Quantum computing, Seifert surfaces and singular fibers

Abstract

The fundamental group ${\pi }_1(L)$ of a knot or link $L$ may be used to generate magic states appropriate for performing universal quantum computation and simultaneously for retrieving complete information about the processed quantum states. In this paper, one defines braids whose closure is the $L$ of such a quantum computer model and computes their Seifert surfaces and the corresponding Alexander polynomial. In particular, some $d$-fold coverings of the trefoil knot, with $d=3$, $4$, $6$ or $12$, define appropriate links $L$ and the latter two cases connect to the Dynkin diagrams of $E_6$ and $D_4$, respectively. In this new context, one finds that this correspondence continues with the Kodaira's classification of elliptic singular fibers. The Seifert fibered toroidal manifold ${\Sigma }^{\prime }$, at the boundary of the singular fiber $\widetilde{E_8}$, allows possible models of quantum computing.

arXiv comment11 pages, 5 figures

References in corpus (9)

• Non-Abelian Anyons and Topological Quantum Computation
• Universal Quantum Computation with ideal Clifford gates and noisy ancillas
• The Resource Theory of Stabilizer Computation
• Quantifying magic for multi-qubit operations
• How to obtain a cosmological constant from small exotic R^4
• Magic informationally complete POVMs with permutations
• The Poincaré half-plane for informationally complete POVMs
• Anyons in Geometric Models of Matter
• On the coverings of Euclidian manifolds ${\mathcal{B}}_1$ and ${\mathcal{B}}_2$

Cited by in corpus (4)

• Topological Quantum Computing and 3-Manifolds
• Group geometrical axioms for magic states of quantum computing
• Quantum computation and measurements from an exotic space-time R4
• 3D Topological Quantum Computing