The study of Opinion Dynamics, which explores how individual opinions and beliefs evolve and how societal consensus is formed, has been examined across social science, physics, and mathematics. Historically based on statistical physics models like the Ising model, recent research integrates quantum information theory concepts, such as Graph States, Stabilizer States, and Toric Codes. These quantum approaches offer fresh perspectives for analyzing complex relationships and interactions in opinion formation, such as modeling local interactions, using topological features for error resistance, and applying quantum mechanics for deeper insights into opinion polarization and entanglement. However, these applications face challenges in complexity, interpretation, and empirical validation. Quantum concepts are abstract and not easily translated into social science contexts, and direct observation of social opinion processes differs significantly from quantum experiments, leading to a gap between theoretical models and real-world applicability. Despite its potential, the practical use of the Toric Code Hamiltonian in Opinion Dynamics requires further exploration and research.
The study of Opinion Dynamics, which explores how individual opinions and beliefs evolve and how societal consensus is formed, has been examined across social science, physics, and mathematics. Historically based on statistical physics models like the Ising model, recent research integrates quantum information theory concepts, such as Graph States, Stabilizer States, and Toric Codes. These quantum approaches offer fresh perspectives for analyzing complex relationships and interactions in opinion formation, such as modeling local interactions, using topological features for error resistance, and applying quantum mechanics for deeper insights into opinion polarization and entanglement. However, these applications face challenges in complexity, interpretation, and empirical validation. Quantum concepts are abstract and not easily translated into social science contexts, and direct observation of social opinion processes differs significantly from quantum experiments, leading to a gap between theoretical models and real-world applicability. Despite its potential, the practical use of the Toric Code Hamiltonian in Opinion Dynamics requires further exploration and research.
Kitaev's toric code is an exactly solvable model with Z2-topological order, which has potential applications in quantum computation and error correction. However, a direct experimental realization remains an open challenge. Here, we propose a building block for Z2 lattice gauge theories coupled to dynamical matter and demonstrate how it allows for an implementation of the toric-code ground state and its topological excitations. This is achieved by introducing separate matter excitations on individual plaquettes, whose motion induce the required plaquette terms. The proposed building block is realized in the second-order coupling regime and is well suited for implementations with superconducting qubits. Furthermore, we propose a pathway to prepare topologically nontrivial initial states during which a large gap on the order of the underlying coupling strength is present. This is verified by both analytical arguments and numerical studies. Moreover, we outline experimental signatures of the ground-state wave function and introduce a minimal braiding protocol. Detecting a $\ensuremath{\pi}$-phase shift between Ramsey fringes in this protocol reveals the anyonic excitations of the toric-code Hamiltonian in a system with only three triangular plaquettes. Our work paves the way for realizing non-Abelian anyons in analog quantum simulators.
Abstract The Kitaev honeycomb model is an approximate topological quantum error correcting code in the same phase as the toric code, but requiring only a 2-body Hamiltonian. As a frustrated spin model, it is well outside the commuting models of topological quantum codes that are typically studied, but its exact solubility makes it more amenable to analysis of effects arising in this noncommutative setting than a generic topologically ordered Hamiltonian. Here we study quantum error correction in the honeycomb model using both analytic and numerical techniques. We first prove explicit exponential bounds on the approximate degeneracy, local indistinguishability, and correctability of the code space. These bounds are tighter than can be achieved using known general properties of topological phases. Our proofs are specialized to the honeycomb model, but some of the methods may nonetheless be of broader interest. Following this, we numerically study noise caused by thermalization processes in the perturbative regime close to the toric code renormalization group fixed point. The appearance of non-topological excitations in this setting has no significant effect on the error correction properties of the honeycomb model in the regimes we study. Although the behavior of this model is found to be qualitatively similar to that of the standard toric code in most regimes, we find numerical evidence of an interesting effect in the low-temperature, finite-size regime where a preferred lattice direction emerges and anyon diffusion is geometrically constrained. We expect this effect to yield an improvement in the scaling of the lifetime with system size as compared to the standard toric code.
The superposition principle of quantum mechanics implies that the amount of classical (i.e., non-quantum) information needed to describe a quantum system is in general exponentially large in the size of the system. This makes simulating quantum physics on a classical computer quickly infeasible, even for moderately-sized systems and using the best present-day supercomputers in the world. This is unfortunate, since systems for which quantum effects are relevant are of interest in many areas of science and engineering, ranging from drug design to materials research. In 1982, Richard Feynman gave birth to the idea of a quantum computer, when he realized that simulating quantum physics could be achieved much more efficiently with a computer that was itself a quantum system. Since then, the number of problems for which a quantum computer is known to enjoy an advantage over a classical computer has steadily increased, ranging now far beyond the task of simulating quantum physics. A quantum computation of any interest will need to create complicated superposition states, for otherwise it would not achieve anything that could not be simulated on a classical computer. These superposition states, however, are highly fragile, and this fragility is the main obstacle that the scientific community faces on the road towards a useful quantum computer. Quantum computing will only be possible if it can be performed in a fault-tolerant way. It is due to seminal work by Alexei Kitaev from 1997 that the modern theory of quantum fault-tolerance is closely related to the term topology. Topology, as a sub-field of mathematics, is concerned with the properties of space that are preserved under continuous deformations. Topological order, a purely quantum phenomenon, refers to states which cannot be distinguished or evolved into each other locally, yet are globally distinct. The information that distinguishes between these states is stored in non-local degrees of freedom. If it were possible to store quantum information in these non-local degrees of freedom, this would be hugely attractive from a practical perspective, as it would mean that the information stored this way is immune to many forms of local errors. Two-dimensional topologically ordered states support excitations known as anyons. These are exotic quasi-particles that defy the dichotomy between fermions and bosons that applies to quantum particles in three spatial dimensions. In a topological quantum computer, the non-locally stored quantum information is processed by braiding these anyons around each other. Such a computation would be insensitive to small perturbations of the path along which anyons are braided, but would only depend on its topological properties. An open problem is whether topological order can persist at finite temperature. Equivalently, one can ask whether it is possible to build a system in which quantum information can be stored in a stable manner for arbitrarily long times, without performing active error correction and despite constant influence of a thermal environment. Such a system would constitute a self-correcting quantum memory and would extend the concept of a hard disk drive to the quantum realm. Whether nature allows for such a system to be built is of tremendous interest from both a fundamental and a practical perspective. Two-dimensional topologically ordered systems – those hosting anyons – do not qualify as self-correcting quantum memories; any finite temperature corrupts them in a time that is independent of the size of the system. In the present thesis, we propose and study systems in which the thermal stability of a simple toy model of an anyonic system, Kitaev’s toric code, is enhanced in various ways. We consider coupling it to optical cavity modes, to bosonic particles, or to a ferromagnet. These auxiliary systems then induce long-ranged interactions between the anyons, which allow to increase the finite-temperature lifetime of the stored quantum information arbitrarily by increasing the size of the system. While a topological quantum computer is naturally immune to many forms of imperfections and perturbations, accidental creation of anyonic quasi-particles due to coupling to an external environment is a form of error that requires active correction. This problem has seen surprisingly little attention until recently and is still poorly understood. In this thesis, we develop algorithms that are able to perform this task and provide the first proofs of its in-principle feasibility. Furthermore, we develop a system with the rare property that it both supports anyons that can be used for topological quantum computing, and allows for error correction with well-established techniques. Historically, the first proposals for building a fault-tolerant quantum \ncomputer involved an array of qubits (quantum bits) and a set of elementary operations that can be performed on individual and pairs of qubits. Fault-tolerant qubit-based quantum computing has been inspired tremendously from the insights gained in the study of topological quantum information processing. The surface code, which combines a qubit-based architecture with topological methods, is now at the forefront of the quest towards a fault-tolerant, scalable quantum computer. In the surface code, a large number of measurements are performed on a continuous basis to get some information about what errors have occurred. This information then needs to be converted by a classical algorithm into a prescription for performing error correction. In this thesis, we develop such an algorithm that, at the time of its publication, was the best efficient algorithm known for the surface code. Error correction for qubit-based quantum computers is typically studied with simplistic error models in which the errors on each qubit are independent from each other. We study what kinds of spatial and temporal correlations between the errors arise when a surface code is coupled to a typical model of an environment, and how they affect its correctability.
Quantum spin liquids and anyons, used to be subjects of condensed matter physics, now are realized in various platforms of qubits, offering unprecedented opportunities to investigate fundamental physics of many-body quantum entangled states. Qubits are inevitably exposed to environment effects such as decoherence and dissipation, which are believed to be detrimental to many-body entanglement. Here, we argue that unlike the common belief decoherence and dissipation can give rise to novel topological phenomena in quantum spin liquids. We study open quantum systems of the Kitaev spin liquid and the toric code via the Lindblad master equation approach. By using exact solutions and numerical approaches, we show the dynamical occurrence of anyon condensation by decoherence and dissipation, which results in a topological transition from the initial state spin liquid to the steady state spin liquid. The mechanism of the anyon condensation transition by the Lindblad dynamics is elucidated. We also provide an insight into the relationship between the Kitaev spin liquid and the toric code in the picture of anyon condensation. Our work suggests open quantum systems to be a new venue for topological phenomena of quantum spin liquids and anyons.
In this paper, the authors apply models extracted from the Many-Body Quantum Mechanics to understand how knowledge production is correlated to the innovation potential of a work team. This study is grounded in key assumtpions. First, complexity theory applied to social science suggests that it is of paramount importance to consider elements of non-objectivity and non-determinism in the statistical description of socio-economic phenomena. Second, a typical factor of indeterminacy in the explanation of these phenomena lead to the need to apply the instruments of quantum physics to formally describe social behaviours. In order to experiment the validity of the proposed mathematic model, the research intends to: 1) model nodes and interactions; 2) simulate the network behaviour starting from specific defined models; 3) visualize the macroscopic results emerging during the analysis/simulation phases through a digital representation of the social network.
Quantum mechanics is traditionally associated with microscopic systems; however, quantum concepts have also been successfully applied to a diverse range of macroscopic systems both within and outside of physics. This book describes how complex systems from a variety of fields can be modelled using quantum mechanical principles; from biology and ecology, to sociology and decision-making. The mathematical basis of these models is covered in detail, furnishing a self-contained and consistent approach. This book provides unique insight into the dynamics of these macroscopic systems and opens new interdisciplinary research frontiers. It will be an essential resource for students and researchers in applied mathematics or theoretical physics who are interested in applying quantum mechanics to dynamical systems in the social, biological or ecological sciences.