A journey into the world of inverse problems in quantum mechanics

Abstract

Technology has come a long way since the birth of quantum mechanics. The science has led to computers, and now, the scientists are pushing the fundamentals further to eventually be able to construct a quantum computer from the bottom up. Quantum tomography has a vital role in this ambitious endeavour: it’s the study of how one can retrieve the values describing a quantum state, like finding coordinates on a map for a given position. The challenge that the quantum tomographer faces lies in the shear number of these values, which grows exponentially with the components of quantum system. This is the curse of dimensionality and cannot be avoided with classical means. Therefore, the tomographer is forced to come up with algorithms that scale well with the number of components, either via prior information or by reducing the problem to its simplest form. In this thesis, we devise algorithms for retrieving quantum states using a little of both approaches. We develop a direct way of retrieving quantum state-vector values by assuming that the state is pure, which is often the case in optics. In addition, we show that a simple optimisation technique, projected gradient descent, can outperform all other methods for retrieving general quantum states. Our contribution to the field is thus to provide tools that enable the tomographer to work on larger quantum states and that hopefully help her create the building blocks of a quantum computer. We touch on other somewhat related subjects such as image denoising and imaging quantum correlations.