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Quantum graphs are commonly used as models of complex quantum systems, for example molecules, networks of wires, and states of condensed matter. We consider quantum statistics for indistinguishable spinless particles on a graph,…
This article presents a novel and succinct algorithmic framework via alternating quantum walks, unifying quantum spatial search, state transfer and uniform sampling on a large class of graphs. Using the framework, we can achieve exact…
In this contribution we review the theory of integrability of quantum systems in one spatial dimension. We introduce the basic concepts such as the Yang-Baxter equation, commuting currents, and the algebraic Bethe ansatz. Quite extensively…
The article surveys quantization schemes for metric graphs with spin. Typically quantum graphs are defined with the Laplace or Schrodinger operator which describe particles whose intrinsic angular momentum (spin) is zero. However, in many…
We extend the graphical calculus developed in the first part of this paper to the parametrising spaces of quantum vertex operators. This involves a graphical implementation of the dynamical twist functor, which is a strict monoidal functor…
A novel algebra underlying integrable systems is shown to generate and unify a large class of quantum integrable models with given $R$-matrix, through reductions of an ancestor Lax operator and its different realizations. Along with known…
A quantum integrable spin chain model associated with the $G_2$ exceptional Lie algebra is studied. By using the fusion technique, the closed recursive relations among the fused transfer matrices are obtained. These identities allow us to…
Graphical Models have various applications in science and engineering which include physics, bioinformatics, telecommunication and etc. Usage of graphical models needs complex computations in order to evaluation of marginal functions,so…
Matrix representations of quantum operators are computationally complete but often obscure the structural topology of information flow within a quantum circuit \cite{nielsen2000}. In this paper, we introduce a generalized graph-theoretic…
Using Reshetikhin's construction for multiparametric quantum algebras we obtain the associated multiparametric quantum spin chains. We show that under certain restrictions these models can be mapped to quantum spin chains with twisted…
Quantum graphs are a paradigmatic model for quantum chaos as well as for spectral theory. We give a concise didactical introduction to quantum graphs, or Schr\"odinger Hamiltonians on metric graphs, with a focus on results related to…
These lectures discuss the formulation of quantum mechanics with fractional spin and statistics in 2+1 dimensions in a relativistic setting, emphasizing the path-integral approach. The non-relativistic theory is reviewed from a…
We study Ising spin models on finitely connected random interaction graphs which are drawn from an ensemble in which not only the degree distribution $p(k)$ can be chosen arbitrarily, but which allows for further fine-tuning of the topology…
The different quantum phases appearing in strongly correlated systems as well as their transitions are closely related to the entanglement shared between their constituents. In 1D systems, it is well established that the entanglement…
We suggest the notion of perfect integrability for quantum spin chains and conjecture that quantum spin chains are perfectly integrable. We show the perfect integrability for Gaudin models associated to simple Lie algebras of all finite…
We develop a framework to simulate quantum field theories (QFTs) with boundaries in $(1+1)$-dimenmsional curved spacetimes by employing open spin systems. Building upon our previous work that established a mapping from spin systems to QFTs…
The following work is an exploration into certain topics in the broad world of integrable models, both classical and quantum, and consists of two main parts of roughly equal length. The first part, consisting of chapters 1-3, concerns…
Presented are several example quantum computing representations of quantum systems with a relativistic energy relation. Basic unitary representations of free Dirac particles and BCS superconductivity are given. Then, these are combined into…
A general analytical approach to the statistical description of quantum graph spectra based on the exact periodic orbit expansions of quantum levels is discussed. The exact and approximate expressions obtained in \cite{Anima} for the…
We construct commuting transfer matrices for models describing the interaction between a single quantum spin and a single bosonic mode using the quantum inverse scattering framework. The transfer matrices are obtained from certain…