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:1707.05490 · Physical Review Letters

Universal Quantum Computation with Gapped Boundaries

Iris Cong, Meng Cheng, Zhenghan Wang

Published Oct 25, 2017• 46 citations
arXivHTMLPDF

Abstract

This Letter discusses topological quantum computation with gapped boundaries of two-dimensional topological phases. Systematic methods are presented to encode quantum information topologically using gapped boundaries, and to perform topologically protected operations on this encoding. In particular, we introduce a new and general computational primitive of topological charge measurement and present a symmetry-protected implementation of this primitive. Throughout the Letter, a concrete physical example, the Z_{3} toric code [D(Z_{3})], is discussed. For this example, we have a qutrit encoding and an abstract universal gate set. Physically, gapped boundaries of D(Z_{3}) can be realized in bilayer fractional quantum Hall 1/3 systems. If a practical implementation is found for the required topological charge measurement, these boundaries will give rise to a direct physical realization of a universal quantum computer based on a purely Abelian topological phase.

DOI
10.1103/physrevlett.119.170504

Related papers

Universal Gates from Braiding and Fusing Anyons on Quantum Hardware

Chiu Fan Bowen Lo, Anasuya Lyons, Dan Gresh +8 more

Topological quantum computation encodes quantum information in the internal fusion space of non-Abelian anyonic quasiparticles, whose braiding implements logical gates. This goes beyond Abelian topological order (TO) such as the toric code, as its anyons lack internal structure. However, the simplest non-Abelian generalizations of the toric code do not support universality via braiding alone. Here we demonstrate that such minimally non-Abelian TOs can be made universal by treating anyon fusion as a computational primitive. We prepare a 54-qubit TO wavefunction associated with the smallest non-Abelian group, $S_3$, on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes $S_3$ TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.

quant-pharXiv:2601.20956•Nature•Jan 30, 2026•0 citations

Topological Quantum Computation with Gapped Boundaries

Iris Cong, Meng Cheng, Zhenghan Wang

This paper studies fault-tolerant quantum computation with gapped boundaries. We first introduce gapped boundaries of Kitaev's quantum double models for Dijkgraaf-Witten theories using their Hamiltonian realizations. We classify the elementary excitations on the boundary, and systematically describe the bulk-to-boundary condensation procedure. We also provide a commuting Hamiltonian to realize defects between boundaries in any quantum double model. Next, we present the algebraic/categorical structure of gapped boundaries and boundary defects, which will be used to describe topologically protected operations and obtain quantum gates. To demonstrate a potential physical realization, we provide quantum circuits for surface codes that can perform all basic operations on gapped boundaries. Finally, we show how gapped boundaries of the abelian theory $\mathfrak{D}(\mathbb{Z}_3)$ can be used to perform universal quantum computation.

arXiv:1609.02037•Sep 7, 2016•24 citations

Measurements in quantum topological phases of matter and quantum computing

Gabrielle Tournaire

Quantum measurements often destroy the “quantumness” of a state. Combining measurements and entanglement, however, unlocks new opportunities in quantum computing, as it transcends the boundaries between quantum phases. Measurement-based quantum computation is a perfect example: from an array of locally entangled qubits, individual measurements teleport and modify some logical information. Since quantum operations are prone to errors, this information is encoded in the topological Toric code, thus enabling fault-tolerant computation. Since the Toric code has a 2-dimensional structure, the resource state needed to perform measurement-based quantum computation is necessarily 3-dimensional and is called the 3D cluster state. In this thesis, we provide new tools to enlarge the capabilities of topological measurement-based quantum computation and reduce its operational cost. First, we introduce a lattice defect in the underlying cluster state that improves the expressivity of the computation. Second, we develop general ideas towards circuit compaction and quantify the resources used by the new methods. Furthermore, here, the Toric code is created thanks to local measurements, while it cannot be created with local unitary operations only. We investigate the mathematical roots of this intriguing property, and we show that long-range entanglement arises from locally measuring some general short-range entangled state. Specifically, we assume that the initial short-range entangled state is in a non-trivial symmetry protected topological phase for an abelian group on an infinite spin chain. That is, there is a global operation generated by local unitary operators labeled by the element of the group that leaves the state invariant. On infinite quantum spin chains, these symmetry protected topological phases have been completely characterized. We then perform on-site measurements on increasing intervals of the chain. This creates a sequence of measured states that are eigenstates of the local unitaries associated with elements of a subgroup of the symmetry group. Using the algebraic properties of the initial state, we show that this sequence of measured states has increasing long-range correlations. The entanglement bounds necessarily deteriorate with the size of the measured intervals; thus, the measured states cannot be uniformly short-range entangled.

•Aug 28, 2026•0 citations

A universal circuit set using the S3 quantum double

Liyuan Chen, Yuanjie Ren, Ruihua Fan +1 more

One potential route toward fault-tolerant universal quantum computation is to use non-Abelian topological codes. In this work, we investigate how to achieve this goal with the quantum double model $${\mathcal{D}}({S}_{3})$$ . By embedding each on-site Hilbert space into a qubit-qutrit pair, we explicitly construct circuits for creating, moving, and locally measuring all non-trivial anyons. We also design a specialized anyon interferometer to remotely measure the total charge of well-separated anyons; this avoids fusion, which compromises fault tolerance. These protocols enable the implementation of a universal gate set proposed by Cui et al. and active correction of the circuit-level noise during computation. To further suppress errors, we encode each physical degree of freedom of $${\mathcal{D}}({S}_{3})$$ into a novel error-correcting code, enabling fault-tolerant realization, at the logical level, of all gates in the anyon manipulation circuits. Our proposal offers a promising path to realize robust universal topological quantum computation in the NISQ era.

arXiv:2411.09697•npj Quantum Information•Jul 3, 2025•5 citations

Measurement-Based Quantum Computation in Symmetry-Enriched Topological Phases

Paul Herringer, Vir B. Bulchandani, Younes Javanmard +2 more

We present the first examples of topological phases of matter with uniform power for measurement-based quantum computation (MBQC). This is possible due to a new framework for analyzing the computational properties of phases of matter that is more general than previous constructions, which have been limited to short-range entangled phases in one dimension. We show that ground states of the toric code in an anisotropic magnetic field yield a natural, albeit noncomputationally universal, application of our framework. We then present a new model with topological order the ground states of which are universal resources for MBQC. Both topological models are enriched by subsystem symmetries and these symmetries protect their computational power. Our framework greatly expands the range of physical models that can be analyzed from the computational perspective.

PRX Quantum•Aug 13, 2025•2 citations

Topological Quantum Computation—From Basic Concepts to First Experiments

Ady Stern, Netanel H. Lindner

Quantum computation requires controlled engineering of quantum states to perform tasks that go beyond those possible with classical computers. Topological quantum computation aims to achieve this goal by using non-Abelian quantum phases of matter. Such phases allow for quantum information to be stored and manipulated in a nonlocal manner, which protects it from imperfections in the implemented protocols and from interactions with the environment. Recently, substantial progress in this field has been made on both theoretical and experimental fronts. We review the basic concepts of non-Abelian phases and their topologically protected use in quantum information processing tasks. We discuss different possible realizations of these concepts in experimentally available solid-state systems, including systems hosting Majorana fermions, their recently proposed fractional counterparts, and non-Abelian quantum Hall states.

Science•Mar 7, 2013•339 citations
This Letter discusses topological quantum computation with gapped boundaries of two-dimensional topological phases. Systematic methods are presented to encode quantum information topologically using gapped boundaries, and to perform topologically protected operations on this encoding. In particular, we introduce a new and general computational primitive of topological charge measurement and present a symmetry-protected implementation of this primitive. Throughout the Letter, a concrete physical example, the Z_{3} toric code [D(Z_{3})], is discussed. For this example, we have a qutrit encoding and an abstract universal gate set. Physically, gapped boundaries of D(Z_{3}) can be realized in bilayer fractional quantum Hall 1/3 systems. If a practical implementation is found for the required topological charge measurement, these boundaries will give rise to a direct physical realization of a universal quantum computer based on a purely Abelian topological phase.
Topological quantum computation encodes quantum information in the internal fusion space of non-Abelian anyonic quasiparticles, whose braiding implements logical gates. This goes beyond Abelian topological order (TO) such as the toric code, as its anyons lack internal structure. However, the simplest non-Abelian generalizations of the toric code do not support universality via braiding alone. Here we demonstrate that such minimally non-Abelian TOs can be made universal by treating anyon fusion as a computational primitive. We prepare a 54-qubit TO wavefunction associated with the smallest non-Abelian group, S3S_3S3​, on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes S3S_3S3​ TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.
This paper studies fault-tolerant quantum computation with gapped boundaries. We first introduce gapped boundaries of Kitaev's quantum double models for Dijkgraaf-Witten theories using their Hamiltonian realizations. We classify the elementary excitations on the boundary, and systematically describe the bulk-to-boundary condensation procedure. We also provide a commuting Hamiltonian to realize defects between boundaries in any quantum double model. Next, we present the algebraic/categorical structure of gapped boundaries and boundary defects, which will be used to describe topologically protected operations and obtain quantum gates. To demonstrate a potential physical realization, we provide quantum circuits for surface codes that can perform all basic operations on gapped boundaries. Finally, we show how gapped boundaries of the abelian theory D(Z3)\mathfrak{D}(\mathbb{Z}_3)D(Z3​) can be used to perform universal quantum computation.
Quantum measurements often destroy the “quantumness” of a state. Combining measurements and entanglement, however, unlocks new opportunities in quantum computing, as it transcends the boundaries between quantum phases. Measurement-based quantum computation is a perfect example: from an array of locally entangled qubits, individual measurements teleport and modify some logical information. Since quantum operations are prone to errors, this information is encoded in the topological Toric code, thus enabling fault-tolerant computation. Since the Toric code has a 2-dimensional structure, the resource state needed to perform measurement-based quantum computation is necessarily 3-dimensional and is called the 3D cluster state. In this thesis, we provide new tools to enlarge the capabilities of topological measurement-based quantum computation and reduce its operational cost. First, we introduce a lattice defect in the underlying cluster state that improves the expressivity of the computation. Second, we develop general ideas towards circuit compaction and quantify the resources used by the new methods. Furthermore, here, the Toric code is created thanks to local measurements, while it cannot be created with local unitary operations only. We investigate the mathematical roots of this intriguing property, and we show that long-range entanglement arises from locally measuring some general short-range entangled state. Specifically, we assume that the initial short-range entangled state is in a non-trivial symmetry protected topological phase for an abelian group on an infinite spin chain. That is, there is a global operation generated by local unitary operators labeled by the element of the group that leaves the state invariant. On infinite quantum spin chains, these symmetry protected topological phases have been completely characterized. We then perform on-site measurements on increasing intervals of the chain. This creates a sequence of measured states that are eigenstates of the local unitaries associated with elements of a subgroup of the symmetry group. Using the algebraic properties of the initial state, we show that this sequence of measured states has increasing long-range correlations. The entanglement bounds necessarily deteriorate with the size of the measured intervals; thus, the measured states cannot be uniformly short-range entangled.
One potential route toward fault-tolerant universal quantum computation is to use non-Abelian topological codes. In this work, we investigate how to achieve this goal with the quantum double model D(S3){\mathcal{D}}({S}_{3})D(S3​) . By embedding each on-site Hilbert space into a qubit-qutrit pair, we explicitly construct circuits for creating, moving, and locally measuring all non-trivial anyons. We also design a specialized anyon interferometer to remotely measure the total charge of well-separated anyons; this avoids fusion, which compromises fault tolerance. These protocols enable the implementation of a universal gate set proposed by Cui et al. and active correction of the circuit-level noise during computation. To further suppress errors, we encode each physical degree of freedom of D(S3){\mathcal{D}}({S}_{3})D(S3​) into a novel error-correcting code, enabling fault-tolerant realization, at the logical level, of all gates in the anyon manipulation circuits. Our proposal offers a promising path to realize robust universal topological quantum computation in the NISQ era.
We present the first examples of topological phases of matter with uniform power for measurement-based quantum computation (MBQC). This is possible due to a new framework for analyzing the computational properties of phases of matter that is more general than previous constructions, which have been limited to short-range entangled phases in one dimension. We show that ground states of the toric code in an anisotropic magnetic field yield a natural, albeit noncomputationally universal, application of our framework. We then present a new model with topological order the ground states of which are universal resources for MBQC. Both topological models are enriched by subsystem symmetries and these symmetries protect their computational power. Our framework greatly expands the range of physical models that can be analyzed from the computational perspective.
Quantum computation requires controlled engineering of quantum states to perform tasks that go beyond those possible with classical computers. Topological quantum computation aims to achieve this goal by using non-Abelian quantum phases of matter. Such phases allow for quantum information to be stored and manipulated in a nonlocal manner, which protects it from imperfections in the implemented protocols and from interactions with the environment. Recently, substantial progress in this field has been made on both theoretical and experimental fronts. We review the basic concepts of non-Abelian phases and their topologically protected use in quantum information processing tasks. We discuss different possible realizations of these concepts in experimentally available solid-state systems, including systems hosting Majorana fermions, their recently proposed fractional counterparts, and non-Abelian quantum Hall states.