Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2102.02281

Hypergraph States in SU(N)1, N odd prime, Chern-Simons Theory

Howard J. Schnitzer

Published Feb 3, 2021• 0 citations
arXivHTMLPDF

Abstract

Graph states and hypergraph states can be constructed from products of basic operations that appear in SU(N)1. The level-rank dual of a theorem of Salton, Swingle, and Walter implies that these operations can be prepared topologically in the n-torus Hilbert space of Chern-Simons theory for N neq 5 mod 4. For SU(N)1, N = 5 mod 4, only stabilizer states can be prepared on the n-torus Hilbert space, which restricts the construction to graph states.

DOI
10.48550/arxiv.2102.02281

Related papers

SU(N)1 Chern-Simons theory, the Clifford group, and Entropy Cone

Howard J. Schnitzer

Entropy cones for SU(N)1 Chern-Simons theory are discussed. It is shown that stabilizer states can be constructed from topological operators in SU(N)1 for N odd prime, but not for SU(N)K; K >= 2. This implies that the topological entropy cone is properly contained in the stabilizer entropy cone for SU(N)K; K >= 2.

arXiv:2008.02406•Aug 6, 2020•0 citations

Level-rank duality of SU(2)k Chern-Simons theory, and of hypergraph and magic states

Howard J. Schnitzer

The level-rank duality of SU(2)k Chern-Simons theory is discussed, and applied to graph, hypergraph, and magic states.

arXiv:2105.11498•May 24, 2021•1 citation

Clifford group and stabilizer states from Chern-Simons theory

Howard J. Schnitzer

The construction of generators of the Clifford group and of stabilizer states from Chern-Simons theory is presented for the Kac-Moody algebras SU(2)1, U(N)N,N(K+N) with N = 2 and K = 1, and SU(N)1 extending results of Salton, et. al.

arXiv:1903.06789•Mar 15, 2019•1 citation

Entanglement from topology in Chern-Simons theory

Grant Salton, Brian G. Swingle, Michael Walter

The way in which geometry encodes entanglement is a topic of much recent interest in quantum many-body physics and the AdS/CFT duality. This relation is particularly pronounced in the case of topological quantum field theories, where topology alone determines the quantum states of the theory. In this work, we study the set of quantum states that can be prepared by the Euclidean path integral in three-dimensional Chern-Simons theory. Specifically, we consider arbitrary three-manifolds with a fixed number of torus boundaries in both Abelian $U(1)$ and non-Abelian $SO(3)$ Chern-Simons theory. For the Abelian theory, we find that the states that can be prepared coincide precisely with the set of stabilizer states from quantum information theory. This constrains the multipartite entanglement present in this theory, but it also reveals that stabilizer states can be described by topology. In particular, we find an explicit expression for the entanglement entropy of a many-torus subsystem using only a single replica, as well as a concrete formula for the number of GHZ states that can be distilled from a tripartite state prepared through path integration. For the non-Abelian theory, we find a notion of ``state universality,'' namely that any state can be prepared to an arbitrarily good approximation. The manifolds we consider can also be viewed as toy models of multiboundary wormholes in AdS/CFT.

arXiv:1611.01516•Physical review. D/Physical review. D.•May 30, 2017•35 citations

Topological preparation of non-stabilizer states and Clifford evolution in SU(2)1 Chern-Simons theory

William Munizzi, Howard J. Schnitzer

A bstract We develop a topological framework for preparing families of non-stabilizer states, and computing their entanglement entropies, in SU (2) 1 Chern-Simons theory. Using the Kac-Moody algebra, we construct Pauli and Clifford operators as path integrals over 3-manifolds with Wilson loop insertions, enabling an explicit topological realization of W n and Dicke states, as well as their entanglement properties. For the genus-1 case, the modular SL (2 , ℤ) generators on a single torus boundary realize the standard Clifford generators. We further conjecture a higher-genus extension of this correspondence specifically for the Clifford orbits of W and Dicke states, in which the orbit of W n is represented by a sum over genus- g handlebodies with Dehn twists applied to each torus. Our results extend the known correspondence between Clifford structures and modular transformations, and demonstrate that non-stabilizer states can be understood as controlled deformations within the SU (2) 1 modular tensor category, providing a concrete example of how non-stabilizer quantum resources emerge from purely topological data.

arXiv:2510.15067•Journal of High Energy Physics•Sep 1, 2026•0 citations

Entangling gates for the SU(N) anyons

Sergey Mironov, Andrey Morozov

The model of a topological quantum computer is a promising one due to its natural resistance to noise and other errors. Operations in such a computer are implemented by braiding the trajectories of anyons. While it is easy to understand how to build one-qubit operations, two-qubit operations are more difficult. In arXiv:2412.20931 we suggested an approach to build such operations for a topological quantum computer based on SU(2) Chern-Simons theory with arbitrary level using cabling of knots. In this paper we discuss how this approach should be generalized to the SU(N) case, what the differences are, and which new problems arise.

hep-thCross-listedarXiv:2605.04016•May 6, 2026•0 citations
Graph states and hypergraph states can be constructed from products of basic operations that appear in SU(N)1. The level-rank dual of a theorem of Salton, Swingle, and Walter implies that these operations can be prepared topologically in the n-torus Hilbert space of Chern-Simons theory for N neq 5 mod 4. For SU(N)1, N = 5 mod 4, only stabilizer states can be prepared on the n-torus Hilbert space, which restricts the construction to graph states.
The way in which geometry encodes entanglement is a topic of much recent interest in quantum many-body physics and the AdS/CFT duality. This relation is particularly pronounced in the case of topological quantum field theories, where topology alone determines the quantum states of the theory. In this work, we study the set of quantum states that can be prepared by the Euclidean path integral in three-dimensional Chern-Simons theory. Specifically, we consider arbitrary three-manifolds with a fixed number of torus boundaries in both Abelian U(1)U(1)U(1) and non-Abelian SO(3)SO(3)SO(3) Chern-Simons theory. For the Abelian theory, we find that the states that can be prepared coincide precisely with the set of stabilizer states from quantum information theory. This constrains the multipartite entanglement present in this theory, but it also reveals that stabilizer states can be described by topology. In particular, we find an explicit expression for the entanglement entropy of a many-torus subsystem using only a single replica, as well as a concrete formula for the number of GHZ states that can be distilled from a tripartite state prepared through path integration. For the non-Abelian theory, we find a notion of ``state universality,'' namely that any state can be prepared to an arbitrarily good approximation. The manifolds we consider can also be viewed as toy models of multiboundary wormholes in AdS/CFT.
A bstract We develop a topological framework for preparing families of non-stabilizer states, and computing their entanglement entropies, in SU (2) 1 Chern-Simons theory. Using the Kac-Moody algebra, we construct Pauli and Clifford operators as path integrals over 3-manifolds with Wilson loop insertions, enabling an explicit topological realization of W n and Dicke states, as well as their entanglement properties. For the genus-1 case, the modular SL (2 , ℤ) generators on a single torus boundary realize the standard Clifford generators. We further conjecture a higher-genus extension of this correspondence specifically for the Clifford orbits of W and Dicke states, in which the orbit of W n is represented by a sum over genus- g handlebodies with Dehn twists applied to each torus. Our results extend the known correspondence between Clifford structures and modular transformations, and demonstrate that non-stabilizer states can be understood as controlled deformations within the SU (2) 1 modular tensor category, providing a concrete example of how non-stabilizer quantum resources emerge from purely topological data.
The model of a topological quantum computer is a promising one due to its natural resistance to noise and other errors. Operations in such a computer are implemented by braiding the trajectories of anyons. While it is easy to understand how to build one-qubit operations, two-qubit operations are more difficult. In arXiv:2412.20931 we suggested an approach to build such operations for a topological quantum computer based on SU(2) Chern-Simons theory with arbitrary level using cabling of knots. In this paper we discuss how this approach should be generalized to the SU(N) case, what the differences are, and which new problems arise.