On topological quantum computing with mapping class group representations

Bloomquist, Wade; Wang, Zhenghan

doi:10.1088/1751-8121/aaeea1

Cited by 5 publications

(5 citation statements)

References 14 publications

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“…We will simply use Z r (f ) to denote Z r (M f ). It is in general very hard to compute directly, for example see [7] for the explicit formula for the set of Dehn twists generating the mapping class group, but for some special mapping class it is easy to write down the matrix. When f = T γ is the right (left) Dehn tiwst along a curve γ, then M f can be presented by surgery on the curve γ × { 1 2 } ⊂ Σ g × I, which is ∓1 framed relative to the surface Σ g × { 1 2 } (denoted by γ ∓ ).…”

Section: Figure 7 the F-movementioning

confidence: 99%

Projective representations of Hecke groups from Topological quantum field theory

Ruan¹

2022

Preprint

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We construct projective (unitary) representations of Hecke groups from the vector spaces associated with Witten-Reshetikhin-Turaev topological quantum field theory of higher genus surfaces. In particular, we generalize the modular data of Temperley-Lieb-Jones modular categories. We also study some properties of the representation. We show the image group of the representation is infinite at low levels in genus 2 by explicit computations. We also show the representation is reducible with at least three irreducible summands when the level equals 4l + 2 for l ≥ 1.

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Section: Figure 7 the F-movementioning

confidence: 99%

Projective representations of Hecke groups from Topological quantum field theory

Ruan¹

2022

Preprint

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“…For the notations and terminologies, see for example [56]. The mapping class group representations are explicitly described in [80].…”

Section: Jhep10(2020)129mentioning

confidence: 99%

“…To understand the representations of MCGs Γ g from the Ising TQFT, we will use four different bases of the Hilbert spaces V iTQFT (Σ g ): the defining basis {e a b }, the standard basis {v i k }, the geometric basis {u i k }, and the spin basis {ω m l } -the last three bases are defined and used in [41], and the defining basis is used in [80]. The defining basis and standard basis consist of labeled fusion graphs using {1, σ, ψ}.…”

Section: Jhep10(2020)129mentioning

confidence: 99%

Establishing strongly-coupled 3D AdS quantum gravity with Ising dual using all-genus partition functions

Jian

Ludwig

Luo

et al. 2020

J. High Energ. Phys.

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We study 3D pure Einstein quantum gravity with negative cosmological constant, in the regime where the AdS radius l is of the order of the Planck scale. Specifically, when the Brown-Henneaux central charge c = 3l/2GN (GN is the 3D Newton constant) equals c = 1/2, we establish duality between 3D gravity and 2D Ising conformal field theory by matching gravity and conformal field theory partition functions for AdS spacetimes with general asymptotic boundaries. This duality was suggested by a genus-one calculation of Castro et al. [Phys. Rev. D85 (2012) 024032]. Extension beyond genus-one requires new mathematical results based on 3D Topological Quantum Field Theory; these turn out to uniquely select the c = 1/2 theory among all those with c < 1, extending the previous results of Castro et al. Previous work suggests the reduction of the calculation of the gravity partition function to a problem of summation over the orbits of the mapping class group action on a “vacuum seed”. But whether or not the summation is well-defined for the general case was unknown before this work. Amongst all theories with Brown-Henneaux central charge c < 1, the sum is finite and unique only when c = 1/2, corresponding to a dual Ising conformal field theory on the asymptotic boundary.

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“…Usually, some simplified methods or one and two transformations are used to get the approximate solution, for example the mean field method which considers the far particle-particle interaction as a field effect, and the Jordan-Wigner transformation which maps the interaction of particles onto a fermion representation [1,2]. Intermediate statistics is a hot topic in physics these decades, e.g., in topological quantum computation [3][4][5][6][7], quantum material [8,9], and quantum information [10]. If the interacting manybody system is intermediate statistical system, getting the approximate solution or even the low dimensional exact solution becomes more difficult.…”

Section: Introductionmentioning

confidence: 99%

A group method solving many-body systems in intermediate statistical representation

Shen,

Zhou,

Dai

et al. 2021

Preprint

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The exact solution of the interacting many-body system is important and is difficult to solve. In this paper, we introduce a group method to solve the interacting many-body problem using the relation between the permutation group and the unitary group. We prove a group theorem first, then using the theorem, we represent the Hamiltonian of the interacting many-body system by the Casimir operators of unitary group. The eigenvalues of Casimir operators could give the exact values of energy and thus solve those problems exactly. This method maps the interacting many-body system onto an intermediate statistical representation. We give the relation between the conjugacy-class operator of permutation group and the Casimir operator of unitary group in the intermediate statistical representation, called the Gentile representation. Bose and Fermi cases are two limitations of the Gentile representation. We also discuss the representation space of symmetric and unitary group in the Gentile representation and give an example of the Heisenberg model to demonstrate this method. It is shown that this method is effective to solve interacting many-body problems.

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On topological quantum computing with mapping class group representations

Cited by 5 publications

References 14 publications

Projective representations of Hecke groups from Topological quantum field theory

Projective representations of Hecke groups from Topological quantum field theory

Establishing strongly-coupled 3D AdS quantum gravity with Ising dual using all-genus partition functions

A group method solving many-body systems in intermediate statistical representation

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