Classical, quantum and numerical aspects of modified theories of gravity

Abstract
This thesis considers modifications of some specific and well-known gravity theories. In particular, linearised infinite-derivative gravity theories are studied in both four-dimensional and two-dimensional space-time. For the four-dimensional case, some specific quantum aspects of infinite-derivative gravity are examined. This includes an examination of the non-static potential that arises in such a theory when two spinless particles exchange a graviton. Infinite-derivative modifications of two specific two-dimensional gravity theories that have a dilaton field are also constructed in the linearised regime. Non-local modifications to the linearised black-hole solutions of the local theories are then obtained. It is found that the obtained linearised non-local solutions are free of the singular nature that is present in the local case. Finally, a numerical relativity code is constructed to study the evolution of a massless scalar field in the context of the four-dimensional Starobinsky gravity model which is a modification of Einstein’s theory of General Relativity.

The power of one qubit in quantum simulation algorithms

Abstract
This thesis focuses on developing new quantum algorithms, targeting some of the key challenges in the simulation of complex quantum systems.The techniques introduced in this thesis span from quantum state preparation to mitigation of hardware and algorithmic noise, from efficient expectation value measurement to noise-resilient applications in quantum chemistry. 
Quantum computing is an emerging technology, which holds the potential to simulate complex quantum systems beyond the reach of classical numerical methods.Despite recent formidable advancements in quantum hardware, constructing a quantum computer capable of performing useful calculations remains challenging.In the absence of a reliable quantum computer, the study of potential applications relies on mathematical methods, ingenious approximations, and heuristics derived from the fields of application.
A common thread connecting all these algorithms is the introduction of a single auxiliary qubit – a fundamental unit of quantum information – which has an active and distinctive role in the task at hand.

Random Geometry and Quantum Spacetime: From scale-invariant random geometries and asymptotic safety to random hyperbolic surfaces and JT gravity

Abstract
This thesis is driven by a central question: “What can we learn from random geometries about the structure of quantum spacetime?” In Chapter 2, we provide a partial review of the mathematical foundation of this thesis, random geometry. In Chapter 3, we use a construction coming from random geometry called Mating of Trees to build scale-invariant random geometries that appear in Liouville Quantum Gravity and have the potential to implement the UV fixed point predicted by Asymptotic Safety in two and three dimensions. In Chapter 4 we explore the random geometry formulation of JT gravity and how our understanding of random critical maps yields the discovery of a new family of deformations of JT gravity. Furthermore, the connection between JT gravity and matrix models leads us to delve deeper into the link between discrete geometry and hyperbolic surfaces, building upon the geometry of metric maps and irreducible metric maps in Chapter 5.