# Universal Quantum Computing and Three-Manifolds

^{1}

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## Abstract

**:**

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 alternate universes.(William Thurston [1]).

## 1. Introduction

#### 1.1. From Poincaré Conjecture to UQC

#### 1.2. Minimal Informationally Complete POVMs and UQC

#### 1.3. Organization of the Paper

## 2. Quantum Information from the Modular Group $\Gamma $ and the Related Trefoil Knot ${T}_{1}$

#### 2.1. Cyclic Branched Coverings over the Trefoil Knot

#### 2.2. Irregular branched coverings over the trefoil knot

## 3. Quantum Information from Universal Knots and Links

#### 3.1. Three-Manifolds Pertaining to the Figure-of-Eight Knot

#### A Two-Qubit Tetrahedral Manifold

#### 3.2. Three-Manifolds Pertaining to the Whitehead Link

#### 3.3. A Few Three-Manifolds Pertaining to Borromean Rings

## 4. A Few Dehn Fillings and Their POVMs

#### 4.1. A Few Surgeries on the Trefoil Knot

#### The Poincaré Homology Sphere

#### 4.2. The Seifert Fibered Toroidal Manifold ${\Sigma}^{\prime}$

#### 4.3. Akbulut’s Manifold ${\Sigma}_{Y}$

#### 4.4. The Hyperbolic Manifold ${\Sigma}_{120e}$

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**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 operators that act on the fiducial state, as shown.

**Figure 2.**(

**a**) The figure-of-eight knot: $K4a1$ = otet${02}_{00001}=m004$, (

**b**) the Whitehead link $L5a1$ = ooct${01}_{00001}=m129$, (

**c**) Borromean rings $L6a4$ = ooct${02}_{00005}=t12067$.

**Figure 3.**(

**a**) The trefoil knot ${T}_{1}=K3a1={3}_{1}$, (

**b**) the link $L7n1$ associated to the Hesse SIC, (

**c**) the link $L6a3$ associated to the two-qubit IC.

**Figure 5.**(

**a**) The link $L12n1741$ associated to the qutrit Hesse SIC, (

**b**) The octahedral manifold ooct${03}_{00014}$ associated to the 2-qubit IC.

**Table 1.**Coverings of degree d over the trefoil knot found from SnapPy [28]. The related subgroup of modular group $\Gamma $ and the corresponding IC-POVM [16] (when applicable) is in the right column. The covering is characterized by its type ty, homology group hom (where 1 means $\mathbb{Z}$), the number of cusps cp, the number of generators gens of the fundamental group, the Chern-Simons invariant CS and the type of link it represents (as identified in SnapPy). The links L7n1 (shown in Figure 3b) and L6a3 (shown in Figure 3c) correspond to the Hesse SIC and the two-qubit IC, respectively. 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.

d | ty | hom | cp | Gens | CS | Link | Type in [16] |
---|---|---|---|---|---|---|---|

2 | cyc | $\frac{1}{3}+1$ | 1 | 2 | −1/6 | ||

3 | irr | $1+1$ | 2 | 2 | 1/4 | L7n1 | ${\Gamma}_{0}\left(2\right)$, Hesse SIC |

. | cyc | $\frac{1}{2}+\frac{1}{2}+1$ | 1 | 3 | . | ${A}_{4}$ | |

4 | irr | $1+1$ | 2 | 2 | 1/6 | L6a3 | ${\Gamma}_{0}\left(3\right)$, 2QB IC |

. | irr | $\frac{1}{2}+1$ | 1 | 3 | . | $4{A}^{0}$, 2QB-IC | |

. | cyc | $\frac{1}{3}+1$ | 1 | 2 | . | ${S}_{4}$ | |

5 | cyc | 1 | 1 | 2 | 5/6 | ${A}_{5}$ | |

. | irr | $\frac{1}{3}+1$ | 1 | 3 | . | $5{A}^{0}$, 5-dit IC | |

6 | reg | $1+1+1$ | 3 | 3 | 0 | L8n3 | $\Gamma \left(2\right)$, 6-dit IC |

. | cyc | $1+1+1$ | 1 | 3 | . | ${\Gamma}^{\prime}$, 6-dit IC | |

. | irr | $1+1+1$ | 3 | 3 | . | ||

. | irr | $\frac{1}{2}+1+1$ | 2 | 3 | . | $3{C}^{0}$, 6-dit IC | |

. | irr | $\frac{1}{2}+1+1$ | 2 | 3 | . | ${\Gamma}_{0}\left(4\right)$, 6-dit IC | |

. | irr | $\frac{1}{2}+1+1$ | 2 | 3 | . | ${\Gamma}_{0}\left(5\right)$, 6-dit IC | |

. | irr | $\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+1$ | 1 | 4 | . | ||

. | irr | $\frac{1}{3}+\frac{1}{3}+1$ | 1 | 3 | . | ||

7 | cyc | 1 | 1 | 2 | −5/6 | ||

. | irr | $1+1$ | 2 | 3 | . | NC 7-dit IC | |

. | irr | $\frac{1}{2}+\frac{1}{2}+1$ | 1 | 4 | . | $7{A}^{0}$ 7-dit IC | |

8 | irr | $1+1$ | 2 | 2 | −1/6 | ||

. | cyc | $\frac{1}{3}+1$ | 2 | 2 | . | ||

. | cyc | $\frac{1}{3}+1+1$ | 2 | 3 | . | ||

. | cyc | $\frac{1}{6}+1$ | 1 | 4 | . | $8{A}^{0}$, ∼8-dit IC |

**Table 2.**Table of 3-manifolds ${M}^{3}$ found from subgroups of finite index d of the fundamental group ${\pi}_{1}({S}^{3}\backslash {K}_{0})$ (alias the d-fold coverings of ${K}_{0}$). The terminology in column 3 is that of Snappy [28]. The identified ${M}^{3}$ is made of $2d$ tetrahedra and has cp cusps. When the rank $rk$ of the POVM Gram matrix is ${d}^{2}$ the corresponding IC-POVM shows $pp$ distinct values of pairwise products as shown.

d | ty | ${\mathit{M}}^{3}$ | cp | rk | pp | Comment |
---|---|---|---|---|---|---|

2 | cyc | otet${04}_{00002}$, $m206$ | 1 | 2 | ||

3 | cyc | otet${06}_{00003}$, $s961$ | 1 | 3 | ||

4 | irr | otet${08}_{00002}$, $L10n46$, ${t}_{12840}$ | 2 | 4 | Mom-4s [36] | |

cyc | otet${08}_{00007}$, $t12839$ | 1 | 16 | 1 | 2-qubit IC | |

5 | cyc | otet${10}_{00019}$ | 1 | 21 | ||

irr | otet${10}_{00006}$, $L8a20$ | 3 | 15, 21 | |||

irr | otet${10}_{00026}$ | 2 | 25 | 1 | 5-dit IC | |

6 | cyc | otet${12}_{00013}$ | 1 | 28 | ||

irr | otet${12}_{00041}$ | 2 | 36 | 2 | 6-dit IC | |

irr | otet${12}_{00039}$, otet${12}_{00038}$ | 1 | 31 | |||

irr | otet${12}_{00017}$ | 2 | 33 | |||

irr | otet${12}_{00000}$ | 2 | 36 | 2 | 6-dit IC | |

7 | cyc | otet${14}_{00019}$ | 1 | 43 | ||

irr | otet${14}_{00002}$, $L14n55217$ | 3 | 49 | 2 | 7-dit IC | |

irr | otet${14}_{00035}$ | 1 | 49 | 2 | 7-dit IC |

**Table 3.**A few 3-manifolds ${M}^{3}$ found from subgroups of the fundamental group associated to the Whitehead link. For $d\ge 4$, only the ${M}^{3}$’s leading to an IC are listed.

d | ty | ${\mathit{M}}^{3}$ | cp | rk | pp | Comment |
---|---|---|---|---|---|---|

2 | cyc | ooct${02}_{00003}$, $t12066$, $L8n5$ | 3 | 2 | Mom-4s [36] | |

cyc | ooct${02}_{00018}$, $t12048$ | 2 | 2 | Mom-4s [36] | ||

3 | cyc | ooct${03}_{00011}$, $L10n100$ | 4 | 3 | ||

cyc | ooct${03}_{00018}$ | 2 | 3 | |||

irr | ooct${03}_{00014}$, $L12n1741$ | 3 | 9 | 1 | qutrit Hesse SIC | |

4 | irr | ooct${04}_{00058}$ | 4 | 16 | 2 | 2-qubit IC |

irr | ooct${04}_{00061}$ | 3 | 16 | 2 | 2-qubit IC | |

5 | irr | ooct${05}_{00092}$ | 3 | 25 | 1 | 5-dit IC |

irr | ooct${05}_{00285}$ | 2 | 25 | 1 | 5-dit IC | |

irr | ooct${05}_{00098}$, $L13n11257$ | 4 | 25 | 1 | 5-dit IC | |

6 | cyc | ooct${06}_{06328}$ | 5 | 36 | 2 | 6-dit IC |

irr | ooct${06}_{01972}$ | 3 | 36 | 2 | 6-dit IC | |

irr | ooct${06}_{00471}$ | 4 | 36 | 2 | 6-dit IC |

**Table 4.**Coverings of degrees 2 to 4 branched over the Borromean rings. The identification of the corresponding hyperbolic 3-manifold ${M}^{3}$ is at the 5th column. Only two types of 3-manifolds allow the building of the Hesse SIC. The two 3-manifolds of degree 4 allow the construction of the two-qubit MIC to be identified by the cardinality structure of their subgroups/coverings.

d | ty | hom | cp | ${\mathit{M}}^{3}$ | Comment |
---|---|---|---|---|---|

2 | cyc | $\frac{1}{2}+\frac{1}{2}+1+1+1$ | 3 | ooct04${}_{00259}$ | |

. | . | $\frac{1}{2}+1+1+1+1$ | 4 | ooct04${}_{00055}$ | |

. | . | $1+1+1+1+1$ | 5 | ooct04${}_{00048},\phantom{\rule{3.33333pt}{0ex}}L12n2226$ | |

3 | cyc | $\frac{1}{3}+\frac{1}{3}+1+1+1$ | 3 | ooct06${}_{07427}$ | |

. | . | $\frac{1}{3}+1+1+1+1+1+1$ | 5 | ooct06${}_{00463}$ | |

. | . | $1+1+1+1+1+1+1+1$ | 7 | ooct06${}_{00411}$ | |

. | irr | $1+1+1+1$ | 4 | ooct06${}_{00466}$ | Hesse SIC |

. | . | $1+1+1+1+1+1$ | 4 | ooct06${}_{00398}$ | Hesse SIC |

. | . | $1+1+1+1+1+1$ | 5 | ooct06${}_{00407}$, L14n63856 | |

4 | irr | $\frac{1}{2}+\frac{1}{2}+1+1+1+1$ | 4 | $\{63,300,10747\cdots \}$ | 2QB MIC |

. | . | $\frac{1}{2}+1+1+1+1+1+1$ | 4 | $\{127,2871,478956,\cdots \}$ | 2QB MIC |

**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 [16] and Section 2 for the ICs corresponding to ${T}_{1}$).

T | Name | ${\mathit{\eta}}_{\mathit{d}}\left(\mathit{T}\right)$ |
---|---|---|

${T}_{1}$ | trefoil | {1,1,2,3,2, 8,7,10,10,28, 27,88,134,171,354} |

${T}_{1}(-1,1)$ | $\Sigma (2,3,5)$ | {1,0,0,0,1, 1,0,0,0,1, 0,1,0,0,1} |

${T}_{1}(1,1)$ | $\Sigma (2,3,7)$ | {1,0,0,0,0, 0,2,1,1,0, 0,0,0,9,3} |

${T}_{1}(0,1)$ | ${\Sigma}^{\prime}$ | {1,1,2,2,1, 5,3,2,4,1, 1,12,3,3,4} |

${K}_{0}(0,1)$ | ${\Sigma}_{Y}$ | {1,1,1,2,2, 5,1,2,2,4, 3,17,1,1,2} |

${v}_{2413}(-3,2)$ | ${\Sigma}_{120e}$ | {1,1,1,4,1, 7,2,25,3,10, 10,62,1,30,23} |

**Table 6.**The index 120 torsion-free subgroups of ${\Gamma}_{min}^{+}$ and their relation to the single isosahedron 3-manifolds [40]. The icosahedral symmetry is broken for ${\Sigma}_{120e}$ (see the text for details).

Manifold T | Subgroup | ${\mathit{\eta}}_{\mathit{d}}\left(\mathit{T}\right)$ |
---|---|---|

oicocld${01}_{00001}=s897(-3,2)$ | ${\Sigma}_{120a}$ | {1,0,0,0,0, 8,2,1,1,8} |

oicocld${01}_{00000}=s900(-3,2)$ | ${\Sigma}_{120b}$ | {1,0,0,0,5, 8,10,15,5,24} |

oicocld${01}_{00003}=v2051(-3,2)$ | ${\Sigma}_{120c}$ | {1,0,0,0,0, 4,8,12,6,6} |

oicocld${01}_{00002}=s890(3,2)$ | ${\Sigma}_{120d}$ | {1,0,1,5,0, 9,0,35,9,2} |

$v2413(-3,2)\ne $ oicocld${01}_{00004}$ | ${\Sigma}_{120e}$ | {1,1,1,4,1, 7,2,25,3,10} |

oicocld${01}_{00005}=v3215(1,2)$ | ${\Sigma}_{120f}$ | {1,0,0,0,0, 14,10,5,10,17} |

v3318(−1, 2) | ${\Sigma}_{120g}$ | {1,3,1,2,0, 11,0,23,12,14} |

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**MDPI and ACS Style**

Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K.
Universal Quantum Computing and Three-Manifolds. *Symmetry* **2018**, *10*, 773.
https://doi.org/10.3390/sym10120773

**AMA Style**

Planat M, Aschheim R, Amaral MM, Irwin K.
Universal Quantum Computing and Three-Manifolds. *Symmetry*. 2018; 10(12):773.
https://doi.org/10.3390/sym10120773

**Chicago/Turabian Style**

Planat, Michel, Raymond Aschheim, Marcelo M. Amaral, and Klee Irwin.
2018. "Universal Quantum Computing and Three-Manifolds" *Symmetry* 10, no. 12: 773.
https://doi.org/10.3390/sym10120773