Topological Quantum Computing and 3-Manifolds

Asselmeyer-Maluga, Torsten

doi:10.3390/quantum3010009

Cited by 9 publications

(10 citation statements)

References 24 publications

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“…We explored the connection of such real surfaces to the character variety of some two-and three-bridge links. We pointed out their relationship to Painlevé VI transcendents through Okamoto Equation (9). While possible experimental directions remain open for further investigation, recent advances in the field are noteworthy [34][35][36][37].…”

Section: Discussionmentioning

confidence: 99%

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Fricke Topological Qubits

et al. 2022

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We recently proposed that topological quantum computing might be based on SL(2,C) representations of the fundamental group π1(S3\K) for the complement of a link K in the three-sphere. The restriction to links whose associated SL(2,C) character variety V contains a Fricke surface κd=xyz−x2−y2−z2+d is desirable due to the connection of Fricke spaces to elementary topology. Taking K as the Hopf link L2a1, one of the three arithmetic two-bridge links (the Whitehead link 512, the Berge link 622 or the double-eight link 632) or the link 732, the V for those links contains the reducible component κ4, the so-called Cayley cubic. In addition, the V for the latter two links contains the irreducible component κ3, or κ2, respectively. Taking ρ to be a representation with character κd (d<4), with |x|,|y|,|z|≤2, then ρ(π1) fixes a unique point in the hyperbolic space H3 and is a conjugate to a SU(2) representation (a qubit). Even though details on the physical implementation remain open, more generally, we show that topological quantum computing may be developed from the point of view of three-bridge links, the topology of the four-punctured sphere and Painlevé VI equation. The 0-surgery on the three circles of the Borromean rings L6a4 is taken as an example.

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Section: Discussionmentioning

confidence: 99%

“…In this paper, following our recent proposal [8] (see also [9]), we propose a nonanyonic theory of a topological quantum computer based on surfaces in a three-dimensional topological space. Such surfaces are part of the SL(2, C) character variety underlying the symmetries of a properly chosen manifold.…”

Section: Introductionmentioning

confidence: 99%

Fricke Topological Qubits

et al. 2022

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“…In Reference [7], the author proposes the representation π 1 → SU (2) ⊗ SU (2) as a model of 2-qubit quantum computing in which each factor is associated to a single qubit located on each component of the Hopf link. Our project expands this idea by taking the representation π 1 → SL(2, C) and the attached character variety Σ H as a model of topological quantum computing.…”

Section: Prolegomenamentioning

confidence: 99%

“…Such surfaces are candidates for a new type of topological quantum computing different from anyons [6]. Related ideas are in References [7,8]. A previous work of our group [9,10] proposed to relate the fundamental group of some 3-manifolds to quantum computing but did not employ the representation theory.…”

Section: Introductionmentioning

confidence: 99%

Character varieties and algebraic surfaces for the topology of quantum computing

Planat,

Amaral,

Fang

et al. 2022

Preprint

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It is shown that the representation theory of some finitely presented groups thanks to their SL2(C) character variety is related to algebraic surfaces. We make use of the Enriques-Kodaira classification of algebraic surfaces and the related topological tools to make such surfaces explicit. We study the connection of SL2(C) character varieties to topological quantum computing (TQC) as an alternative to the concept of anyons. The Hopf link H, whose character variety is a Del Pezzo surface fH (the trace of the commutator), is the kernel of our view of TQC. Qutrit and two-qubit magic state computing, derived from the trefoil knot in our previous work, may be seen as TQC from the Hopf link. The character variety of some two-generator Bianchi groups as well as that of the fundamental group for the singular fibers Ẽ6 and D4 contain fH . A surface birationally equivalent to a K3 surface is another compound of their character varieties.

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“…In this paper, following our recent proposal [4] (see also [5]), we propose a nonanyonic theory of a topological quantum computer based on surfaces in a three-dimensional topological space. Such surfaces are part of the SL(2, C) character variety underlying the symmetries of a properly chosen manifold.…”

Section: Introductionmentioning

confidence: 99%

Fricke Topological Qubits

Planat¹,

Chester²,

Amaral³

et al. 2022

Preprint

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We recently proposed that topological quantum computing might be based on $SL(2,\mathbb{C})$ representations of the fundamental group $\pi_1(S^3\setminus K)$ for the complement of a link $K$ in the $3$-sphere. The restriction to links whose associated $SL(2,\mathbb{C})$ character variety $\mathcal{V}$ contains a Fricke surface $\kappa_d=xyz -x^2-y^2-z^2+d$ is desirable due to the connection of Fricke spaces to elementary topology. Taking $K$ as the Hopf link $L2a1$, one of the three arithmetic two-bridge links [the Whitehead link $5_1^2$, the Berge link $6_2^2$, the double-eight link $6_3^2$] or the link $7_3^2$, the $\mathcal{V}$ for those links contains the reducible component $\kappa_4$, the so-called Cayley cubic. In addition, the $\mathcal{V}$ for the later two links contains the irreducible component $\kappa_3$, or $\kappa_2$, respectively. Taking $\rho$ to be a representation with character $\kappa_d$ ($d<4$), with $|x|,|y|,|z| \le 2$, then $\rho(\pi_1)$ fixes a unique point in the hyperbolic space $\mathcal{H}_3$ and is conjugate to a $SU(2)$ representation (a qubit). Even though details on the physical implementation remain open, more generally, we show that topological quantum computing may be developed from the point of view of three-bridge links, the topology of the $4$-punctured sphere and Painlev\'e VI equation. The $0$-surgery on the $3$ circles of the Borromean rings L6a4 is taken as an example.

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Topological Quantum Computing and 3-Manifolds

Cited by 9 publications

References 24 publications

Fricke Topological Qubits

Fricke Topological Qubits

Character varieties and algebraic surfaces for the topology of quantum computing

Fricke Topological Qubits

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