Esteban Martinez

Lattice gauge theories describe fundamental phenomena in nature, but calculating their real-time dynamics on classical computers is notoriously difficult. In a recent publication [Nature 534, 516 (2016)], we proposed and experimentally demonstrated a digital quantum simulation of the paradigmatic Schwinger model, a U(1)-Wilson lattice gauge theory describing the interplay between fermionic matter and gauge bosons. Here, we provide a detailed theoretical analysis of the performance and the potential of this protocol. Our strategy is based on analytically integrating out the gauge bosons, which preserves exact gauge invariance but results in complicated long-range interactions between the matter fields. Trapped-ion platforms are naturally suited to implementing these interactions, allowing for an efficient quantum simulation of the model, with a number of gate operations that scales only polynomially with system size. Employing numerical simulations, we illustrate that relevant phenomena can be observed in larger experimental systems, using as an example the production of particle--antiparticle pairs after a quantum quench. We investigate theoretically the robustness of the scheme towards generic error sources, and show that near-future experiments can reach regimes where finite-size effects are insignificant. We also discuss the challenges in quantum simulating the continuum limit of the theory. Using our scheme, fundamental phenomena of lattice gauge theories can be probed using a broad set of experimentally accessible observables, including the entanglement entropy and the vacuum persistence amplitude.

We study the thermodynamical properties of crystals of trapped ions which are laser cooled to two different temperatures in two separate regions. We show that these properties strongly depend on the structure of the ion crystal. Such structure can be changed by varying the trap parameters and undergoes a series of phase transitions from linear to zig-zag or helicoidal configurations. Thus, we show that these systems are ideal candidates to observe and control the transition from anomalous to normal heat transport. All structures behave as `heat superconductors', with a thermal conductivity increasing linearly with system size and a vanishing thermal gradient inside the system. However, zig-zag and helicoidal crystals turn out to be hyper sensitive to disorder having a linear temperature profile and a length independent conductivity. Interestingly, disordered 2D ion crystals are heat insulators. Sensitivity to disorder is much smaller in the 1D case.

Quantum computers hold the promise to solve certain problems exponentially faster than their classical counterparts. Trapped atomic ions are among the physical systems in which building such a computing device seems viable. In this work we present a small-scale quantum information processor based on a string of $^{40}$Ca${^+}$ ions confined in a macroscopic linear Paul trap. We review our set of operations which includes non-coherent operations allowing us to realize arbitrary Markovian processes. In order to build a larger quantum information processor it is mandatory to reduce the error rate of the available operations which is only possible if the physics of the noise processes is well understood. We identify the dominant noise sources in our system and discuss their effects on different algorithms. Finally we demonstrate how our entire set of operations can be used to facilitate the implementation of algorithms by examples of the quantum Fourier transform and the quantum order finding algorithm.