Thesis 1: Quantum Algorithm for Handling Missing Data, Thesis 2: Simulation Method for Open Quantum Systems

Abstract

Thesis 1: In the last few years, we have seen significant progress in the quantum computing field. What seemed distant is now becoming a reality. Therefore, an important task in this field is to find applications where a quantum computer could give an advantage (such as speed or accuracy) compared to our conventional computers, also known as classical computers. In this thesis I present such an application applied to handling missing data. The motivation for creating a quantum algorithm for missing data has two parts: (1) The problem of missing data is large and extends many disciplines. If not handled correctly, it can lead to insufficient analysis of the data. It is an important problem to tackle. (2) One way to find promising applications of quantum computers is to look at what quantum computers are particularly good at compared to classical computers. Quantum computers can have special forms of correlations such as superposition, interference and entanglement. Therefore, you can map such correlations in a quantum computer to the problem of finding correlations in missing data. I have created a quantum algorithm that can be applied to handling missing data. This quantum algorithm is more accurate in predicting the probability distribution of complete data compared to the probability distribution of incomplete data (data with missing values). In this work, I give a brief overview of quantum computing knowledge relevant to this thesis. Then I show different types of missing data and current methods to handle missing data. After that, I present the implementation of the quantum algorithm and intuition on why it works. Next I show how this quantum algorithm can be simulated on our current computers. Then I apply the algorithm to real world data: a dataset that consists of attributes relevant to heart disease and the diagnosis. I show how the quantum algorithm can be used to get advantages in knowing the probability distributions when the data is missing. Lastly, I show how this work can be extended in other fields such as machine learning and I provide other potential research questions. This thesis presents several contributions: (1) there is not much work on this particular application of quantum computers, so this thesis provides several steps in the direction of using quantum computers for handling missing data. (2) I show a quantum algorithm that can be used for this task and how it can be applied to real world datasets. (3) The quantum algorithm performs better on predicting the correct distribution of the complete data compared to the distribution of the missing data. (4) This particular algorithm can also be applied to other fields such as machine learning and deep learning and could have a variety of other applications. In particular, the quantum algorithm presented in this thesis uses a set of parameters that can be adjusted to fit a probability distribution. Our results show that the quantum algorithm is more accurate in predicting a complete data probability distribution compared to the missing data probability distribution, even when the quantum algorithm is trained on the missing data probability distribution. Thesis 2: Plants convert sunlight into chemical energy through photosynthesis. During photosynthesis, plants want to collect as much sunlight as possible in order to obtain more energy. They are able to achieve this through light-harvesting complexes, which can direct energy transfer to its reaction center (which executes the primary energy conversion reactions). Remarkably, they are able to do this with high efficiency. If we are able to understand how this mechanism works, we could potentially create quantum devices that can harness noisy environments. Noisy environments are usually viewed as a hindrance to quantum effects (for example researchers are trying to build quantum computers that have very little interaction with the environment), but this shows that they could be an advantage in particular cases. This mechanism can be explained through an open quantum system: a quantum system that interacts with the environment. In this thesis, we explore a computational method for simulating certain types of open quantum systems on our classical computers (our conventional computers). A better understanding of these open quantum systems, which our simulation provides, could potentially give us insight into building better quantum devices and simulating chemistry and physics problems. In general, this research contributes to a better understanding of quantum transport in open systems when coupling to the environment is strong. We contribute to this area by (1) presenting a more efficient way to get similar results of current methods, (2) better predictions of certain types of phenomena and (3) better understanding of optimal quantum transport through our models and simulations. In particular, we use a Stochastic Discrete Schrodinger process to more efficiently simulate an open quantum system's time evolution. We are able to find optimal quantum transport by simulating dissipation as a function of decoherence. Finally, we are able to better predict the relaxation of excitons to lower energy states which is important for light-harvesting-systems. We could potentially use similar mechanisms to improve today's technology.