arXiv:1802.04196v1 [quant-ph] 12 Feb 2018UNIVERSAL QUANTUM COMPUTING
AND THREE-MANIFOLDS
MICHEL PLANAT†, RAYMOND ASCHHEIM‡,
MARCELO M. AMARAL‡ AND KLEE IRWIN‡
Abstract. A single qubit may be represented on the Bloch sphere or
similarly on the 3-sphere S3. Our goal is to dress this correspondence by
converting the language of universal quantum computing (uqc) to that
of 3-manifolds. A magic state and the Pauli group acting on it define
a model of uqc as a POVM that one recognizes to be a 3-manifold
M3. E. g., the d-dimensional POVMs defined from subgroups of finite
index of the modular group PSL(2,Z) in [2] correspond to d-fold M3-
coverings of the trefoil knot. In this paper, one also investigates quantum
information on a few ‘universal’ knots and links such as the figure-of-
eight knot, the Whitehead link and Borromean rings [3], making use
of the catalog of platonic manifolds available on SnapPy [4]. Further
connections between POVMs based uqc and M3’s obtained from Dehn
fillings are explored.
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, 81P50, 57M25, 57R65, 14H30, 20E05, 57M12
Keywords: quantum computation, IC-POVMs, knot theory, three-manifolds, branch
coverings, Dehn surgeries.
Manifolds are around us in many guises.
As observers in a three-dimensional world, we are most familiar with two-
manifolds: the surface of a ball or a doughnut or a pretzel, the surface of a
house or a tree or a volleyball net...
Three-manifolds may be harder to understand at first. But as actors and
movers in a three-dimensional world, we can learn to imagine them as al-
ternate universes. [1]
1. Introduction
Mathematical concepts pave the way for improvements in technology. As
far as topological quantum computation is concerned, non abelian anyons
have been proposed as an attractive (fault tolerant) alternative to standard
quantum computing which is based on a universal set of quantum gates
[5, 6, 7, 8]. Anyons are two-dimensional quasiparticles with world lines
forming braids in space-time. Whether non abelian anyons do exist in the
real world and/or would be easy to create artificially is still open to discus-
sion. Three-dimensional topological quantum computing beyond anyons [9]
still is not well developed although, as it will be shown in this essay, it is
a straightforward consequence of a set of ideas belonging to standard uni-
versal quantum computation (uqc) and simultaneously to three-dimensional
topology. For quantum computation, we have in mind the concepts of magic
states and the related positive-operator valued measures (POVMs) that were
1
2MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡ANDKLEE IRWIN‡
investigated in detail in [10, 11, 12]. For three-dimensional topology, the
starting point consists of the Poincaré and Thurston conjectures, now theo-
rems [1]. Three-dimensional topological quantum computing would federate
the foundations of quantum mechanics and cosmology, a recurrent dream of
many physicists. Three-dimensional topology was already investigated by
several groups in the context of quantum information [13, 14], high energy
physics [15, 16], biology [17] and consciousness studies [18].
1.1. From Poincaré conjecture to uqc. Poincaré conjecture is the ele-
mentary (but deep) statement that every simply connected, closed 3-manifold
is homeomorphic to the 3-sphere S3 [19]. Having in mind the correspondence
between S3 and the Bloch sphere that houses the qubits ψ = a |0〉 + b |1〉,
a, b ∈ C, |a|2+ |b|2 = 1, one would desire a quantum translation of this state-
ment. For doing this, one may use the picture of the Riemann sphere C∪∞
in parallel to that of the Bloch sphere and follow F. Klein lectures on the
icosahedron to perceive the platonic solids within the landscape [20]. This
picture fits well the Hopf fibrations [21], their entanglements described in
[22, 23] and quasicrystals [25, 24]. But we can be more ambitious and dress
S3 in an alternative way that reproduces the historic thread of the proof
of Poincaré conjecture. Thurston’s geometrization conjecture, from which
Poincaré conjecture follows, dresses S3 as a 3-manifold not homeomorphic
to S3. The wardrobe of 3-manifolds M3 is huge but almost every dress is
hyperbolic and W. Thurston found the recipes for them [1]. Every dress is
identified thanks to a signature in terms of invariants. For our purpose, the
fundamental group π1 of M
3 does the job.
The three-dimensional space surrounding a knot K -the knot complement
S3 \ K- is an example of a three-manifold [1, 26]. We will be especially
interested by the trefoil knot that underlies work of the first author [2]
as well as the figure-of-eight knot, the Whitehead link and the Borromean
rings because they are universal (in a sense described below), hyperbolic and
allows to build 3-manifolds from platonic manifolds [4]. Such manifolds carry
a quantum geometry corresponding to quantum computing and (possibly
informationally complete: IC) POVMs identified in our earlier work [2, 11,
12].
According to [3], the knot K and the fundamental group G = π1(S
3 \K)
are universal if every closed and oriented 3-manifold M3 is homeomorphic
to a quotient H/G of the hyperbolic 3-space H by a subgroup H of finite
index d of G. The figure-of-eight knot and the Whitehead link are universal.
The catalog of the finite index subgroups of their fundamental group G and
of the corresponding 3-manifolds defined from the d-fold coverings [27] can
easily been established up to degree 8, using the software SnapPy [28].
In paper [2] of the first author, it has been found that d-dimensional
IC-POVMs may be built from finite index subgroups of the modular group
Γ = PSL(2,Z). To an IC is associated a subgroup of index d of Γ, a funda-
mental domain in the Poincaré upper-half plane and a signature in terms of
genus, elliptic points and cusps as summarized in [2, Fig. 1]. There exists a
relationship between the modular group Γ and the trefoil knot T1 since the
fundamental group π1(S
3 \ T1) of the knot complement is the braid group
B3, the central extension of Γ. But the trefoil knot and the corresponding
UNIVERSAL QUANTUM COMPUTING
AND THREE-MANIFOLDS
3
Figure
1. (a)
The
figure-of-eight
knot:
K4a1=otet0200001 = m004,
(b) the Whitehead link
L5a1 =ooct0100001 = m129,
(c) Borromean rings
L6a4 =ooct0200005 = t12067.
braid group B3 are not universal [29] which forbids the relation of the finite
index subgroups of B3 to all three-manifolds.
It is known that two coverings of a manifold M with fundamental group
G = π1(M) are equivalent if there exists a homeomorphism between them.
Besides, a d-fold covering is uniquely determined by a subgroup of index d
of the group G and the inequivalent d-fold coverings of M correspond to
conjugacy classes of subgroups of G [27].
In this paper we will fuse the
concepts of a three-manifold M3 attached to a subgroup H of index d and
the POVM, possibly informationally complete (IC), found from H (thanks
to the appropriate magic state and related Pauli group factory).
Figure 2. (a) The trefoil knot T1 = 31, (b) the link L7n1
associated to the Hesse SIC, (c) the link L6a3 associated to
the two-qubit IC.
1.2. Minimal informationally complete POVMs and uqc. In our ap-
proach [12, 2], minimal informationally complete (IC) POVMs are derived
4MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡ANDKLEE IRWIN‡
from appropriate fiducial states under the action of the (generalized) Pauli
group. The fiducial states also allow to perform universal quantum compu-
tation [11].
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 [30, eq. (29)].
New minimal IC-POVMs (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
[2]. A SIC is equiangular with |A| = 1 and a1 =
1
√
d+1
. The states en-
countered 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 [12].
1.3. Organization of the paper. The paper runs as follows. Sec. 2 deals
about the relationship between quantum information seen from the mod-
ular group Γ and from the trefoil knot 3-manifold. Sec. 3 deals about
the (platonic) 3-manifolds related to coverings of the figure-of-eight knot,
Whitehead link and Borromean rings, and how they relate to a few known
IC-POVMs. Sec. 4 describes the important role played by Dehn fillings
for describing the many types of 3-manifolds that may relate to topological
quantum computing.
2. Quantum information from the modular group Γ and the
related trefoil knot T1
In this section, we describe the results established in [2] in terms of the 3-
manifolds corresponding to coverings of the trefoil knot complement S3 \T1.
Let us introduce to the group representation of a knot complement π1(S
3\
K). A Wirtinger representation is a finite representation of π1 where the
relations are the form wgiw
−1 = gj where w is a word in the k generators
{g1, · · · , gk}. For the trefoil knot T1, a Wirtinger representation is
π1(S
3 \ T1) = 〈x, y|yxy = xyx〉 or equivalently π1 =
〈
x, y|y2 = x3
〉
.
In the rest of the paper, the number of d-fold coverings of the manifold
M3 corresponding to the knot T will be displayed as the ordered list ηd(T ),
d ∈ {1..10 . . .}. For T1 it is
ηd(T1) = {1, 1, 2, 3, 2, 8, 7, 10, 18, 28, . . .}.
Details about the corresponding d-fold coverings are in Table 1. As ex-
pected, the coverings correspond to subgroups of index d of the fundamental
group associated to the trefoil knot T1 = 31.
UNIVERSAL QUANTUM COMPUTING
AND THREE-MANIFOLDS
5
d
ty
hom
cp gens
CS link
type in [2]
2 cyc
1
3
+ 1
1 2
-1/6
D4
3
irr
1 + 1
2 2
1/4 L7n1 Γ0(2), Hesse SIC
.
cyc
1
2
+ 1
2
+ 1
1 3
.
A4
4
irr
1 + 1
2 2
1/6 L6a3
Γ0(3), 2QB IC
.
irr
1
2
+ 1
1 3
.
4A0, 2QB-IC
.
cyc
1
3
+ 1
1 2
.
S4
5 cyc
1
1 2
5/6
A5
.
irr
1
3
+ 1
1 3
.
5A0, 5-dit IC
6 reg
1 + 1 + 1
3 3
0
Γ′, 6-dit IC
.
irr
1 + 1 + 1
2 3
. L6n1
Γ(2), 6-dit IC
.
irr
1
2
+ 1 + 1
2 3
. L6n1
Γ0(4), 6-dit IC
.
irr
1
2
+ 1 + 1
2 3
.
3C0, 6-dit IC
.
irr
1
2
+ 1 + 1
2 3
.
Γ0(5), 6-dit IC
.
cyc
1 + 1 + 1
1 3
.
.
irr
1
2
+ 1
2
+ 1
2
+ 1
1 4
.
.
irr
1
3
+ 1
3
+ 1
1 3
.
7 cyc
1
1 2
-5/6
.
irr
1 + 1
2 3
.
NC 7-dit IC
.
irr
1
2
+ 1
2
+ 1
1 4
.
7A0 7-dit IC
8
irr
1 + 1
2 2
-1/6
.
cyc
1
3
+ 1
2 2
.
.
cyc
1
3
+ 1 + 1
2 3
.
.
cyc
1
6
+ 1
1 4
.
8A0, ∼ 8-dit IC
Table 1. Covers of degree d over the fundamental group of
the trefoil knot found from SnapPy [28]. The related sub-
group of modular group Γ and the corresponding IC-POVM
[2] (when applicable) is in the right column. The cover is
characterized by its type ty, homology group hom (where
1 means Z), the number of cusps cp, the number of gen-
erators gens of the fundamental group, the Chern-Simons
invariant CS and the type of link it represents (as identi-
fied in SnapPy). The case of cyclic coverings corresponds to
Brieskorn 3-manifolds as explained in the text: the spherical
groups for these manifolds is given at the right hand side
column.
Cyclic branched covers over the trefoil knot. Let p, q, r be three positive inte-
gers (with p ≤ q ≤ r), the Brieskorn 3-manifold Σ(p, q, r) is the intersection
in C3 of the 5-sphere S5 with the surface of equation zp1 + z
q
2 + z
r
3 = 1.
In [32], it is shown that a r-fold cyclic covering over S3 branched along
a torus knot or link of type (p, q) is a Brieskorn 3-manifold Σ(p, q, r) (see
also Sec 4.1). For the spherical case p−1 + q−1 + r−1 > 1, the group as-
sociated to a Brieskorn manifold is either dihedral [that is the group Dr
for the triples (2, 2, r)], tetrahedral [that is A4 for (2, 3, 3)], octahedral [that
is S4 for (2, 3, 4)] or icosahedral [that is A5 for (2, 3, 5)]. The Euclidean
6MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡ANDKLEE IRWIN‡
case p−1 + q−1 + r−1 = 1 corresponds to (2, 3, 6), (2, 4, 4) or (3, 3, 3). The
remaining cases are hyperbolic.
The cyclic branched coverings with spherical groups for the trefoil knot
(2, 3) are identified in right hand side column of Table 1.
Irregular branched covers over the trefoil knot. The right hand side column of
Table 1 shows the subgroups of Γ identified in [2, Table 1] as corresponding
to an IC-POVM. In particular, the Hesse SIC [already found associated
to the congruence subgroup Γ0(2)] corresponds to the 7-crossing link L7n1
while the two-qubit IC [already found associated to the congruence subgroup
Γ0(3)] corresponds to the 6- crossing link L6a3. The trefoil knot and the
former two links are pictured in Fig. 2.
3. Quantum information from universal knots and links
3.1. Three-manifolds pertaining to the figure-of-eight knot. The
fundamental group for the figure-of-eight knot K0 is
π1(S
3 \K0) =
〈
x, y|y ∗ x ∗ y−1xy = xyx−1yx
〉
.
and the number of d-fold coverings is in the list
ηd(K0) = {1, 1, 1, 2, 4, 11, 9, 10, 11, 38, . . .}.
Table 2 establishes the list of 3-manifolds corresponding to subgroups
of index d ≤ 7 of the universal group G = π1(S
3 \ K0). The manifolds
are labeled otetNn in [4] because they are oriented and built from N = 2d
tetrahedra, with n an index in the table. The identification of 3-manifolds of
finite index subgroups of G was first obtained by comparing the cardinality
list ηd(H) of the corresponding subgroup H to that of a fundamental group
of a tetrahedral manifold in SnapPy table [28]. But of course there is more
straightforward way to perform this task by identifying a subgroup H to a
degree d cover of K0 [27]. The full list of d-branched covers of the figure
eight knot up to degree 8 is available in SnapPy. Extra invariants of the
corresponding M3 may be found there. In addition, the lattice of branched
covers over K0 was investigated in [33].
Let us give more details about the results summarized in Table 2. Us-
ing Magma, the conjugacy class of subgroups of index 2 in the fundamen-
tal group G is represented by the subgroup on three generators and two
relations as follows H =
〈
x, y, z|y−1zx−1zy−1x−2, z−1yxz−1yz−1xy
〉
, from
which the sequence of subgroups of finite index can be found as ηd(M
3) =
{1, 1, 5, 6, 8, 33, 21, 32, · · · }. The manifoldM3 corresponding to this sequence
is found in Snappy as otet0400002 , alias m206.
The conjugacy class of subgroups of index 3 in G is represented as
H =
〈
x, y, z|x−2zx−1yz2x−1zy−1, z−1xz−2xz−2y−1x−2zy
〉
,
with ηd(M
3) = {1, 7, 4, 47, 19, 66, 42, 484, · · · } corresponding to the manifold
otet0600003 , alias s961.
As shown in Table 2, there are two conjugacy classes of subgroups of
index 4 in G corresponding to tetrahedral manifolds otet0800002 (the per-
mutation group P organizing the cosets is Z4) and otet0800007 (the permu-
tation group organizing the cosets is the alternating group A4). The latter
group/manifold has fundamental group
UNIVERSAL QUANTUM COMPUTING
AND THREE-MANIFOLDS
7
d ty
M3
cp
rk pp
comment
2 cyc
otet0400002 , m206
1
2
3 cyc
otet0600003 , s961
1
3
4
irr
otet0800002 , L10n46, t12840
2
4
Mom-4s [34]
cyc
otet0800007 , t123839
1
16 1
2-qubit IC
5 cyc
otet1000019
1
21
irr
otet1000006 , L8a20
3
15,21
irr
otet1000026
2
25 1
5-dit IC
6 cyc
otet1200013
1
28
irr
otet1200041
2
36 2
6-dit IC
irr
otet1200039 , otet1200038
1
31
irr
otet1200017
2
33
irr
otet1200000
2
36 2
6-dit IC
7 cyc
otet1400019
1
43
irr
otet1400002 , L14n55217
3
49 2
7-dit IC
irr
otet1400035
1
49 2
7-dit IC
Table 2. Table of 3-manifolds M3 found from subgroups of
finite index d of the fundamental group π1(S
3 \K0) (alias the
d-fold coverings of K0). The terminology in column 3 is that
of Snappy [28]. The identified M3 is made of 2d tetrahedra
and has cp cusps. When the rank rk of the POVM Gram
matrix is d2 the corresponding IC-POVM shows pp distinct
values of pairwise products as shown.
Figure 3. Two platonic three-manifolds leading to the con-
struction of the two-qubit IC-POVM. Details are given in
Tables 2 and 3.
H =
〈
x, y, z|yx−1y−1z−1xy−2xyzx−1y, zx−1yx−1yx−1zyx−1y−1z−1xy−1
〉
,
with cardinality sequences of subgroups as ηd(M
3) = {1, 3, 8, 25, 36, 229, 435 · · · }.
To H is associated an IC-POVM [12, 2] which follows from the action of the
8MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡ANDKLEE IRWIN‡
two-qubit Pauli group on a magic/fiducial state of type (0, 1,−ω6, ω6 − 1),
with ω6 = exp(2iπ/6) a six-root of unity.
For index 5, there are three types of 3-manifolds corresponding to the
subgroups H. The tetrahedral manifold otet1000026 of sequence ηd(M
3) =
{1, 7, 15, 88, 123, 802, 1328 · · · }, is associated to a 5-dit equianguler IC-POVM,
as in [12, Table 5].
For index 6, the 11 coverings define six classes of 3-manifolds and two of
them: otet1200041 and otet1200000 are related to the construction of ICs. For
index 7, one finds three classes of 3-manifolds with two of them: otet1400002
(alias L14n55217) and otet1400035 are related to ICs. Finally, for index 7,
3 types of 3-manifolds exist, two of them relying on the construction of the
7-dit (two-valued) IC. For index 8, there exists 6 distinct 3-manifolds (not
shown) none of them leading to an IC.
d ty
M3
cp
rk pp
comment
2 cyc ooct0200003, t12066, L8n5
3
2
Mom-4s[34]
cyc
ooct0200018, t12048
2
2
Mom-4s[34]
3
cyc
ooct0300011, L10n100
4
3
cyc
ooct0300018
2
3
irr
ooct0300014, L12n1741
3
9 1
qutrit Hesse SIC
4
irr
ooct0400058
4
16 2
2-qubit IC
irr
ooct0400061
3
16 2
2-qubit IC
5
irr
ooct0500092
3
25 1
5-dit IC
irr
ooct0500285
2
25 1
5-dit IC
irr
ooct0500098, L13n11257
4
25 1
5-dit IC
6 cyc
ooct0606328
5
36 2
6-dit IC
irr
ooct0601972
3
36 2
6-dit IC
irr
ooct0600471
4
36 2
6-dit IC
Table 3. A few 3-manifolds M3 found from subgroups of
the fundamental group associated to the Whitehead link. For
d ≥ 4, only the M3’s leading to an IC are listed.
A two-qubit tetrahedral manifold. The tetrahedral three-manifold otet0800007
is remarkable in the sense that it corresponds to the subgroup of index 4
of G that allows the construction of the two-qubit IC-POVM. The corre-
sponding hyperbolic polyhedron taken from SnapPy is shown in Fig. 3a. Of
the 29 orientable tetrahedral manifolds with at most 8 tetrahedra, 20 are
two-colorable and each of those has at most 2 cusps. The 4 three-manifolds
(with at most 8 tetrahedra) identified in Table 2 belong to the 20’s and
the two-qubit tetrahedral manifold otet0800007 is one with just one cusp [35,
Table 1].
3.2. Three-manifolds pertaining to the Whitehead link. One could
also identify the 3-manifold substructure of another universal object, viz the
Whitehead link L0 [31].
The cardinality list corresponding to the Whitehead link group π1(L0) is
ηd(L0) = {1, 3, 6, 17, 22, 79, 94, 412, 616, 1659 . . .},
UNIVERSAL QUANTUM COMPUTING
AND THREE-MANIFOLDS
9
Figure 4.
(a) The link L12n1741 associated to the qutrit
Hesse SIC, (b) The octahedral manifold ooct0300014 associ-
ated to the 2-qubit IC.
Table 3 shows that the identified 3-manifolds for index d subgroups of π1(L0)
are aggregates of d octahedra. In particular, one finds that the qutrit Hesse
SIC can be built from ooct0300014 and that the two-qubit IC-POVM may be
buid from ooct0400058. The hyperbolic polyhedron for the latter octahedral
manifold taken from SnapPy is shown on Fig. 3b. The former octahedral
manifold follows from the link L12n1741 shown in Fig. 4a and the corre-
sponding polyhedron taken from SnapPy is shown in Fig. 4b.
d ty
hom
cp
M3
comment
2 cyc
1
2
+ 1
2
+ 1 + 1 + 1
3
ooct0400259
.
.
1
2
+ 1 + 1 + 1 + 1
4
ooct0400055
.
.
1 + 1 + 1 + 1 + 1
5
ooct0400048, L12n2226
3 cyc
1
3
+ 1
3
+ 1 + 1 + 1
3
ooct0607427 Hesse SIC
.
.
1
3
+ 1 + 1 + 1 + 1 + 1
5
ooct0600463 Hesse SIC
.
.
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 7
ooct0600411
.
irr
1 + 1 + 1 + 1
4
ooct0600466 Hesse SIC
.
.
1 + 1 + 1 + 1 + 1 + 1
4
ooct0600398
.
.
1 + 1 + 1 + 1 + 1 + 1
5
ooct0600407, L14n63856
Table 4. Covers of degrees 2 and 3 over the M3 branched
along the Borromean rings. The identification of the corre-
sponding hyperbolic 3-manifold is at the 5th column. It is
seen at the right hand side column that only three types of
3-manifolds allow to build the Hesse SIC.
3.3. A few three-manifolds pertaining to Borromean rings. Three-manifolds
corresponding to coverings of degree 2 and 3 of the 3-manifold branched along the
Borromean rings L6a4 [that is a not a (3,3)-torus link but an hyperbolic link] (see
10MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡ANDKLEE IRWIN‡
Fig. 1c) are given in Table 4. The identified manifolds are hyperbolic octahedral
manifolds of volume 14.655 (for the degree 2) and 21.983 (for the degree 3).
4. A few Dehn fillings and their POVMs
To summarize our findings of the previous section, we started from a building
block, a knot (viz the trefoil knot T1) or a link (viz the figure-of-eight knot K0)
whose complement in S3 is a 3-manifold M3. Then a d-fold covering of M3 was
used to build a d-dimensional POVM, possibly an IC. Now we apply a kind of
‘phase surgery’ on the knot or link that transforms M3 and the related coverings
while preserving some of the POVMs in a way to be determined. We will start
with our friend T1 and arrive at a few standard 3-manifolds of historic importance,
the Poincaré homology sphere [alias the Brieskorn sphere Σ(2, 3, 5)], the Brieskorn
sphere Σ(2, 3, 7) and a Seifert fibered toroidal manifold Σ′. Then we introduce
the 3-manifold ΣY resulting from 0-surgery on the figure-of-eight knot [40]. Later
in this section, we will show how to use the {3, 5, 3} Coxeter lattice and surgery
to arrive at a hyperbolic 3-manifold Σ120e of maximal symmetry whose several
coverings (and related POVMs) are close to the ones of the trefoil knot [36].
T
Name
ηd(T )
T1
trefoil
{1,1,2,3,2, 8,7,10,10,28, 27,88,134,171,354}
T1(−1, 1)
Σ(2, 3, 5)
{1,0,0,0,1, 1,0,0,0,1, 0,1,0,0,1}
T1(1, 1)
Σ(2, 3, 7)
{1,0,0,0,0, 0,2,1,1,0, 0,0,0,9,3}
T1(0, 1)
Σ′
{1,1,2,2,1, 5,3,2,4,1, 1,12,3,3,4}
K0(0, 1)
ΣY
{1,1,1,2,2, 5,1,2,2,4, 3,17,1,1,2}
v2413(−3, 2)
Σ120e
{1,1,1,4,1, 7,2,25,3,10 10,62,1,30,23}
Table 5. A few surgeries (column 1), their name (column 2)
and the cardinality list of d-coverings (alias conjugacy classes
of subgroups). Plain characters are used to point out the
possible construction of an IC-POVM in at least one the
corresponding three-manifolds (see [2] and Sec. 2 for the ICs
corresponding to T1).
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 [1,
p. 275],[26, p 259],[37]. According to Lickorish’s theorem, every closed, orientable,
connected 3-manifold is obtained by performing Dehn surgery on a link in the
3-sphere.
4.1. A few surgeries on the trefoil knot.
The Poincaré homology sphere. The Poincaré dodecahedral space (alias the Poincaré
homology sphere) was the first example of a 3-manifold not the 3-sphere. It can be
obtained from (−1, 1) surgery on the trefoil knot T1 [38].
Let p, q, r be three positive integers and mutually coprime, the Brieskorn sphere
Σ(p, q, r) is the intersection in C3 of the 5-sphere S5 with the surface of equation
zp1 + z
q
2 + z
r
3 = 1. The homology of a Brieskorn sphere is that of the sphere S
3.
UNIVERSAL QUANTUM COMPUTING
AND THREE-MANIFOLDS
11
A Brieskorn sphere is homeomorphic but not diffeomorphic to S3. The sphere
Σ(2, 3, 5) may be identified to the Poincaré homology sphere. The sphere Σ(2, 3, 7)
[40] may be obtained from (1, 1) surgery on T1. Table 5 provides the sequences ηd
for the corresponding surgeries (±1, 1) on T1. Plain digits in these sequences point
out the possibility of building ICs of the corresponding degree. This corresponds
to a considerable filtering of the ICs coming from T1.
For instance, the smallest IC from Σ(2, 3, 5) has dimension five and is precisely
the one coming from the congruence subgroup 5A0 in Table 1. But it is built from a
non modular (fundamental) group whose permutation representation of the cosets
is the alternating group A5 ∼= 〈(1, 2, 3, 4, 5), (2, 4, 3)〉 (compare [12, Sec. 3.3]).
The smallest dimensional IC derived from Σ(2, 3, 7) is 7-dimensional and two-
valued, the same than the one arising from the congruence subgroup 7A0 given in
Table 1. But it arises from a non modular (fundamental) group with the permuta-
tion representation of cosets as PSL(2, 7) ∼= 〈(1, 2, 4, 6, 7, 5, 3), (2, 5, 3)(4, 6, 7)〉.
4.2. The Seifert fibered toroidal manifold Σ′. An hyperbolic knot (or link)
in S3 is one whose complement is 3-manifold M3 endowed with a complete Rie-
mannian metric of constant negative curvature, i.e. it has a hyperbolic geometry
and finite volume. A Dehn surgery on a hyperbolic knot is exceptional if it is re-
ducible, toroidal or Seifert fibered (comprising a closed 3-manifold together with a
decomposition into a disjoint union of circles called fibers). All other surgeries are
hyperbolic. These categories are exclusive for a hyperbolic knot. In contrast, a non
hyperbolic knot such as the trefoil knot admits a toroidal Seifert fiber surgery Σ′
obtained by (0, 1) Dehn filling on T1 [39].
The smallest dimensional ICs built from Σ′ are the Hesse SIC that is obtained
from the congruence subgroup Γ0(2) (as for the trefoil knot) and the two-qubit IC
that comes from a non modular fundamental group [with cosets organized as the
alternating group A4 ∼= 〈(2, 4, 3), (1, 2, 3)〉].
4.3. Akbulut’s manifold ΣY . Exceptional Dehn surgery at slope (0, 1) on the
figure-of-eight knot K0 leads to a remarkable manifold ΣY found in [40] in the
context of 3-dimensional integral homology spheres smoothly bounding integral
homology balls. Apart from its topological importance, we find that some of its
coverings are associated to already discovered ICs and those coverings have the
same fundamental group π1(ΣY ).
The smallest IC-related covering (of degree 4) occurs with integral homology Z
and the congruence subgroup Γ0(3) also found from the trefoil knot (see Table 1).
Next, the covering of degree 6 and homology Z
5
+Z leads to the 6-dit IC of type 3C0
(also found from the trefoil knot). The next case corresponds to the (non-modular)
11-dimensional (3-valued) IC.
4.4. The hyperbolic manifold Σ120e. The hyperbolic manifold closest to the
trefoil knot manifold known to us was found in [36]. The goal in [36] is the search
of -maximally symmetric- fundamental groups of 3-manifolds. In two dimensions,
maximal symmetry groups are called Hurwitz groups and arise as quotients of
the (2, 3, 7)-triangle groups.
In three dimensions, the quotients of the minimal
co-volume lattice Γmin of hyperbolic isometries, and of its orientation preserving
subgroup Γ+min, play the role of Hurwitz groups. Let C be the {3, 5, 3} Coxeter
group, Γmin the split extension C ⋊ Z2 and Γ
+
min one of the index two subgroups
of Γmin of presentation
Γ+min =
〈
x, y, z|x3, y2, z2, (xyz)2, (xzyz)2, (xy)5
〉
.
12MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡ANDKLEE IRWIN‡
According to [36, Corollary 5], all torsion-free subgroups of finite index in Γ+min
have index divisible by 60. There are two of them of index 60, called Σ60a and
Σ60b, obtained as fundamental groups of surgeries m017(−3, 2) and m016(−3, 2).
Torsion-free subgroups of index 120 in Γ+min are given in Table 6. It is remarkable
that these groups are fundamental groups of oriented three-manifolds built with a
single icosahedron except for Σ120e.
manifold T
subgroup
ηd(T )
oicocld0100001 = s897(−3, 2)
Σ120a
{1,0,0,0,0, 8,2,1,1,8}
oicocld0100000 = s900(−3, 2)
Σ120b
{1,0,0,0,5, 8,10,15,5,24}
oicocld0100003 = v2051(−3, 2)
Σ120c
{1,0,0,0,0, 4,8,12,6,6}
oicocld0100002 = s890(3, 2)
Σ120d
{1,0,1,5,0, 9,0,35,9,2}
v2413(−3, 2)
6= oicocld0100004
Σ120e
{1,1,1,4,1, 7,2,25,3,10}
oicocld0500005 = v3215(1, 2)
Σ120f
{1,0,0,0,0, 14,10,5,10,17}
Table 6. The index 120 torsion free subgroups of Γ+min and
their relation to the single isosahedron 3-manifolds [36]. The
icosahedral symmetry is broken for Σ120e (see the text for
details).
Σ120e is also special in the sense that many small dimensional ICs may be built
from it in contrast to the other groups in Table 6. The smallest ICs that may be
build from Σ120e are the Hesse SIC coming from the congruence subgroup Γ0(2), the
two-qubit IC coming the congruence subgroup 4A0 and the 6-dit ICs coming from
the congruence subgroups Γ(2), 3C0 or Γ0(4) (see [2, Sec. 3 and Table 1]). Higher
dimensional ICs found from Σ120e does not come from congruence subgroups.
5. Conclusion
The relationship between 3-manifolds and universality in quantum computing
has been explored in this work. Earlier work of the first author already pointed out
the importance of hyperbolic geometry and the modular group Γ for deriving the
basic small dimensional IC-POVMs. In Sec. 2, the move from Γ to the trefoil knot
T1 (and the braid group B3) to non hyperbolic 3-manifolds could be investigated by
making use of the d-fold coverings of T1 that correspond to d-dimensional POVMs
(some of them being IC). Then, in Sec. 3, we passed to universal links (such as
the figure-of-eight knot, Whitehead link and Borromean rings) and the related hy-
perbolic platonic manifolds as new models for quantum computing based POVMs.
Finally, In Sec. 4, Dehn fillings on T1 were used to explore the connection of quan-
tum computing to important exotic 3-manifolds [i.e. Σ(2, 3, 5) and Σ(2, 3, 7)], to
the toroidal Seifert fibered Σ′, to Akbulut’s manifold ΣY and to a maximum sym-
metry hyperbolic manifold Σ120e slightly breaking the icosahedral symmetry. It is
expected that our work will have importance for new ways of implementing quan-
tum computing and for the understanding of the link between quantum information
theory and cosmology [41, 42, 43].
Funding
The first author acknowledges the support by the French “Investissements d’Avenir”
program, project ISITE-BFC (contract ANR-15-IDEX-03). The other ressources
came from Quantum Gravity Research.
UNIVERSAL QUANTUM COMPUTING
AND THREE-MANIFOLDS
13
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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