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.
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.
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), 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.
Gauge theories are ubiquitous in physics. Many intriguing phenomena in condensed matter physics owe to the action of the electromagnetic field, which is an Abelian gauge theory. The numerical treatment of many-body systems is inherently complex due to the exponentially growing size of the Hilbert space. While in one dimension an area law guarantees that numerical methods on classical computers can deal with strongly correlated systems, in higher dimensions the quantum simulation comes as the panacea for the many-body problem. The present thesis comprises the elaboration of experimentally feasible methods for the quantum simulation of dynamical Abelian gauge fields with ultra-cold gases of neutral atoms and the theoretical analysis of the related model Hamiltonians. As neutral atoms do not interact with external vector potentials like charged particles would do, the gauge fields have to be artificially engineered. The elements of a gauge theory that need to be replicated on a quantum simulator vary depending on the subject of investigation. The key ingredient at the root of many condensed matter phenomena, from the quantum Hall effect to superconductivity and chiral topological insulators, is the Berry phase. Whilst artificial static gauge fields have been widely explored, much remains to do regarding the realization of artificial dynamical gauge fields. In Chapter 3 we present a method based on the amplitude modulation of a one-dimensional optical lattice, which allows for an unprecedented degree of control over a wide range of parameters. The method also comprises the generation of a density-dependent complex phase, fundamental to the creation of anyonic pseudo-particles. The anyons are amenable of observation through interferometric measurement, realizable with the same experimental set-up. With regard to gauge theories, the Berry phase is just the visible tip of the iceberg. Below the waterline, there is more to consider in order to comprehensively reproduce a gauge theory, like the electric and magnetic fields in quantum electrodynamics. Moreover, a full account for the inherent symmetry is crucial to investigate phenomena proper of non-Abelian gauge theories in the context of high-energy physics, such as confinement. For this collection of topics, one can turn to lattice gauge theories. In Chapter 5, we consider a class of lattice gauge theories particularly suitable for quantum simulation, the Quantum Link Model. The study of the Abelian U(1) Quantum Link Model on a ladder geometry reveals a highly non-trivial phase diagram, featuring a symmetry-protected topological phase. In both Chapters, innovative solutions for the experimental realization of the model Hamiltonians are designed and proposed. To gain numerical access to the ground-state properties and the dynamics of the systems investigated we make use of state-of-the-art numerical methods based on Tensor Networks. The elements of the numerical analysis carried out throughout this thesis are presented in Chapter 6. In the last part we offer an outlook on research perspectives related to the topics discussed in the thesis.
Chaos in many-body quantum systems is of great importance to both many-body physics as well as black hole physics. In the field of many-body physics, interactions and disorder in the system can lead to various dynamic phenomena, be it thermalisation, many-body localisation, chaotic behavior and scrambling of quantum information. In the field of high energy physics, the holographic principle connects chaos and scrambling in many-body quantum systems with the information-theoretic properties of black holes. Theoretical quantum physics provides the framework for the models, methods, and results in this thesis, while black hole physics partly provides some motivation and inspiration. First, we introduce a new model for a many-body quantum system based on random quantum circuits. These are a popular framework for theoretic study of disordered spin chains. By drawing the random unitaries in the circuit from different ensembles, we can adjust the disorder strength in the interactions, which in turn leads to a thermal/many-body localisation phase transition. Next, we study the Brownian SYK model, a disordered model of Majorana fermions with all-to-all interactions, motivated by its link to the holographic principle. We develop a new numerical method based on an effective permutational symmetry to reduce computational cost from exponential to linear or quadratic in system size N. As a consequence, we can compute scrambling quantifiers in detail, and find a log N scrambling time, as conjectured in the context of fast scrambling for black holes. Finally, we develop a model based on the continuous-time limit of a random quantum circuit. It serves as a microscopic toy model for the evaporation of a black hole. With a similar method as developed for the Brownian SYK model, we can analyse its information theoretic properties. In particular, we follow established protocols for information retrieval from the Hawking radiation. We find a separation of time scales for entanglement growth and information retrieval, related to the intrinsic black hole dynamics (∝log N) and the coupling to the environment (∝N).
Abstract Quantum many-body systems with a non-Abelian topological order can host anyonic quasiparticles. It has been proposed that anyons could be used to encode and manipulate information in a topologically protected manner that is immune to local noise, with quantum gates performed by braiding and fusing anyons. Unfortunately, realizing non-Abelian topologically ordered states is challenging, and it was not until recently that the signatures of non-Abelian statistics were observed through digital quantum simulation approaches. However, not all forms of topological order can be used to realize universal quantum computation. Here we use a superconducting quantum processor to simulate non-Abelian topologically ordered states of the Fibonacci string-net model and demonstrate braidings of Fibonacci anyons featuring universal computational power. We demonstrate the non-trivial topological nature of the quantum states by measuring the topological entanglement entropy. In addition, we create two pairs of Fibonacci anyons and demonstrate their fusion rule and non-Abelian braiding statistics by applying unitary gates on the underlying physical qubits. Our results establish a digital approach to explore non-Abelian topological states and their associated braiding statistics with current noisy intermediate-scale quantum processors.
(English) In recent years, the evolution of quantum technologies has resulted in an unprecedented control over individual quantum particles and many-body systems. This remarkable progress has given rise to a new era, marked by the convergence of classical and quantum methodologies to investigate fundamental questions concerning the nature of quantum matter, improving our understanding of the role of entanglement in solid-state systems or the mechanisms behind high-energy physics. From analog quantum simulators to digital quantum computers, these advancements hold potential across diverse domains. This thesis explores the interplay between classical and quantum methods in understanding topological phases of matter. We concentrate on three distinct directions: non-conventional topological superconductors, interaction-induced topological phases in ultracold atom quantum simulators, and applications of variational quantum algorithms. Each trajectory relies on the combination of different techniques with the aim of understanding and characterizing topological phenomena in different settings. Exploring non-conventional topological superconductors involves extending the paradigmatic Kitaev chain model by incorporating additional terms in the Hamiltonian such as long-range interactions and quasi-periodic potentials. This investigation is relevant to better understand the impact of real-world conditions on the both the topological and localization properties of systems hosting non-local Majorana modes, which are promising candidates for topological quantum computation. In the realm of interacting systems, we explore the realization of interaction-induced topological phases in systems of ultracold atoms in optical lattices, both in one and two dimensions. The remarkable control and versatility of such platforms enable the simulation of both theoretical topological models and strongly correlated physics. Notably, the interplay between interactions and topology can give rise to intriguing phenomena, such as delocalized fractional charges and gapless topological phases,challenging existing intuition. We employ advanced numerical methods based on tensor networks to benchmark the experimental proposals that open the door to the realization and detection of novel many-body phases of matter, including topological quantum critical points and a higher-order topological Peierls insulator in Bose-Hubbard models with long-range interactions. Variational quantum algorithms, conversely, have the potential to efficiently tackle a wide range of problems, including ground state search, phase classification or accessing topological invariants. Despite current limitations in trainability and scalability, these hybrid classical-quantum algorithms provide practical insights into current quantum hardware capabilities and can inspire future architectures. We explore the application of variational quantum algorithms to shed light on topological phenomena, raising questions about their ability to discern topological phase transitions and compute topological invariants in situations where classical approaches fail. This thesis presents a comprehensive exploration of distinct approaches to topological quantum matter by leveraging quantum technologies and quantum-inspired classical algorithms. Our results not only advance our understanding of quantum systems but also pave the way for the realization and discovery of novel physics extending to quantum information processing, materials science, and beyond.