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High spin correlations in topological and rainbow lattices

Lucy Sarah Byles

Published Sep 1, 2025• 0 citations
OpenAlex

Abstract

Quantum simulations are an invaluable tool in condensed matter physics, enabling the precise generation and control of physical phenomena abstracted from the complexity of the underlying many-body system. At the heart of such complexity lies entanglement, the intrinsically quantum correlations that have attracted renewed interest as a fundamental resource in quantum computation and information. In this thesis, we explore the use of one- and two-dimensional lattice models to simulate and harness such correlations for next-generation quantum technologies. We begin in one dimension with a spin-chain model that allows complete control over bipartite correlation structures. Specifically, we introduce a model of nearest-neighbour couplings and transverse field terms, whose exact solution yields a q-deformed rainbow ground state formed from a tensor product of q-deformed singlets, each with directly tunable entanglement. This framework acts as a universal simulator of free-fermion entanglement spectra, with numerical studies confirming high-fidelity realisations across a range of examples. We then turn to strong correlations in two dimensions, where topologically ordered systems host exotic quasiparticles known as anyons. Braiding non-Abelian anyons enacts unitary transformations on a logically encoded fusion space, offering a natural pathway to fault-tolerant quantum computation. Using Kitaev’s quantum double model D(G), we focus on the smallest non-Abelian instance, D(S3), realised on a lattice of d=6 qudits. We develop a minimal protocol in which braiding and fusion are simulated solely through operators that create and measure anyons, demonstrating that D(S3) anyons generate magic states, a vital resource for universal quantum computation. Finally, a dense encoding scheme is presented, rendering the minimal realisation of this protocol accessible with current quantum technologies and a circuit model for realisation within the framework of linear optics is presented. Notably, error analysis of this encoding scheme reveals an inherent passive protection provided by the topological encoding, extending beyond conventional notions of topological fault tolerance.

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Liyuan Chen

A central challenge at the intersection of quantum condensed-matter physics and quantum information science is to understand how exotic topological phases and quantum resources govern the power of quantum computation and simulation---and to translate that understanding into practical architectures. This thesis contributes to this program along three interconnected directions: enabling fault-tolerant universal quantum computation with non-Abelian topological order, establishing the fundamental computational reach of near-term analog quantum simulators, and delineating the boundary between classical and quantum computational power through the lenses of topology and magic. In the first direction, drawing on the structure of non-Abelian topological order, we address the dominant resource overhead in current quantum computing architectures---magic-state distillation---by moving beyond the standard stabilizer formalism entirely. We construct a comprehensive framework for fault-tolerant universal quantum computation based on the non-Abelian quantum double model $\mathcal{D}(S_3)$, with all operations realized on qubit--qutrit degrees of freedom in a planar, geometrically local architecture feasible on all current experimental platforms. This work addresses the long-standing open question of realizable, fault-tolerant universal topological quantum computation. In the second direction, we establish the computational universality of analog quantum simulators operating under global control---platforms that circumvent the demanding requirements of digitization and individual addressability. We prove a necessary and sufficient condition for universality in globally controlled qubit chains, showing that widely used platforms such as Rydberg atom arrays and trapped ions are generically universal once lattice reflection symmetry is broken. We further derive sufficient conditions for universality in fermionic and bosonic optical superlattices. As an experimental demonstration of this framework, we realize the dynamics of the cluster-state $ZXZ$ model---an exemplar of symmetry-protected topological order---in Rydberg atom arrays, validating the theoretical foundation. In the third direction, we sharpen the classical--quantum computational boundary from both sides. We show that non-Abelian Majorana braiding realized in classical mechanical metamaterials admits a faithful classical simulation, establishing that braiding-only architectures are classically simulable without supplementary non-Clifford resources. Conversely, we prove that random matrix product states carry magic growing exponentially with system size, demonstrating that generic one-dimensional many-body states harbor ample non-stabilizer resources. Together, these results delineate precisely where genuine quantum advantage emerges in topological and many-body systems. Taken as a whole, this thesis leverages the convergence of topological matter and quantum information to remove key bottlenecks toward fault-tolerant quantum computation, to place near-term quantum simulation on rigorous footing, and to clarify the fundamental resources that separate classical from quantum computational power.

•Jan 1, 2026•0 citations

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

Quantum simulations: 1D black holes and non-abelian anyons

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Numerical simulation is typically used to study the dynamics of complex quantum systems, namely 1-dimensional many-body spin systems and 2-dimensional spin lattice models. Systems of this nature tend to have large or complicated to experimentally realise Hilbert spaces. This results in a difficulty in producing realistic experimental methods of creating interacting spin systems and non-abelian topological phenomena. It also gives a limited success in numerical simulation, requiring advanced methods of quantum simulation to achieve results on an interesting scale. In this thesis we verify known theoretical properties and uncover new features of two such models: a many-body XX Hamiltonian modified with an interacting chiral term and a quantum double model utilised in the manipulation of topological particles. The interacting chiral spin chain has previously shown a rich variety of properties, from the semi-classical black hole background found in its continuum limit, to measurements of Hawking radiation and maximal information scrambling abilities. Our work builds upon these findings. First by analysing the phase order transition from the dominance of the chiral coupling and identifying the role interactions play in the chiral phase that define the interior of a black hole. The interactions of this model are then broken down with the Trotter-Suzuki method into a set of local evolutionary ’gates’. This quantum circuit decomposition of the chiral many-body dynamics gives rise to a potential method of experimental realisation, both of black hole geometry and of its maximal scrambling. Then a lattice model capable of manipulating topological particles is introduced. With this model we produce a methodology of extracting the braiding statistics of a simple non-abelian set of topological quasiparticles based on the S3 group with minimal resources. Within the limitations of this model, we manage to produce the result that the braiding operator this particle set has the ability to produce quantum magic states. This indicates a capability to perform universal quantum computation that can be demonstrated in the laboratory with current technology.

•Sep 1, 2025•0 citations

Quantum computing with anyons is fault tolerant

Anasuya Lyons, Benjamin J. Brown

In seminal work (arxiv:quant-ph/9707021) Alexei Kitaev proposed topological quantum computing (arXiv:cond-mat/0010440, arxiv:quant-ph/9707021, arXiv:quant-ph/0001108, arXiv:0707.1889), whereby logic gates of a quantum computer are conducted by creating, braiding and fusing anyonic particles on a two-dimensional plane. Furthermore, he showed the proposal is inherently robust to local perturbations (arXiv:cond-mat/0010440, arxiv:quant-ph/9707021, arXiv:1001.0344, arXiv:1001.4363) when anyons are created as quasiparticle excitations of a topologically ordered lattice model prepared at zero temperature. Over the decades following this proposal there have been considerable technological developments towards the construction of a fault-tolerant quantum computer. Rather than maintaining some target ground state at zero temperature, a modern approach is to actively correct the errors a target state experiences, where we use noisy quantum circuit elements to identify and subsequently correct for deviations from the ideal state. We present an error-correction scheme that enables us to carry out robust universal quantum computation by braiding anyons. We show that our scheme can be carried out on a suitably large device with an arbitrarily small failure rate assuming circuit elements are below some threshold level of local noise. The error-corrected scheme we have developed therefore enables us to carry out fault-tolerant topological quantum computation using modern quantum hardware that is now under development.

quant-pharXiv:2602.11258•Feb 13, 2026•0 citations

Anyon braiding statistics： From fundamental physics to scalable quantum computing

HE Jianhong, GAO Jie

Topological quantum computation aims to encode quantum information in the nonlocal degrees of freedom of non-Abelian anyons and manipulate it through braiding and fusion operations， offering a promising pathway to fault-tolerant quantum computation. Non-Abelian anyons， with their nontrivial braiding statistics and intrinsic topological fault tolerance， are regarded as a key building block for realizing scalable and decoherence-resistant quantum computing. This review systematically surveys the key developments in anyon physics over the past four decades， from the early theoretical prediction of fractional statistics to direct experimental verification of anyon braiding in fractional quantum Hall states. The physical principles and key findings of essential experimental schemes， including Fabry-Pérot interferometers， Mach-Zehnder interferometers， and anyon colliders， are comprehensively covered. Based on these foundations， the characterization of topological order in even-denominator fractional quantum Hall states， exemplified by the ν=5/2 state， and transport and interferometric probes of non-Abelian statistics are discussed in detail. Furthermore， recent advances in topological quantum states in novel material platforms， including graphene and twisted transition metal dichalcogenides， are reviewed， together with artificial quantum platforms that simulate non-Abelian braiding using superconducting quantum processors. Finally， the key challenges and future directions on the path from anyon physics to scalable topological quantum computing prototype devices are outlined.

•Sep 1, 2026•0 citations

Quantum Field Theories, Topological Materials, and Topological Quantum Computing

Ilyas, Muhammad

A quantum computer can perform exponentially faster than its classical counterpart. It works on the principle of superposition. But due to the decoherence effect, the superposition of a quantum state gets destroyed by the interaction with the environment. It is a real challenge to completely isolate a quantum system to make it free of decoherence. This problem can be circumvented by the use of topological quantum phases of matter. These phases have quasiparticles excitations called anyons. The anyons are charge-flux composites and show exotic fractional statistics. When the order of exchange matters, then the anyons are called non-Abelian anyons. Majorana fermions in topological superconductors and quasiparticles in some quantum Hall states are non-Abelian anyons. Such topological phases of matter have a ground state degeneracy. The fusion of two or more non-Abelian anyons can result in a superposition of several anyons. The topological quantum gates are implemented by braiding and fusion of the non-Abelian anyons. The fault-tolerance is achieved through the topological degrees of freedom of anyons. Such degrees of freedom are non-local, hence inaccessible to the local perturbations. In this dissertation, we provide a comprehensive review of the fundamentals of logic design in topological quantum computing. The braid group and knot invariants in the skein theory are discussed. The physical insight behind the braiding is explained by the geometric phases and the gauge transformation. The mathematical models for the fusion and braiding are presented in terms of the category theory and the quantum deformation of the recoupling theory. The topological phases of matter are described by the topology of band structure. The wave function of quasiparticles in the quantum Hall effect and the theory of Majorana fermions in topological superconductors are also discussed. The dynamics of the charge-flux composites and their Hilbert space are expressed through the Chern-Simons theory and the two-dimensional topological quantum field theory. The Ising and Fibonacci anyonic models for binary gates are briefly given. Ternary logic gates are more compact than their binary counterparts and naturally arise in a type of anyonic model called the metaplectic anyons. We reduced the quantum cost of the existing ternary quantum arithmetic gates and proposed that these gates can be implemented with the metaplectic anyons.

arXiv:2208.09707•Dec 15, 2021•1 citation
Quantum simulations are an invaluable tool in condensed matter physics, enabling the precise generation and control of physical phenomena abstracted from the complexity of the underlying many-body system. At the heart of such complexity lies entanglement, the intrinsically quantum correlations that have attracted renewed interest as a fundamental resource in quantum computation and information. In this thesis, we explore the use of one- and two-dimensional lattice models to simulate and harness such correlations for next-generation quantum technologies. We begin in one dimension with a spin-chain model that allows complete control over bipartite correlation structures. Specifically, we introduce a model of nearest-neighbour couplings and transverse field terms, whose exact solution yields a q-deformed rainbow ground state formed from a tensor product of q-deformed singlets, each with directly tunable entanglement. This framework acts as a universal simulator of free-fermion entanglement spectra, with numerical studies confirming high-fidelity realisations across a range of examples. We then turn to strong correlations in two dimensions, where topologically ordered systems host exotic quasiparticles known as anyons. Braiding non-Abelian anyons enacts unitary transformations on a logically encoded fusion space, offering a natural pathway to fault-tolerant quantum computation. Using Kitaev’s quantum double model D(G), we focus on the smallest non-Abelian instance, D(S3), realised on a lattice of d=6 qudits. We develop a minimal protocol in which braiding and fusion are simulated solely through operators that create and measure anyons, demonstrating that D(S3) anyons generate magic states, a vital resource for universal quantum computation. Finally, a dense encoding scheme is presented, rendering the minimal realisation of this protocol accessible with current quantum technologies and a circuit model for realisation within the framework of linear optics is presented. Notably, error analysis of this encoding scheme reveals an inherent passive protection provided by the topological encoding, extending beyond conventional notions of topological fault tolerance.
A central challenge at the intersection of quantum condensed-matter physics and quantum information science is to understand how exotic topological phases and quantum resources govern the power of quantum computation and simulation---and to translate that understanding into practical architectures. This thesis contributes to this program along three interconnected directions: enabling fault-tolerant universal quantum computation with non-Abelian topological order, establishing the fundamental computational reach of near-term analog quantum simulators, and delineating the boundary between classical and quantum computational power through the lenses of topology and magic. In the first direction, drawing on the structure of non-Abelian topological order, we address the dominant resource overhead in current quantum computing architectures---magic-state distillation---by moving beyond the standard stabilizer formalism entirely. We construct a comprehensive framework for fault-tolerant universal quantum computation based on the non-Abelian quantum double model D(S3)\mathcal{D}(S_3)D(S3​), with all operations realized on qubit--qutrit degrees of freedom in a planar, geometrically local architecture feasible on all current experimental platforms. This work addresses the long-standing open question of realizable, fault-tolerant universal topological quantum computation. In the second direction, we establish the computational universality of analog quantum simulators operating under global control---platforms that circumvent the demanding requirements of digitization and individual addressability. We prove a necessary and sufficient condition for universality in globally controlled qubit chains, showing that widely used platforms such as Rydberg atom arrays and trapped ions are generically universal once lattice reflection symmetry is broken. We further derive sufficient conditions for universality in fermionic and bosonic optical superlattices. As an experimental demonstration of this framework, we realize the dynamics of the cluster-state ZXZZXZZXZ model---an exemplar of symmetry-protected topological order---in Rydberg atom arrays, validating the theoretical foundation. In the third direction, we sharpen the classical--quantum computational boundary from both sides. We show that non-Abelian Majorana braiding realized in classical mechanical metamaterials admits a faithful classical simulation, establishing that braiding-only architectures are classically simulable without supplementary non-Clifford resources. Conversely, we prove that random matrix product states carry magic growing exponentially with system size, demonstrating that generic one-dimensional many-body states harbor ample non-stabilizer resources. Together, these results delineate precisely where genuine quantum advantage emerges in topological and many-body systems. Taken as a whole, this thesis leverages the convergence of topological matter and quantum information to remove key bottlenecks toward fault-tolerant quantum computation, to place near-term quantum simulation on rigorous footing, and to clarify the fundamental resources that separate classical from quantum computational power.
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.
Numerical simulation is typically used to study the dynamics of complex quantum systems, namely 1-dimensional many-body spin systems and 2-dimensional spin lattice models. Systems of this nature tend to have large or complicated to experimentally realise Hilbert spaces. This results in a difficulty in producing realistic experimental methods of creating interacting spin systems and non-abelian topological phenomena. It also gives a limited success in numerical simulation, requiring advanced methods of quantum simulation to achieve results on an interesting scale. In this thesis we verify known theoretical properties and uncover new features of two such models: a many-body XX Hamiltonian modified with an interacting chiral term and a quantum double model utilised in the manipulation of topological particles. The interacting chiral spin chain has previously shown a rich variety of properties, from the semi-classical black hole background found in its continuum limit, to measurements of Hawking radiation and maximal information scrambling abilities. Our work builds upon these findings. First by analysing the phase order transition from the dominance of the chiral coupling and identifying the role interactions play in the chiral phase that define the interior of a black hole. The interactions of this model are then broken down with the Trotter-Suzuki method into a set of local evolutionary ’gates’. This quantum circuit decomposition of the chiral many-body dynamics gives rise to a potential method of experimental realisation, both of black hole geometry and of its maximal scrambling. Then a lattice model capable of manipulating topological particles is introduced. With this model we produce a methodology of extracting the braiding statistics of a simple non-abelian set of topological quasiparticles based on the S3 group with minimal resources. Within the limitations of this model, we manage to produce the result that the braiding operator this particle set has the ability to produce quantum magic states. This indicates a capability to perform universal quantum computation that can be demonstrated in the laboratory with current technology.
In seminal work (arxiv:quant-ph/9707021) Alexei Kitaev proposed topological quantum computing (arXiv:cond-mat/0010440, arxiv:quant-ph/9707021, arXiv:quant-ph/0001108, arXiv:0707.1889), whereby logic gates of a quantum computer are conducted by creating, braiding and fusing anyonic particles on a two-dimensional plane. Furthermore, he showed the proposal is inherently robust to local perturbations (arXiv:cond-mat/0010440, arxiv:quant-ph/9707021, arXiv:1001.0344, arXiv:1001.4363) when anyons are created as quasiparticle excitations of a topologically ordered lattice model prepared at zero temperature. Over the decades following this proposal there have been considerable technological developments towards the construction of a fault-tolerant quantum computer. Rather than maintaining some target ground state at zero temperature, a modern approach is to actively correct the errors a target state experiences, where we use noisy quantum circuit elements to identify and subsequently correct for deviations from the ideal state. We present an error-correction scheme that enables us to carry out robust universal quantum computation by braiding anyons. We show that our scheme can be carried out on a suitably large device with an arbitrarily small failure rate assuming circuit elements are below some threshold level of local noise. The error-corrected scheme we have developed therefore enables us to carry out fault-tolerant topological quantum computation using modern quantum hardware that is now under development.
Topological quantum computation aims to encode quantum information in the nonlocal degrees of freedom of non-Abelian anyons and manipulate it through braiding and fusion operations， offering a promising pathway to fault-tolerant quantum computation. Non-Abelian anyons， with their nontrivial braiding statistics and intrinsic topological fault tolerance， are regarded as a key building block for realizing scalable and decoherence-resistant quantum computing. This review systematically surveys the key developments in anyon physics over the past four decades， from the early theoretical prediction of fractional statistics to direct experimental verification of anyon braiding in fractional quantum Hall states. The physical principles and key findings of essential experimental schemes， including Fabry-Pérot interferometers， Mach-Zehnder interferometers， and anyon colliders， are comprehensively covered. Based on these foundations， the characterization of topological order in even-denominator fractional quantum Hall states， exemplified by the ν=5/2 state， and transport and interferometric probes of non-Abelian statistics are discussed in detail. Furthermore， recent advances in topological quantum states in novel material platforms， including graphene and twisted transition metal dichalcogenides， are reviewed， together with artificial quantum platforms that simulate non-Abelian braiding using superconducting quantum processors. Finally， the key challenges and future directions on the path from anyon physics to scalable topological quantum computing prototype devices are outlined.
A quantum computer can perform exponentially faster than its classical counterpart. It works on the principle of superposition. But due to the decoherence effect, the superposition of a quantum state gets destroyed by the interaction with the environment. It is a real challenge to completely isolate a quantum system to make it free of decoherence. This problem can be circumvented by the use of topological quantum phases of matter. These phases have quasiparticles excitations called anyons. The anyons are charge-flux composites and show exotic fractional statistics. When the order of exchange matters, then the anyons are called non-Abelian anyons. Majorana fermions in topological superconductors and quasiparticles in some quantum Hall states are non-Abelian anyons. Such topological phases of matter have a ground state degeneracy. The fusion of two or more non-Abelian anyons can result in a superposition of several anyons. The topological quantum gates are implemented by braiding and fusion of the non-Abelian anyons. The fault-tolerance is achieved through the topological degrees of freedom of anyons. Such degrees of freedom are non-local, hence inaccessible to the local perturbations. In this dissertation, we provide a comprehensive review of the fundamentals of logic design in topological quantum computing. The braid group and knot invariants in the skein theory are discussed. The physical insight behind the braiding is explained by the geometric phases and the gauge transformation. The mathematical models for the fusion and braiding are presented in terms of the category theory and the quantum deformation of the recoupling theory. The topological phases of matter are described by the topology of band structure. The wave function of quasiparticles in the quantum Hall effect and the theory of Majorana fermions in topological superconductors are also discussed. The dynamics of the charge-flux composites and their Hilbert space are expressed through the Chern-Simons theory and the two-dimensional topological quantum field theory. The Ising and Fibonacci anyonic models for binary gates are briefly given. Ternary logic gates are more compact than their binary counterparts and naturally arise in a type of anyonic model called the metaplectic anyons. We reduced the quantum cost of the existing ternary quantum arithmetic gates and proposed that these gates can be implemented with the metaplectic anyons.