As the realisation of a fully scalable quantum computer remains distant, the simulation of many-body quantum systems becomes an intermediate goal of great practical importance. This thesis focusses on the simulation of continuous quantum systems by application of variational methods. Such methods rely on a choice of variational ansatz states, which when constructed appropriately can be used to successfully determine properties of physical systems and simulate their dynamics. Applications of variational methods in this thesis employ the family of continuous matrix product states (cMPS) for quantum field theories and continuum models in one spatial dimension as a variational class.
We develop a variational method exploiting the natural physics of cavity quantum elec- trodynamics (cavity QED) architectures to simulate interacting quantum fields. The natural interpretation of the output of cavity QED apparatuses as a cMPS is exploited, allowing for an analogue quantum simulation procedure using current technology. We demonstrate that the paradigmatic cavity QED system comprising a single trapped atom coupled to a single cavity mode is capable of simulating the ground state physics of an equally paradigmatic quantum field, namely the Lieb-Liniger model. We find that varying the adjustable parame- ters of the cavity QED system within an experimentally feasible parameter regime, and in the presence of losses, allows for the quantum simulation of Lieb-Liniger ground state physics. The scheme can also be extended to simulate systems of entangled multi-component fields, beyond the reach of existing classical simulation methods.
Furthermore, we develop an algorithm that allows for the simulation of the dynamics of a continuous quantum system under the action of a random potential. The exact simulation of the dynamics of such a continuous quantum random system (cQRS) would usually require an infinite number of evolutions, corresponding to each realisation of the random potential. We avoid this impracticable task by introducing an auxiliary system, such that a mapping between the cQRS and an interacting, non-random, system is established. By means of an extension of the time-dependent variational principle using multi-component cMPS, we explicitly derive the equations of motion determining the time evolution of the cMPS variational parameters, allowing for numerical simulation of the dynamics of the interacting quantum fields. In a simple one-dimensional case we obtain the Gross Pitaevskii equation and thereby verify our results.