In this thesis, a variation on the nonlinear Schrödinger (NLS) equation with multiplicative noise is studied. In particular, we consider a stochastic version of the parametrically-forced nonlinear Schrödinger equation (PFNLS), which models the effect of linear loss and the compensation thereof by phase-sensitive amplification in pulse propagation through optical fibers. We establish global existence and uniqueness of mild solutions for initial data in L2(R) and H1(R).
The proof is an adaptation of a fixed-point argument employed by de Bouard and Debussche [Comm. Math. Phys., 205:161-181, 1999] for the nonlinear Schrödinger equation with multiplicative noise. The fixed-point argument relies on space-time estimates on the semigroup generated by the linear parametrically-forced Schrödinger operator. We prove these so-called Strichartz estimates, originally proven for the Schrödinger operator, using Fourier methods. A key difference between the Schrödinger operator and its parametrically-forced version is that the latter is not self-adjoint. We overcome this complication by establishing fixed-time estimates on the semigroup and its adjoint, based on their Fourier representations. We also briefly discuss possible future research in the direction of stability of solitary standing wave solutions of the PFNLS equation under the influence of multiplicative noise. Using informal calculations, we demonstrate an approach to track the displacement of a soliton due to small stochastic forcing.

In this work, a quantum self-test which certifies that measurements of a quantum device have a large Gowers norm of order k is presented and analysed. The test protocol is described as a two-player quantum game, in which players provide answers based on measurements on subsystems of a maximally entangled
bipartite state. The protocol makes use of 2k +1 subtests. It is shown that strategies that succeed in the test with success rate 1-E must have a Gowers norm larger than 1-O(E). This test generalizes measurement tests that certify measurements based on the second order Gowers norm.

In this thesis we introduce a variation on the quantum random walk to discuss shifts in an arbitrary range. The concept of Hadamard coin was therefore generalised to a higher order. By a Fourier transform method and a tensor product decomposition of the evolution matrix the long-range quantum random walk was found to converge in distribution to a random variable, different for every range. The limiting random variable consists of three parts: one part fast decaying with the range size, a non-convergent part and a convergent part. Lastly, an introduction was made into the topic of trapped quantum random walks. As a starting point, the survival probability of such a walk on a 3-cycle was calculated and found to scale as 2^(-n), as does the classical trapped random walk on this topology.