Classical and quantum simulation of quantum systems

Despite the increasing power of classical computers, the computational simulation of quantum systems remains in general a difficult and hard problem. The main reason for the difficulties is the exponential growth of state space of multiple quantum systems, which makes a classical treatment of even a relatively small number of quantum systems intractable and e.g. prevents us from a deeper understanding of strongly correlated systems. Possible approaches to overcome this limitation include the truncation of Hilbert space to a subspace of polynomial size (as in DMRG) in such a way that the resulting states are sufficiently close to the exact one, and the usage of quantum systems (which itself have an exponentially large state space) to simulate other quantum systems. This research project follows two lines of investigation. The first line is concerned with new methods to achieve an efficient classical simulation of quantum systems. To this aim, families of states -so called weighted graph states- will be studied with respect to their entanglement properties and used for state modelling. That is, states of this kind will be constructed in such a way that they resemble characteristic entanglement properties of (critical) spin systems, and a connection to the ground states of such systems will be drawn. We expect that this shines new light on the role of entanglement in quantum phase transitions and allow for the development of new ways to describe such states and simulate the corresponding spin systems efficiently. These states will also be used to develop an efficient classical simulation of the entanglement properties of certain quantum systems, including toy models such as the "semi-quantal Boltzmann gas" or more realistic models such as "quantum lattice gases", which could be realized experimentally, e.g. with ultracold atoms in optical lattices. The second line of investigations is concerned with novel ways of using finite dimensional quantum systems to simulate the dynamics of continous variable quantum systems by tayloring local interactions. Optimal ways of discretization and encoding will be studied, and the tractability of different potentials with this method will be investigated. It is conceivable that such quantum simulators will be used in the future to investigate mathematical models of complex classical and quantum phenomena.

Despite the increasing power of classical computers, the computational simulation of quantum systems remains in general a difficult and hard problem. The main reason for the difficulties is the exponential growth of state space of multiple quantum systems, which makes a classical treatment of even a relatively small number of quantum systems intractable and e.g. prevents us from a deeper understanding of strongly correlated systems. Possible approaches to overcome this limitation include the truncation of Hilbert space to a subspace of polynomial size (as in DMRG) in such a way that the resulting states are sufficiently close to the exact one, and the usage of quantum systems (which itself have an exponentially large state space) to simulate other quantum systems. This research project follows two lines of investigation. The first line is concerned with new methods to achieve an efficient classical simulation of quantum systems. To this aim, families of states - so called weighted graph states - will be studied with respect to their entanglement properties and used for state modelling. That is, states of this kind will be constructed in such a way that they resemble characteristic entanglement properties of (critical) spin systems, and a connection to the ground states of such systems will be drawn. We expect that this shines new light on the role of entanglement in quantum phase transitions and allow for the development of new ways to describe such states and simulate the corresponding spin systems efficiently. These states will also be used to develop an efficient classical simulation of the entanglement properties of certain quantum systems, including toy models such as the "semi-quantal Boltzmann gas" or more realistic models such as "quantum lattice gases", which could be realized experimentally, e.g. with ultracold atoms in optical lattices. The second line of investigations is concerned with novel ways of using finite dimensional quantum systems to simulate the dynamics of continous variable quantum systems by tayloring local interactions. Optimal ways of discretization and encoding will be studied, and the tractability of different potentials with this method will be investigated. It is conceivable that such quantum simulators will be used in the future to investigate mathematical models of complex classical and quantum phenomena.