# Group Geometrical Axioms for Magic States of Quantum Computing

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## Abstract

**:**

## 1. Introduction

#### 1.1. Finite Geometries from Cosets

#### The Kochen–Specker Theorem with a Mermin Square of Two-Qubit Observables

#### 1.2. Magic States in Quantum Computing

#### Building a Two-Qubit MIC from a Subgroup ${\mathsf{\Gamma}}_{s}$ of Index 4 of the Modular Group $\mathsf{\Gamma}$

#### Constructing MICs Thanks to the Fundamental Group of a Knot or a Link

#### 1.3. Coset Geometry of Magic States

## 2. MIC States Pertaining to the Figure-Eight Knot and Its 0-Surgery

#### 2.1. Group Geometrical Axioms Applied to the Fundamental Group ${\pi}_{1}(Y)$

#### 2.2. Group Geometrical Axioms Applied to the Fundamental Group ${\pi}_{1}({S}^{3}\backslash K4a1)$

## 3. MIC States Pertaining to the Trefoil Knot and Its 0-Surgery

#### 3.1. Group Geometrical Axioms Applied to the Fundamental Group ${\pi}_{1}({\tilde{E}}_{8})$

#### 3.2. Group Geometrical Axioms Applied to the Fundamental Group ${\pi}_{1}({S}^{3}\backslash {3}_{1})$

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The generalized quadrangle of order two $GQ(2,2)$. The picture provides a representation of the fifteen $2QB$ observables that are commuting by triples: the lines of the geometry. Bold lines are for an embedded $3\times 3$ grid (also called Mermin square), that is, a basic model of Kochen–Specker theorem (e.g., see ([3], Figure 1) or [10]). The second representation is in terms of the cosets of the permutation group arising from the index 15 subgroup of $G\cong {A}_{6}$ (the 6-letter alternating group).

**Figure 2.**The map (

**a**) leading to Mermin’s square (

**b**). The two-point stabilizer subgroups of the permutation representation P corresponding to the dessin (one for each line) are as follows; ${s}_{1}=(2,3)(4,7)(5,6)$, ${s}_{2}=(1,7)(2,8)(6,9)$, ${s}_{3}=(1,4)(3,8)(5,9)$, ${s}_{4}=(2,6)(3,5)(8,9)$, and ${s}_{5}=(1,9)(4,5)(6,7)$, ${s}_{6}=(1,8)(2,7)(3,4)$, where the points of the square (resp. the edges of the dessin d’enfant) are labeled as $[1,..,9]=[e,a,{a}^{-1},{a}^{2},ab,{a}^{-1}b,{a}^{-2},{a}^{3},aba]$.

**Figure 3.**Representation of ${A}_{4}\cong {\mathsf{\Gamma}}_{0}\left(3\right)$ as a map (

**a**) and as the tiling of the fundamental domain (the two thick vertical lines have to identified) (

**b**). The character * denotes the unique elliptic point (of order 3). The organization of triple products of projectors leads to the generalized quadrangle $GQ(2,2)$ pictured in panel (

**c**).

**Figure 4.**(

**a**) The link L6a3 defining the two-qubit MIC from the trefoil knot and (

**b**) the braid representation. (

**c**) The link L10n46 defining the two-qubit MIC from the figure-eight knot, (

**d**) the corresponding hyperbolic 3-manifold otet${08}_{00002}$ and (

**e**) the braid representation.

**Figure 5.**The contextual geometry associated to the 2QB-MIC and permutation group $P={A}_{4}$ in Table 1. The line/triangle $\{a,ab,a{b}^{-1}\}$ is not made of mutually commuting cosets, thus geometric contextuality occurs.

**Figure 6.**Contextual geometries associated to (i) true and (ii) false for the MICs of the figure-eight knot $K4a1$ listed in Table 1. (

**a**) The Fano plane related to the manifold otet${14}_{00002}$ at index 7, and (

**b**) the Desargues configuration $\left[{10}_{3}\right]$ at index 10. The bold lines are for cosets that are not all mutually commuting. Each line corresponds to pair of cosets with the same stabilizer subgroup isomorphic to ${\mathbb{Z}}_{2}^{2}$.

**Figure 7.**Contextual geometries associated to (i) true and (ii) false for the MICs of the trefoil knot ${3}_{1}$ listed in Table 2. (

**a**) The octahedron, related to the subgroup ${\mathsf{\Gamma}}_{0}\left(4\right)$ of $\mathsf{\Gamma}$ at index 6; (

**b**) three disjoint lines ${K}_{3}^{3}$ at index 9; (

**c**) and the Mermin’s pentagram at index 10. The bold lines are for cosets that are not all mutually commuting.

**Table 1.**Table of subgroups of the fundamental group ${\pi}_{1}[{S}^{3}\backslash K4a1(0,1)]$ with $K4a1(0,1)$, the 0-surgery over the figure-eight knot. The permutation group P organizing the cosets in column 2. If (i) is true, unless otherwise specified, the graph of cosets leading to a MIC is that of the d-simplex and/or the condition (ii) is true: no geometry. The symbol $\mathsf{\Delta}$ means that (ii) fails to be satisfied. When there exists a MIC with (i) true and (ii) false, the geometry is shown in bold characters (here, this occurs in dimension 4, see Figure 5). If it exists, the MIC is $pp$-valued as given in column 4. In addition, $K(2,2,2)$ is the binary tripartite graph (alias the octahedron), and $\overline{\mathcal{L}\left(K\right(4,5\left)\right)}$ means the complement of the line graph of the bipartite graph $K(4,5)$.

d | P | (i) | pp | Geometry |
---|---|---|---|---|

4 | ${A}_{4}$ | yes | 2 | 2QB MIC, $\mathsf{\Delta}$ |

5 | 10 | yes | $\mathsf{\Delta}$ | |

6 | ${A}_{4}$ | no | 2 | 6-dit MIC, $K(2,2,2)$ |

9 | $(36,9)\cong {3}^{2}\u22ca4$ | yes | 2 | 2QT MIC |

11 | $(55,1)=11\u22ca5$, $(\times 2)$ | yes | 3 | 11-dit MIC |

16 | $(48,3)\cong 4\u22ca{A}_{4}$ | yes | $\mathsf{\Delta}$ | |

19 | $(171,3)\cong 19\u22ca9$ | yes | 3 | 19-dit MIC |

20 | $(120,39)\cong 4\u22ca(5\u22ca(6,2\left)\right)$ | yes | $\overline{\mathcal{L}\left(K\right(4,5\left)\right)}$ |

**Table 2.**Table of 3-manifolds ${M}^{3}$ found from subgroups of finite index d of the fundamental group ${\pi}_{1}({S}^{3}\backslash K4a1)$ (alias the d-fold coverings over the figure-eight knot 3-manifold). The covering type “ty” in column 2, the manifold identification “${M}^{3}$” in column 3 and the number of cusps, “cp” in column 4, are from SnapPy [19]. For $d=9$ and 10, SnapPy does not provide results so that we only identify the permutation group $P=$ SmallGroup $(o,k)$ (abbreviated as $(o,k)$), where o is the order and k is the k-th group of order o in the standard notation (that is used in Magma). If it exists, the MIC is “$pp$”-valued. If (i) is true, unless otherwise specified, the graph of cosets leading to a MIC is that of the d-simplex [and/or the condition (ii) is true: no geometry]. The symbol $\mathsf{\Delta}$ means that (ii) fails to be satisfied. When there exists a MIC with (i) true and (ii) false, the geometry is shown in bold characters. The symbol “fd” means a false detection of a MIC when (i) and (ii) are satisfied simultaneously while a MIC does not exist. The abbreviations “Fano”, “d-ortho”, and “$\left[{10}_{3}\right]$” are for the Fano plane, the d-orthoplex, and the Desargues configuration.

d | ty | ${\mathit{M}}^{3}$ (or P) | cp | (i) | pp | Geometry |
---|---|---|---|---|---|---|

2 | cyc | otet${04}_{00002}$, $m206$ | 1 | no | ||

3 | cyc | otet${06}_{00003}$, $s961$ | 1 | no | ||

4 | irr | otet${08}_{00002}$, $L10n46$, ${t}_{12840}$ | 2 | yes | 2 | 2QB MIC, $\mathsf{\Delta}$ |

cyc | otet${08}_{00007}$, $t12839$ | 1 | no | |||

5 | cyc | otet${10}_{00019}$ | 1 | no | ||

irr | otet${10}_{00006}$, $L8a20$ | 3 | yes | $\mathsf{\Delta}$ | ||

irr $(\times 2)$ | otet${10}_{00026}$ | 2 | yes | 1 | 5-dit MIC | |

6 | cyc | otet${12}_{00013}$ | 1 | no | ||

irr | otet${12}_{00039}$ | 1 | no | |||

irr $(\times 2)$ | otet${12}_{00038}$ | 1 | yes | 10 | 6-dit MIC | |

irr | otet${12}_{00041}$ | 2 | no | |||

irr $(\times 2)$ | otet${12}_{00017}$ | 2 | no | |||

irr $(\times 4)$ | otet${12}_{00000}$ | 2 | yes | 2 | 6-dit MIC | |

7 | cyc | otet${14}_{00019}$ | 1 | no | ||

irr $(\times 4)$ | otet${14}_{00002}$, $L14n55217$ | 3 | yes | 2 | 7-dit MIC, $\mathsf{\Delta}:\mathit{Fano}$ | |

irr $(\times 4)$ | otet${14}_{00035}$ | 1 | yes | 2 | 7-dit MIC, $\mathsf{\Delta}:\mathrm{Fano}$ | |

8 | cyc | otet${16}_{00026}$ | 1 | no | ||

irr $(\times 2)$ | otet${16}_{00035}$ | 1 | no | |||

irr $(\times 2)$ | otet${16}_{00079}$ | 2 | yes | fd | ||

irr $(\times 2)$ | otet${16}_{00016}$ | 2 | yes | fd | ||

irr | otet${16}_{00092}$ | 2 | no | |||

irr | otet${16}_{00091}$ | 2 | yes | 16-cell | ||

irr | otet${16}_{00013},L14n17678$ | 2 | no | |||

9 | $(36,9)\cong {3}^{2}\u22ca4$ | yes | 2 | 2QT MIC | ||

$(\times 2)$ | $(504,156)=PSL(2,8)$ | yes | 3 | 2QT MIC | ||

$(\times 2)$ | $(216,153)\cong {3}^{2}\u22ca(24,3)$ | yes | 2 | 2QT MIC | ||

10 | $(\times 6)$ | $(160,234)\cong {2}^{4}\u22ca10$ | yes | 5 | 10-dit MIC | |

$(\times 2)$ | $(120,34)={S}_{5}$ | yes | 4 | 10-dit MIC, $\mathsf{\Delta}:\left[{\mathbf{10}}_{\mathbf{3}}\right]$ | ||

$(\times 2)$ | $(120,34)={S}_{5}$ | no | 7 | 10-dit MIC, 5-ortho | ||

$(360,118)={A}_{6}$ | yes | 5 | 10-dit MIC |

d | P | (i) | pp | Geometry |
---|---|---|---|---|

3 | 6 | yes | 1 | Hesse SIC, $\mathsf{\Delta}$ |

4 | ${A}_{4}$ | yes | 2 | 2QB MIC, $\mathsf{\Delta}$ |

6 | ${A}_{4}$ | no | 2 | 6-dit MIC, $K(2,2,2)$ |

7 | $(42,1)\cong 7\u22ca(6,2)$ | yes | 2 | 7-dit MIC |

9 | $(54,5)\cong {3}^{2}\u22ca(18,3)$, $(\times 2)$ | yes | $K(3,3,3)$ | |

12 | $(72,44)\cong {2}^{2}\u22ca(18,3)$ | yes | $\overline{\mathcal{L}\left(K\right(3,4\left)\right)}$ | |

13 | $(78,1)\cong 13\u22ca(6,2)$, $(\times 2)$ | yes | 4 | 13-dit MIC |

15 | $(150,6)\cong {5}^{2}\u22ca(6,2)$, $(\times 2)$ | no | 6 | 15-dit MIC, $K(5,5,5)$ |

16 | $(96,72)\cong {2}^{3}\u22ca{A}_{4}$ | yes | $K(4,4,4,4)$ | |

19 | $(114,1)\cong 19\u22ca(6,2)$ | yes | 3 | 19-dit MIC |

21 | $(126,9)\cong 7\u22ca(18,3)$, $(\times 2)$ | yes | 5 | 21-dit MIC, $\mathsf{\Delta}$: K(3,3,3,3,3,3,3) |

**Table 4.**Subgroups of index d of the fundamental group ${\pi}_{1}({S}^{3}\backslash {3}_{1})$ (alias the d-fold coverings over the trefoil knot 3-manifold). The meaning of symbols is as in Table 2. When the subgroup in question is a subgroup of the modular group $\mathsf{\Gamma}$, it is identified as a congruence subgroup or by its signature $NC(g,N,{\nu}_{2},{\nu}_{3},[{c}_{i}^{{W}_{i}}])$ (see [14] for the meaning of entries). The permutation group $P=$ SmallGroup $(o,k)$ is abbreviated as $(o,k)$. As in Table 1, if (i) is true, unless otherwise specified, the graph of cosets leading to a MIC is the d-simplex and/or the condition (ii) is true. Exceptions (with geometry identified in bold characters) are for a MIC with (i) true and (ii) false. For indices 9 and 10, some subgroups of large order could not be checked as leading to a MIC or not; they are not shown in the table. The abbreviation octa is for the octahedron, MP is for the Mermin pentagram, and ${K}_{3}^{3}$ means three disjoint triangles.

d | ty | cp | P | (i) | pp | Type in $\mathsf{\Gamma}$ | Geometry |
---|---|---|---|---|---|---|---|

2 | cyc | 1 | (2,1) ≡ 2 | no | |||

3 | cyc | 1 | (3,1) ≡ 3 | no | |||

irr | 2 | (6,1) ≡ 6 | yes | 1 | ${\mathsf{\Gamma}}_{0}\left(2\right)$ | Hesse SIC, $\mathsf{\Delta}$, L7n1 | |

4 | cyc | 1 | (4,1) ≡ 4 | no | |||

irr | 2 | $(12,3)={A}_{4}$ | yes | 2 | ${\mathsf{\Gamma}}_{0}\left(3\right)$ | 2QB MIC, $\mathsf{\Delta}$, L6a3 | |

irr | 1 | $(24,12)={S}_{4}$ | yes | 2 | $4{A}^{0}$ | 2QB MIC | |

5 | cyc | 2 | (5,1) ≡ 5 | no | |||

irr | 3 | $(60,5)={A}_{5}$ | yes | 1 | $5{A}^{0}$ | 5-dit MIC | |

6 | reg | 3 | (6,1) ≡ 6 | no | 2 | $\mathsf{\Gamma}\left(2\right)$ | 6-dit MIC, ${6}_{3}^{3}$ [18] |

cyc | 3 | (6,2) = 3$\times 2$ | no | ${\mathsf{\Gamma}}^{\prime}$ | |||

irr | 2 | ${A}_{4}$ | no | 2 | $3{C}^{0}$ | 6-dit MIC, K(2,2,2) | |

irr | 1 | $(24,13)=3\u22ca8$ | no | $6{B}^{0}$ | |||

irr | 1 | $(18,3)\cong {3}^{2}\u22ca2$ | no | $6{A}^{0}$ | |||

irr | 3 | ${S}_{4}$ | yes | 2 | ${\mathsf{\Gamma}}_{0}\left(4\right)$ | 6-dit MIC, $\mathsf{\Delta}:\mathrm{octa}$ | |

irr | 2 | ${A}_{5}$ | yes | 2 | ${\mathsf{\Gamma}}_{0}\left(5\right)$ | 6-dit MIC | |

irr | 2 | ${S}_{4}$ | yes | 2 | $4{C}^{0}$ | 6-dit MIC, $\mathsf{\Delta}:\mathrm{octa}$ | |

7 | cyc | 1 | (7,1)$\equiv 7$ | no | |||

irr ($\times 2$) | 2 | $(42,1)\cong 7\u22ca(6,2)$ | yes | 2 | $NC(0,6,1,1,\left[{1}^{1}{6}^{1}\right])$ | 7-dit MIC | |

irr $(\times 2)$ | 1 | $(168,42)=PSL(2,7)$ | yes | 2 | $7{A}^{0}$ | 7-dit MIC | |

irr ($\times 2$) | 2 | ${S}_{7}$ (order 5040) | yes | $NC(0,10,1,1,\left[{2}^{1}{5}^{1}\right])$ | 7-dit MIC | ||

8 | cyc | 1 | $(8,1)\equiv 8$ | no | |||

irr | 2 | (24,13) | no | $6{C}^{0}$ | |||

irr | 2 | ${S}_{4}$ | no | $4{D}^{0}$ | |||

irr ($\times 2$) | 2 | $(24,3)\cong 2.{A}_{4}$ | yes | 16-cell | |||

irr | 2 | $PSL(2,7)$ | yes | ${\mathsf{\Gamma}}_{0}\left(7\right)$ | fd | ||

irr ($\times 2$) | 1 | $SL(2,7)$ | yes | $NC(0,8,2,2,\left[{8}^{1}\right])$ | fd | ||

irr ($\times 2$) | 2 | $(48,29)\cong 2.(24,3)$ | yes | $8{A}^{0}$ | 16-cell | ||

9 | $(9,1)\equiv 9$ | no | |||||

2 | (18,3) | no | 7 | $6{D}^{0}$ | 9-dit MIC, K(3,3,3) | ||

2 | $(54,5)\cong {3}^{2}\u22ca(18,3)$ | no | 7 | $NC(0,6,3,0,\left[{3}^{1}{6}^{1}\right])$ | 9-dit MIC, K(3,3,3) | ||

1 | $(324,160)\cong {3}^{3}\u22ca{A}_{4}$ | no | $9{A}^{0}$ | K(3,3,3), ${K}_{3}^{3}$ | |||

3 | (54,5) | yes | 7 | $NC(0,6,1,0,\left[{1}^{1}{2}^{1}{6}^{1}\right])$ | 9-dit MIC | ||

($\times 3$) | $(162,10)\cong {3}^{2}\u22ca6$ | yes | K(3,3,3) | ||||

($\times 2$) | 1 | $(504,156)=PSL(2,8)$ | yes | 3 | $NC(1,9,1,0,\left[{9}^{1}\right])$ | 2QT MIC | |

($\times 2$) | 2 | $(432,734)\cong {3}^{2}\u22ca(48,29)$ | yes | 2 | $NC(0,8,3,0,\left[{8}^{1}{1}^{1}\right])$ | 2QT MIC | |

3 | $(648,703)\cong {3}^{3}\u22ca{S}_{4}$ | yes | 2 | $NC(0,12,1,0,\left[{2}^{1}{3}^{1}{4}^{1}\right])$ | 2QT MIC, $\mathsf{\Delta}:{\mathit{K}}_{\mathbf{3}}^{\mathbf{3}}$ | ||

10 | 1 | $(120,35)\cong 2\u22ca{A}_{5}$ | no | $10{A}^{0}$ | 5-ortho | ||

2 | ${A}_{5}$ | yes | 5 | $5{C}^{0}$ | 10-dit MIC, $\mathsf{\Delta}$: MP | ||

($\times 2$) | 1 | $(720,764)\cong {A}_{6}\u22ca2$ | yes | 9 | $NC(0,10,0,4,\left[{10}^{1}\right])$ | 10-dit MIC |

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**MDPI and ACS Style**

Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K. Group Geometrical Axioms for Magic States of Quantum Computing. *Mathematics* **2019**, *7*, 948.
https://doi.org/10.3390/math7100948

**AMA Style**

Planat M, Aschheim R, Amaral MM, Irwin K. Group Geometrical Axioms for Magic States of Quantum Computing. *Mathematics*. 2019; 7(10):948.
https://doi.org/10.3390/math7100948

**Chicago/Turabian Style**

Planat, Michel, Raymond Aschheim, Marcelo M. Amaral, and Klee Irwin. 2019. "Group Geometrical Axioms for Magic States of Quantum Computing" *Mathematics* 7, no. 10: 948.
https://doi.org/10.3390/math7100948