One possible way of proving the famous and important Riemann hypothesis would be to realize the Riemann zeta zeroes as the eigenvalues of some self-adjoint operator, using self-adjointness to show that the non-trivial zeroes all lie on the critical line. This is known as the Hilbert-Polya conjecture. Interpreting such a self-adjoint operator as an observable of some quantum system, one can try to describe the Riemann zeta zeroes in physical terms, making the Hilbert-Polya conjecture an interesting border case between mathematics and physics.

Major work on the Hilbert-Polya conjecture was done by Alain Connes. In his framework, the prolate spheroidal differential operator plays an important but auxiliary role. Before Connes’ work, this second-order differential operator was mainly known for its use in signal analysis, specifically in Slepian’s work on signals which are both time-limited and band-limited. In a more recent article, Connes and Moscovici show that the relation between the prolate spheroidal operator and the Riemann zeta function might be deeper than it seems. They show that a certain self-adjoint realization of the prolate spheroidal operator on the entire real line has discrete spectrum that is asymptotically similar to the squares of the Riemann zeta zeroes. This surprising discovery allows them to construct an operator which approximately solves the Hilbert-Polya conjecture.

Though Connes and Moscovici’s article is wonderful and inspiring, it is tersely written. Hence, in this bachelor thesis, details have been provided to the reasoning of the article. In addition, the physical interpretations of the Riemann zeta zeroes, which support the Hilbert-Polya conjecture, have also been presented.

In the current thesis we argue that quantum mechanics is best understood as a classical theory in which each measurement hides a set of its states. Our aim will however not be to replace quantum mechanical theory by a classical theory of hidden states or to avoid any of its possibly unfavourable metaphysical implications by replacing the current paradigm by a different formulation of the theory. The current thesis on hidden variables is rather guided by the question of how quantum mechanical theory differs from classical mechanics than any attempt to overcome this difference. In contrast to the standard approach to hidden variables, we argue that quantum mechanics in its current formulation itself is best understood as a hidden states theory.

In his 2019 article, Kalinichenko proposed an alternative way of doing stochastic integration in general separable Banach spaces [12].This way circumvents the usual UMD assumption on our separable Banach space X, and instead imposes a strict condition on the integrating process $\Phi : (0,T)\times\Omega\to \Lll(H,X)$. Namely, we require the existence of an $X$-valued Gaussian $g$ such that almost surely for all $x^*\in X^*$,
\[ \int_0^T \|\Phi(t,\omega)^* x^*\|_H^2 \ dt \leq \EE\langle g,x^*\rangle^2. \]
Most notably, this approach works in any separable Banach space. In this thesis we will take a closer look at the proofs used by Kalinichenko, and place his article in the context of the known theory on stochastic analysis in Banach spaces. We will compare the approach both to the UMD and martingale type 2 situation, and discuss the advantages and disadvantages of either strategy.

Moreover, we will compare the conditions imposed in [12] on the stochastic process to the condition of -radonification as assumed in the UMD case [25].

In this report the goals is to theoretically construct a shift resistant encoding in a rotor system. The report starts with an overview of the rotor space. In the context of which the difference between PVM and POVM is highlighted. After this overview, the shift resistant encodings are discussed. These two concepts are then combined and a shift resistant encoding in the rotor space is presented. Lastly a system which hints at an implementation is presented.. Of this system the Hamiltonian is discussed.

In 2005, Jordan showed how to estimate the gradient of a real-valued function with a high-dimensional domain on a quantum computer. Subsequently, in 2017, it was shown by Gilyén et al. how to do this with a different input model. They also proved optimality of their algorithm for \ell^\infty -approximations of functions satisfying some smoothness conditions.

In this text, we expand the ideas of Gilyén et al., and extend their algorithm such that functions with fewer regularity constraints can be used as input. Moreover, we show that their algorithm is essentially optimal in the query complexity to the phase oracle even for classes of functions that have more stringent smoothness conditions. Finally, we also prove that their algorithm is optimal for approximating gradients with respect
to general \ell^p -norms, where p \in [1,\infty].

Furthermore, we investigate how Gilyén et al.’s algorithm can be used to do reinforcement learning on a quantum computer. We elaborate on Montanaro’s ideas for quantum Monte-Carlo simulation, and show how they can be used to implement quantum value estimation of Markov reward processes. We also show essential optimality of this algorithm in the query complexity of all its oracles. Next, we show how we can construct a quantum policy evaluation algorithm, and how we can use these algorithms as subroutines in Gilyén et al.’s quantum gradient estimation algorithm to perform quantum policy optimization.

The most important open questions remain whether it is possible to improve the query complexity of the extension of Gilyén et al.’s algorithm, when function classes containing functions of Gevrey-type 1 are used as input, as at the moment for this specific parameter setting the algorithm is not better than a very simple classical gradient estimation procedure. Improvement of this result would automatically improve the quantum policy optimization routine as well.

This report is a discussion of the 1984 report 'An automatic proof procedure for several geometries' by Th. Bruyn and H.L. Claasen, inspired by a personal desire to understand the work of Th. Bruyn. See: http://resolver.tudelft.nl/uuid:b768c6ce-f625-4236-9b0b-32a47fab143e
Bruyn and Claasen prove that certain true propositions of the theory of intersections within the two-dimensional projective geometry over the real numbers can be formulated by use of figures. It is proven that figures obtained by manipulating these figures will also correspond to propositions. The method to do so proves that the obtained propositions are a direct consequence of the original propositions and are therefore proven to be true. One of their main results is to use the theorem of Pappus to generate the theorem of Desargues, thereby proving that Desargues follows from Pappus (something that is well known in projective geometry).
This report aims to give a comprehensive explanation of their method as well as a detailed demonstration of their procedure. It is a summary of their work with added explanations and examples.