# Character Varieties and Algebraic Surfaces for the Topology of Quantum Computing

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## Abstract

**:**

## 1. Introduction

## 2. Prolegomena

#### 2.1. Algebraic Surfaces

#### 2.2. The Hopf Link

#### 2.3. Magic State Quantum Computing

## 3. Character Varieties for Fundamental Groups of Three-Manifolds and the Related Algebraic Surfaces

#### 3.1. The $S{L}_{2}\left(\mathbb{C}\right)$ Character Varieties of Knot Groups Whose Reducible Component Is that of the Hopf Link

#### 3.2. The $S{L}_{2}\left(\mathbb{C}\right)$ Character Variety of Singular Fiber $I{V}^{*}={\tilde{E}}_{6}$

#### 3.3. The $S{L}_{2}\left(\mathbb{C}\right)$ Character Variety of Singular Fiber ${I}_{0}^{*}={\tilde{D}}_{4}$

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Culler, M.; Shalen, P.B. Varieties of group representations and splitting of 3-manifolds. Ann. Math.
**1983**, 117, 109–146. [Google Scholar] [CrossRef] - Harada, S. Canonical components of character varieties of arithmetic two-bridge link complements. arXiv
**2012**, arXiv:1112.3441. [Google Scholar] - Ashley, C.; Burelle, J.P.; Lawton, S. Rank 1 character varieties of finitely presented groups. Geom. Dedicata
**2018**, 192, 1–19. [Google Scholar] [CrossRef] [Green Version] - Python Code to Compute Character Varieties. Available online: http://math.gmu.edu/~slawton3/Main.sagews (accessed on 1 May 2021).
- Enriques–Kodaira Classification. Available online: https://en.wikipedia.org/wiki/Enriques–Kodaira_classification (accessed on 1 May 2021).
- Topological Quantum Computer. Available online: https://en.wikipedia.org/wiki/Topological_quantum_computer (accessed on 1 January 2021).
- Pachos, J.K. Introduction to Topological Quantum Computation; Cambridge University Press: Cambridge, UK, 2012. [Google Scholar]
- Wang, Z. Topological Quantum Computation; Number 112; American Mathematical Society: Providence, Rhode Island, 2010. [Google Scholar]
- Asselmeyer-Maluga,T. Topological quantum computing and 3-manifolds. Quant. Rep.
**2021**, 3, 9. [Google Scholar] [CrossRef] - Asselmeyer-Maluga, T. 3D topological quantum computing. Int. J. Quant. Inf.
**2021**, 19, 2141005. [Google Scholar] [CrossRef] - Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K. Universal quantum computing and three-manifolds. Symmetry
**2018**, 10, 773. [Google Scholar] [CrossRef] [Green Version] - Planat, M.; Aschheim, R.; Amaral, M.M. Irwin, Group geometrical axioms for magic states of quantum computing. Mathematics
**2019**, 7, 948. [Google Scholar] [CrossRef] [Green Version] - Bosma, W.; Cannon, J.J.; Fieker, C.; Steel, A. (Eds.) Handbook of Magma Functions. 2017. Available online: http://magma.maths.usyd.edu.au/magma/ (accessed on 1 January 2019).
- Liskovets, V.; Mednykh, A. On the number of connected and disconnected coverings over a manifold. Ars Math. Contemp.
**2009**, 2, 181–189. [Google Scholar] [CrossRef] [Green Version] - Baake, M.; Grimm, U. Aperiodic Order, Vol. I: A Mathematical Invitation; Cambrige University Press: Cambridge, UK, 2013. [Google Scholar]
- Goldman, W.M. Trace coordinates on Fricke spaces of some simple hyperbolic surfaces. In Handbook of Teichmüller Theory; European Mathematical Society: Zürich, Switzerland, 2009; Volume 13, pp. 611–684. [Google Scholar]
- Asselmeyer-Maluga, T. Quantum computing and the brain: Quantum nets, dessins d’enfants and neural networks. EPI Web Conf.
**2019**, 198, 00014. [Google Scholar] [CrossRef] [Green Version] - Bravyi, S.; Kitaev, A. Universal quantum computation with ideal Clifford gates and nosy ancillas. Phys. Rev.
**2005**, A71, 022316. [Google Scholar] [CrossRef] [Green Version] - Planat, M.; Gedik, Z. Magic informationally complete POVMs with permutations. R. Soc. Open Sci.
**2017**, 4, 170387. [Google Scholar] [CrossRef] [Green Version] - Planat, M. The Poincaré half-plane for informationally complete POVMs. Entropy
**2018**, 20, 16. [Google Scholar] [CrossRef] [Green Version] - Grunewald, F.; Schwermer, J. Subgroups of Bianchi groups and arithmettic quotients of hyperbolic 3-space. Trans. Am. Math. Soc.
**1993**, 335, 47–78. [Google Scholar] [CrossRef] [Green Version] - Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K. Quantum computing with Bianchi groups. EPJ Web Conf.
**2018**, 2018 198, 0012. [Google Scholar] - Culler, M.; Dunfield, N.M.; Goerner, M.; Weeks, J.R. SnapPy, a Computer Program for Studying the Geometry and Topology of three-Manifolds. Available online: http://snappy.computop.org (accessed on 1 January 2022).
- Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K. Quantum computing, Seifert surfaces and singular fibers. Quantum Rep.
**2019**, 1, 3. [Google Scholar] [CrossRef] [Green Version] - Scorpian, A. The Wild World of 4-Manifolds; American Mathematical Society: Providence, RI, USA, 2005. [Google Scholar]
- Wu, Y.-Q. Seifert fibered surgery on Montesinos knots. arXiv
**2012**, arXiv:1207.0154. [Google Scholar] - Generalizations of Fibonacci Numbers. Available online: https://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers (accessed on 1 March 2022).
- Planat, M.; Amaral, M.M.; Fang, F.; Chester, D.; Aschheim, R.; Irwin, K. Group theory of syntactical freedom in DNA transcription and genome decoding Curr. Issues Mol. Biol.
**2022**, 44, 95. [Google Scholar] [CrossRef] - Kauffman, L.H.; Lomonacco, S.J., Jr. Topological quantum information theory. Proc. Symp. Appl. Math.
**2010**, 68, 103–176. [Google Scholar] - Assanioussi, M.; Bahr, B. Hopf link volume simplicity constraints in spin foam models. Class. Quant. Grav.
**2020**, 37, 205003. [Google Scholar] [CrossRef] - Rovelli, C.; Vidotto, F. Covariant Loop Quantum Gravity, 1st ed.; Cambridge University Press: Cambridge, UK, 2014. [Google Scholar]
- Paluba, R. Geometry of Complex Character Varieties. Ph.D. Thesis, Université Paris-Saclay, Gif-sur-Yvette, France, 2017. Available online: https://tel.archives-ouvertes.fr/tel-01596075 (accessed on 1 March 2022).
- Boalch, P.; Paluba, R. Symmetric cubic surfaces as G
_{2}character varieties. J. Algebr. Geom.**2016**, 25, 607–631. [Google Scholar] [CrossRef] [Green Version] - Planat, M.; Zainuddin, H. Zoology of atlas-groups: Dessins d’enfants, finite geometries and quantum commutation. Mathematics
**2017**, 5, 6. [Google Scholar] [CrossRef] [Green Version] - Lévay, P.; Saniga, M.; Vrna, P. Three-qubit operators, the split Cayley hexagon of order two, and black holes Phys. Rev. D
**2008**, 78, 124022. [Google Scholar] [CrossRef] [Green Version] - Planat, M.; Saniga, M.; Holweck, F. Distinguished three-qubit ‘magicity’ via automorphisms of the split Cayley hexagon. Quant. Inf. Proc.
**2013**, 12, 2535–2549. [Google Scholar] [CrossRef] [Green Version] - Stacey, B.M. Sporadic SICs and the Normed Division Algebras. Found. Phys.
**2017**, 47, 1665. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**

**Left**: the Hopf link.

**Right**: a 3-dimensional picture of the $S{L}_{2}\left(\mathbb{C}\right)$ character variety ${\Sigma}_{H}$ for the Hopf link complement.

**Figure 2.**The canonical component of character varieties for (

**a**) the Whitehead link L5a1, (

**b**) the Whitehead link sister L13n5885, and (

**c**) the Bergé link L6a2.

**Figure 3.**A few singular fibers in Kodaira’s classification of minimal elliptic surfaces. (

**a**) Fiber ${I}_{0}^{*}$ (alias ${\tilde{D}}_{4}$), (

**b**) fiber $I{V}^{*}$ (alias ${\tilde{E}}_{6}$), and (

**c**) fiber $I{I}^{*}$ (alias ${\tilde{E}}_{8}$).

**Figure 4.**The surfaces ${f}_{1}(x,y,z)$ in (

**a**) and ${f}_{2}(x,y,z)$ in (

**b**) belonging to the character variety of singular fiber $I{V}^{*}={\tilde{E}}_{6}$. Both surfaces are birationally equivalent to ${K}_{3}$ surfaces.

**Table 1.**Character varieties of fundamental groups whose reducible representations are that of the Hopf link. Column 1 identifies the group, as well as the corresponding link and 3- or 4-manifold. Column 2 is the name of the link or the relation it has to magic state quantum computing based on qutrits (QT) or two qubits (2QB). Column 3 is for the relation(s) of the two-generator fundamental groups. When the link is not the Hopf link, Column 4 is for the canonical component(s) of the representations and its (their) type as a surface in the 3-dimensional projective space.

Link L | Name | Rel(s) Link Group ${\mathit{\pi}}_{1}\left(\mathit{L}\right)$ | Character Variety $\mathit{f}(\mathit{x},\mathit{y},\mathit{z})$ |
---|---|---|---|

L2a1 | Hopf | $[a,b]=abAB$ | ${f}_{H}=xyz-{x}^{2}-{y}^{2}-{z}^{2}+4$ |

- | - | - | deg 3 Del Pezzo |

${\Gamma}_{0}\left(2\right)$, L7n1 | QT related | $[a,{b}^{2}]$ | $y{f}_{H}$ |

${\Gamma}_{0}\left(3\right)$, L6a3 | 2QB related | $[a,{b}^{3}]$ | $({y}^{2}-1){f}_{H}$ |

${\Gamma}_{-1}\left(12\right)$, L5a1 | Whitehead | $a{b}^{3}{a}^{2}bA{B}^{3}{A}^{2}B$ | $x{y}^{2}z-{y}^{3}-{x}^{2}y-xz+2y$ |

$ooct{01}_{00001}$ | WL | - | conic bundle, ${K}_{3}$ type |

${\Gamma}_{-1}\left(12\right)$, L13n5885 | sister WL | ${a}^{2}bA{b}^{2}{A}^{2}Ba{B}^{2}$ | ${x}^{2}{y}^{2}-xyz-{x}^{2}+1$ |

$ooct{01}_{00000}$ | - | - | deg 4 Del Pezzo, ${K}_{3}$ type |

${\Gamma}_{-3}\left(24\right)$, L6a2 | Bergé | $a{b}^{3}{a}^{2}{b}^{2}A{B}^{3}{A}^{2}{B}^{2}$ | $x{y}^{3}z-{x}^{2}{y}^{2}-{y}^{4}-xyz+3{y}^{2}-1$ |

$otet{04}_{00001}$ | - | - | conic bundle, general type |

${\Gamma}_{-7}\left(6\right)$, L6a1 | $abAB{a}^{2}BA{b}^{3}ABab{A}^{2}ba{B}^{3}$ | undetermined | |

${\tilde{E}}_{6}$ | $I{V}^{*}$ | ${a}^{3}{b}^{3},a{b}^{2}aB{A}^{2}B$ | $x{y}^{3}-{y}^{2}z-{x}^{2}-2xy+z+2,$ |

- | - | - | ${y}^{4}-{x}^{2}z+xy-4{y}^{2}+z+2$ |

- | - | - | ${K}_{3}$ type |

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**MDPI and ACS Style**

Planat, M.; Amaral, M.M.; Fang, F.; Chester, D.; Aschheim, R.; Irwin, K.
Character Varieties and Algebraic Surfaces for the Topology of Quantum Computing. *Symmetry* **2022**, *14*, 915.
https://doi.org/10.3390/sym14050915

**AMA Style**

Planat M, Amaral MM, Fang F, Chester D, Aschheim R, Irwin K.
Character Varieties and Algebraic Surfaces for the Topology of Quantum Computing. *Symmetry*. 2022; 14(5):915.
https://doi.org/10.3390/sym14050915

**Chicago/Turabian Style**

Planat, Michel, Marcelo M. Amaral, Fang Fang, David Chester, Raymond Aschheim, and Klee Irwin.
2022. "Character Varieties and Algebraic Surfaces for the Topology of Quantum Computing" *Symmetry* 14, no. 5: 915.
https://doi.org/10.3390/sym14050915