROUPS
7
Figure 2. (a) The 3-manifold L14n63788 corresponding to
the 3-fold covering of L6a5 relevant to qutrit universal quan-
tum computing, (b) The 3-manifold corresponding to the 4-
fold covering of L6a5 relevant to two-qubit universal quan-
tum computing.
GQ(2, 2) ( shown in Fig. 1b) obtained with the ‘magic’ link L6a5. The
qutrit uqc follows from a link called L14n63788 (with Poincaré polyhedron
shown in Fig. 2a) and the two-qubit uqc follows from a 3-manifold with
Poincaré polyhedron (shown in Fig. 2b). The former case corresponds to
the 3-fold irregular covering of the fundamental group π1(L6a5) whose 3-
manifold has first homology Z⊕5, 5 cusps and volume ≈ 16.00. The latter
case corresponds to the 4-fold irregular covering whose 3-manifold has first
homology Z2 ⊕ Z⊕4, 4 cusps and volume ≈ 21.34.
For the other links investigated in Table 3 one proceeds in the same way.
4.2. Congruence links in the Bianchi factory. As announced earlier,
there is an interesting sequence of links starting at the Thurston’s link [17]
and ending at the figure-eight knot that one obtains by applying (±1, 1)-
slope Dehn fillings. The Dehn fillings of the last five links called Mi, i = 5..1
were studied in [16]. Observe that M3 is the magic link L6a5, M4 is the
4-link L8n7 and M5 is the 5-link L10n113 of Table 3. The full sequence is
as follows
8MICHEL PLANAT† , RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡ANDKLEE IRWIN‡
Figure 3. The consecutive links from (a) to (h) obtained
from (1,1) Dehn filling at a single cusp [except for (−1, 1)
filling at (e)]. The links (a), (c), (d), (f) are congruence.
The sequence (d) to (h) already appears in [16]. Conversely,
drilling out along the shortest closed geodesic (provided by
SnapPy), one gets the chain (h) → (g) → · · · → (b).
Thurston’s link :
〈
(5 +
√
−3)/2
〉 (1,1)
−→ 7− link L14n64180
(1,1)
−→
〈
2 + 0
√
−1
〉
: 6 link L12n2256
(1,1)
−→
〈
2 + 0
√
−3
〉
: 5− link L10n113
(1,1)
−→ 4− link L8n7
(−1,1)
−→
〈
(1 +
√
−7)/2
〉
: magic 3− link L6a5
(1,1)
−→ Whitehead link L5a1 (1,1)
−→ figure-eight knot K4a1
(1,1)
−→ Poincaré homology sphere.
(1)
It is pictured in Fig. 3. Applying (1, 1)-slope Dehn filling on the figure-
eight knot, the sequence terminates at the spherical manifold M0 that is the
Poincaré homology sphere (also known as Poincaré dodecahedral space [1,
2]). Conversely, drilling out along the shortest closed geodesic, as provided
by SnapPy, an inverse sequence applies as follows (h) → (g) → · · · → (b).
At the next step, (b) transforms into a cyclic 8-link not into the Thurston
link.
QUANTUM COMPUTING WITH BIANCHI GROUPS
9
Many of the links of the sequence correspond to ‘congruence’ manifolds
[18]. A Bianchi subgroup ΓB of PSL(2,Od) is called a ‘congruence’ subgroup
if there exists an ideal I ∈ Od so that ΓB contains the ‘principal congruence
subgroup’ Γ(I) = ker{PSL(2,Od) → PSL(2,Od/I)}. Details about these
links are available on-line at [19]. Of course congruence Bianchi subgroups
generalize congruence subgroups that played a leading role in our approach
of uqc based on Γ [8].
The eight links under scrutiny are pictured in Fig. 3. Four of them are
congruence links and are identified by a pair (−k, I) meaning that they act
on Q(
√
−k) with the ideal I: the Thurston’s link (−3
〈
(5 +
√
−3)/2
〉
), the
6-link L12n2256: (−1,
〈
2 + 0
√
−1
〉
), the 5-link L10n113: (−3,
〈
2 + 0
√
−3
〉
)
and the magic link L6a5: (−1,
〈
(1 +
√
−7)/2
〉
). The link L10n113 is also
otet1000027 and the link L12n2256 is ooct0400042.
The 3-manifolds that we could identify as connected to MICs are in
Table 3. This becomes a cumbersome task for links larger than the 6-
link. For the link L8n7 the cardinality sequence of subgroups is ηd =
[63, 794, 23753, 280162, · · · ], for the link L10n113 it is given by the sequence
ηd = [31, 176, 1987, 7628, 11682, · · · ] and for the 6-link it is given by the se-
quence ηd = [63, 580, 12243, 94274, · · · ]. For the latter link, SnapPy is able
to build the Dirichlet domain (not shown) of symmetry group D6, corre-
sponding to the Hesse SIC while it is not the case for the one attached to
the 5-link.
We expect that the full sequence should have a meaning in the context of
uqc and MICs.
5. Conclusion
It is not yet known which basic piece a practical quantum computer will
be made of. The authors of [4] developed a view of universal quantum
computing (uqc) based on 3-manifolds. In a nutshell, there exists a connec-
tion between the Poincaré conjecture -it states that every simply connected
closed 3-manifold is homeomorphic to the 3-sphere S3- and the Bloch sphere
that houses the qubits. According to this approach, the dressing of qubits
in S3 leads to 3-manifolds (they have been investigated in many details by
W. P. Thurston culminating in the proof of Poincaré conjecture) many of
them corresponding to MICs (minimal informationally complete POVMs)
and the related uqc. In [8] the MICs were based on the modular group Γ that
corresponds to the trefoil knot approach in [4]. In the present paper, the
3-manifolds under investigation derive from subgroups of Bianchi groups, a
generalization of Γ. There seems to exist a nice connection between some
congruence subgroups of Bianchi groups through the chain (1) and quantum
computing that the author started to investigate. At a more practical level,
3-manifolds can also be seen as (three-dimensional) quasiparticles beyond
anyons [20].
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† Université de Bourgogne/Franche-Comté, Institut FEMTO-ST CNRS UMR
6174, 15 B Avenue des Montboucons, F-25044 Besançon, France.
E-mail address: michel.planat@femto-st.fr
‡ Quantum Gravity Research, Los Angeles, CA 90290, USA
E-mail address: raymond@QuantumGravityResearch.org
E-mail address: Klee@quantumgravityresearch.org
E-mail address: Marcelo@quantumgravityresearch.org