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Quantum simulations provide a useful tool to study a broad range of problems in physics, chemistry, and biology. Trapped ions are a versatile quantum simulator and a main contender for a universal circuit model quantum computer. They provide a highly controllable quantum environment, which grants access to measure phenomena in regimes that are not otherwise accessible in nature. Here, our aim is to show particular examples of quantum physics simulations with a trapped ion quantum computer/simulator. On one hand, we will focus on the simulation of para-particle oscillators; that is, a parity-deformed harmonic oscillator characterized by an order parameter. These oscillators generalize the standard Fermi-Dirac and Bose-Einstein statistics associated with fermions and bosons to para-particles. We realize a method for simulating and characterizing these alternative particles using a trapped-ion experiment. The combination of the Jaynes-Cummings and anti-Jaynes Cummings dynamics present in a trapped ion coupled to multiple modes of motion simultaneously allows us to recover effective Hamiltonians which create a system analogous to para-Fermi or para-Bose oscillators. Trapped ions are a versatile quantum simulator and a main contender for a universal circuit model quantum computer. We use both of these flavors in this project, simulating para-Bosons digitally and para-Fermions directly by tailoring the native ion-mode couplings. We discuss the mapping steps and the latest experimental results. On the other hand, quantum walks provide a powerful framework for quantum simulation of physical systems. As a result, the community has been keen to see first implementations demonstrated, including single-particle relativistic quantum mechanics governed by the Dirac equation. Then, continuing with quantum circuit applications, our work presents a first programmable circuit implementation of quantum walks in one-dimensional space on a trapped-ion processor and its time-evolution up to five steps. The connection between quantum walks and Dirac cellular automata, a modeling framework for complex systems made up of simpler units, allows us to recover the characteristic spreading probability of an initially localized relativistic Dirac particle for different values of its mass. Due to a resource-efficient mapping of qubit states to walker positions, we achieve a high simulation performance despite going to deep quantum circuits (of up to 32 entangling gates).

We introduce the cross-cavity quantum Rabi model describing the interaction of a single two-level system with two orthogonal boson fields and propose its quantum simulation by two-dimensional, bichromatic, first-sideband driving of a single trapped ion. Like a first try of provide an analytical solution to our model, we solved the model in the weak coupling regime with Schwinger two-boson representation of SU(2). We introduce the Fulton-Gouterman approach to diagonalize the CCQRM in the two-level system basis, we provide an introductory survey of the model, including its diagonalization in the two-level system basis, numerical spectra and its characteristics in the weak, ultra strong and deep strong coupling regimes are included. We also show that the particular case of degenerate field frequencies and balanced couplings allows us to cast the model as two parity deformed oscillators in any given coupling regime. Presentamos el modelo de Rabi cuántico en cavidades cruzadas que describe un único sistema de dos niveles que interactúa con dos campos ortogonales de bosones y proponemos su simulación cuántica utilizando dos pares de campos biocromáticos ortogonales, sintonizados en la primera banda de transición de un sólo ion atrapado. Como primer intento para dar una solución analítica a nuestro modelo, estudiamos el régimen de acoplamiento débil en la representación de Schwinger de SU(2) de dos bosones. Y, para el caso más general, introducimos el formalismo de Fulton-Gouterman para diagonalizar el Hamiltoniano en la base del qubit, lo que nos permite estudiar numéricamente espectro y sus características en los regímenes de acoplamiento débil, ultra-fuerte y fuerte-profundo. También mostramos que el caso particular de campos degenerados en frecuencia y acoplamientos iguales nos permite ver el modelo como dos osciladores con paridad deformada en cualquier régimen de acoplamiento.