# Universal Quantum Computing and Three-Manifolds

^{1}

^{2}

^{*}

## 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

- Thurston, W.P. Three-Dimensional Geometry and Topology; Princeton University Press: Princeton, NJ, USA, 1997; Volume 1. [Google Scholar]
- Yu Kitaev, A. Fault-tolerant quantum computation by anyons. Ann. Phys.
**2003**, 303, 2–30. [Google Scholar] [CrossRef][Green Version] - Nayak, C.; Simon, S.; Stern, A.; Freedman, M.; Sarma, S.D. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys.
**2008**, 80, 1083. [Google Scholar] [CrossRef] - Wang, Z. Topological Quantum Computation; American Mathematical Soc.: Providence, RI, USA, 2010. [Google Scholar]
- Pachos, J.K. Introduction to Topological Quantum Computation; Cambridge University Press: Cambridge, UK, 2012. [Google Scholar]
- Kauffman, L.H.; Baadhio, R.L. Quantum Topology; Series on Knots and Everything; World Scientific: Singapore, 1993. [Google Scholar]
- Kauffman, L.H. Knot logic and topological quantum computing with Majorana fermions. In Linear and Algebraic Structures in Quantum Computing; Lecture Notes in Logic 45; Chubb, J., Eskandarian, A., Harizanov, V., Eds.; Cambridge Univ. Press: Cambridge, UK, 2016. [Google Scholar]
- Seiberg, N.; Senthil, T.; Wang, C.; Witten, E. A duality web in 2 + 1 dimensions and condensed matter physics. Ann. Phys.
**2016**, 374, 395–433. [Google Scholar] [CrossRef] - Gang, D.; Tachikawa, Y.; Yonekura, K. Smallest 3d hyperbolic manifolds via simple 3d theories. Phys. Rev. D
**2017**, 96, 061701(R). [Google Scholar] [CrossRef] - Lim, N.C.; Jackson, S.E. Molecular knots in biology and chemistry. J. Phys. Condens. Matter
**2015**, 27, 354101. [Google Scholar] [CrossRef] [PubMed][Green Version] - Irwin, K. Toward a Unification of Physics and Number Theory. Available online: https://www.researchgate.net/publication/314209738 (accessed on 1 January 2018).
- Bravyi, S.; Kitaev, A. Universal quantum computation with ideal Clifford gates and noisy ancillas. Phys. Rev.
**2005**, A71, 022316. [Google Scholar] [CrossRef] - Veitch, V.; Mousavian, S.A.; Gottesman, D.; Emerson, J. The resource theory of stabilizer quantum computation. New J. Phys.
**2014**, 16, 013009. [Google Scholar] [CrossRef][Green Version] - Planat, M.; Haq, R.U. The magic of universal quantum computing with permutations. Adv. Math. Phys.
**2017**, 217, 5287862. [Google Scholar] [CrossRef] - 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] - Milnor, J. The Poincaré Conjecture 99 Years Later: A Progress Report (The Clay Mathematics Institute 2002 Annual Report, 2003). Available online: http://www.math.sunysb.edu/$\sim$jack/PREPRINTS/poiproof.pdf (accessed on 1 January 2018).
- Planat, M. On the geometry and invariants of qubits, quartits and octits. Int. J. Geom. Methods Mod. Phys.
**2011**, 8, 303–313. [Google Scholar] [CrossRef] - Manton, N.S. Connections on discrete fiber bundles. Commun. Math. Phys.
**1987**, 113, 341–351. [Google Scholar] [CrossRef] - Mosseri, R.; Dandoloff, R. Geometry of entangled states, Bloch spheres and Hopf fibrations. Int. J. Phys. A Math. Gen.
**2001**, 34, 10243. [Google Scholar] [CrossRef] - Nieto, J.A. Division-Algebras/Poincare-Conjecture Correspondence. J. Mod. Phys.
**2013**, 4, 32–36. [Google Scholar] [CrossRef] - Fang, F.; Hammock, D.; Irwin, K. Methods for calculating empires in quasicrystals. Crystals
**1997**, 7, 304. [Google Scholar] [CrossRef] - Sen, A.; Aschheim, R.; Irwin, K. Emergence of an aperiodic Dirichlet space from the tetrahedral units of an icosahedral internal space. Mathematics
**2017**, 5, 29. [Google Scholar] - Adams, C.C. The Knot Book, An Elementary Introduction to the Mathematical Theory of Knots; W. H. Freeman and Co.: New York, NY, USA, 1994. [Google Scholar]
- Fominikh, E.; Garoufalidis, S.; Goerner, M.; Tarkaev, V.; Vesnin, A. A census of tethahedral hyperbolic manifolds. Exp. Math.
**2016**, 25, 466–481. [Google Scholar] [CrossRef] - Hilden, H.M.; Lozano, M.T.; Montesinos, J.M.; Whitten, W.C. On universal groups and three-manifolds. Invent. Math.
**1987**, 87, 441–445. [Google Scholar] [CrossRef] - Mednykh, A.D. A new method for counting coverings over manifold with finitely generated fundamental group. Dokl. Math.
**2006**, 74, 498–502. [Google Scholar] [CrossRef] - Culler, M.; Dunfield, N.M.; Goerner, M.; Weeks, J.R. SnapPy, a Computer Program for Studying the Geometry and Topology of 3-Manifolds. Available online: http://snappy.computop.org (accessed on 1 January 2018).
- Hilden, H.M.; Lozano, M.T.; Montesinoos, J.M. On knots that are universal. Topology
**1985**, 24, 499–504. [Google Scholar] [CrossRef] - Fuchs, C.A. On the quantumness of a Hibert space. Quant. Inf. Comp.
**2004**, 4, 467–478. [Google Scholar] - Appleby, M.; Chien, T.Y.; Flammia, S.; Waldron, S. Constructing Exact Symmetric Informationally Complete Measurements from Numerical Solutions. arXiv, 2018; arXiv:1703.05981. [Google Scholar] [CrossRef]
- Rolfsen, D. Knots and Links; Mathematics Lecture Series 7; Publish of Perish: Houston, TX, USA, 1990. [Google Scholar]
- Milnor, J. On the 3-dimensional Brieskorn manifolds M(p,q,r). In Knots, Groups and 3-Manifolds; Neuwirth, L.P., Ed.; Princeton Univ. Press: Princeton, NJ, USA, 1975; pp. 175–225. [Google Scholar]
- Bosma, W.; Cannon, J.J.; Fieker, C.; Steel, A. (Eds.) Handbook of Magma Functions; University of Sydney: Sydney, Australia, 2017. [Google Scholar]
- Hempel, J. The lattice of branched covers over the Figure-eight knot. Topol. Appl.
**1990**, 34, 183–201. [Google Scholar] [CrossRef] - Haraway, R.C. Determining hyperbolicity of compact orientable 3-manifolds with torus boundary. arXiv, 2014; arXiv:1410.7115. [Google Scholar]
- Ballas, S.A.; Danciger, J.; Lee, G.S. Convex projective structures on non-hyperbolic three-manifolds. arXiv, 2018; arXiv:1508.04794. [Google Scholar]
- Gabai, D. The Whitehead manifold is a union of two Euclidean spaces. J. Topol.
**2011**, 4, 529–534. [Google Scholar] [CrossRef] - Akbulut, S.; Larson, K. Brieskorn spheres bounding rational balls. arXiv, 2017; arXiv:1704.07739. [Google Scholar] [CrossRef]
- Conder, M.; Martin, G.; Torstensson, A. Maximal symmetry groups of hyperbolic 3-manifolds. N. Z. J. Math.
**2006**, 35, 3762. [Google Scholar] - Gordon, C.M. Dehn Filling: A survey, Knot Theory; Banach Center Publ.: Warsaw, Poland, 1998; Volume 42, pp. 129–144. [Google Scholar]
- Sirag, S.-P. ADEX Theory, How the ADE Coxeter Graphs Unify Mathematics and Physics; World Scientific: Singapore, 2016. [Google Scholar]
- Kirby, R.C.; Scharlemann, M.G. Eight faces of the Poincaré homology 3-sphere. In Geometric Topology; Acad. Press: New York, NY, USA, 1979; pp. 113–146. [Google Scholar]
- Wu, Y. Seifert fibered surgery on Montesinos knots. arXiv, 2012; arXiv:1207.0154. [Google Scholar]
- Chan, K.T.; Zainuddin, H.; Atan, K.A.M.; Siddig, A.A. Computing Quantum Bound States on Triply Punctured Two-Sphere Surface. Chin. Phys. Lett.
**2016**, 33, 090301. [Google Scholar] [CrossRef] - Aurich, R.; Steiner, F.; Then, H. Numerical computation of Maass waveforms and an application to cosmology. In Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology; Jens, B., Frank, S., Eds.; Cambridge Univ. Press: Cambridge, UK, 2012. [Google Scholar]
- Asselmeyer-Maluga, T. Smooth quantum gravity: Exotic smoothness and Quantum gravity. In At the Frontier of Spacetime Scalar-Tensor Theory, Bells Inequality, Machs Principle, Exotic Smoothness; Fundamental Theories of Physics Book Series (FTP); Asselmeyer-Maluga, T., Ed.; Springer: Cham, Switzerland, 2016; pp. 247–308. [Google Scholar]
- Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K. Quantum computing with Bianchi groups. arXiv, 2018; arXiv:1808.06831. [Google Scholar]

**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} |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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