Quantum computing is an emerging field with potential applications in machine learning. This research project aimed to compare the performance of a quantum kernel to that of a classical kernel in machine learning binary classification tasks. Two Support Vector Machines, a popular classification model, was implemented for the respective Variational Quantum kernel and the classical Radial Basis Function kernel and tested on the same sets of artificial quantum-based testing data. The results show that the quantum kernel significantly outperformed the classical kernel for the specific type of data and parameters used in the study. The findings suggest that quantum kernels have the potential to improve machine learning performance for certain types of problems, such as search engines and self-driving vehicles. Further research is, however, needed to confirm their utility in general situations. / Kvantberäkning är ett växande forskningsområde med möjliga tillämpningar inom maskininlärning. I detta forskningsprojekt jämfördes prestandan hos en klassisk kärna med den hos en kvantkärna i binär klassificering för maskininlärninguppgifter, och implikationerna av resultaten diskuterades. Genom att implementera två stödvektormaskiner, en populär klassifikationsmodell, för respektive variabel kvantkärna och klassisk radiell basfunktionskärna kunde vi direkt testa båda kärnorna på samma uppsättning av artificiella kvant-baserad testdata. Resultaten visar på betydande prestandafördelar för kvantkärnan jämfört med den klassiska kärnan när det gäller denna specifika typ av data och de parametrar som användes i vår studie. Vi drar slutsatsen att kvantkärnor inom maskininlärning har potential att överträffa klassiska kärnor, men att mer forskning krävs för att fastställa om detta har någon nytta i allmänna situationer. Om det finns betydande prestandafördelar kan det finnas många tillämpningar, till exempel för sökmotorer och självkörande fordon.

This thesis investigates the impact noise has on Grover’s algorithm when being used to search for multiple items in a database. The main metric being looked at is the probability of the algorithm successfully finding a correct item. The Qiskit framework was used to implement and evaluate the algorithm’s performance in noise-free and noisy environments. Results from the experiments show significant findings. In noiseless tests, the algorithm performs effectively and as expected. However, with the introduction of a noise model, the algorithm’s performance declines noticeably. The probability of it finding a marked item was close to the probability of randomly selecting the same item from the database. This was the case regardless of how many items were marked or the database size. These unexpected outcomes illustrate the disabling effect of noise on Grover’s algorithm. Limitations of the study include noise completely disrupting the algorithm, challenges in accurately modelling quantum noise, and the use of relatively small databases. Further research is needed to explore noise mitigation strategies and assess the algorithm’s robustness in larger-scale scenarios. This research strengthens our understanding of noise’s impact on Grover’s algorithm, showcasing the challenges and limitations of its implementation. It highlights the importance of properly managing noise in quantum computing to fully utilize its potential in efficiently solving complex problems. / Denna avhandling undersöker effekten av brus på Grover’s algoritm för att söka efter flera markerade element i en databas. Huvudfokuset var att undersöka sannolikheten att algoritmen korrekt skulle hitta ett av flera markerade element i en databas. Qiskit-ramverket användes för att utvärdera algoritmens prestanda i brusfria och brusiga miljöer. Resultaten från experimenten var betydelsefulla. I brusfria tester presterar algoritmen effektivt och som förväntat. Men, med införandet av brus minskar algoritmens prestanda avsevärt. Sannolikheten för att algoritmen hittar ett markerat element liknar sannolikheten för att slumpmässigt välja ut samma element från databasen. Detta var fallet oavsett hur många element som var markerade och databasens storlek. Dessa oväntade resultat illustrerar brusets söndrande effekt på Grover’s algoritm. Begränsningar i studien inkluderar att bruset helt får algoritmen att sluta fungera, utmaningar med att noggrant modellera kvantbrus och användningen av relativt små databaser. Vidare forskning behövs för att undersöka strategier för att mitigera brus och bedöma algoritmens robusthet i storskaliga scenarier. Denna forskning stärker vår förståelse för brusets påverkan på Grover’s algoritm och betonar utmaningar och begränsningar vid dess implementering. Den betonar vikten av att hantera brus inom kvantdatorer för att kunna utnyttja deras potential för effektiv lösning av komplexa problem.

Qubits (quantum bits) are what runs quantum computers, like a bit in classical computers. Quantum gates are used to operate on qubits in order to change their states. As such they are what ”programmes” a quantum computer. An unfortunate side effect of quantum physics is that coupling a quantum system (like our qubits) to an outside environment will lead to a certain loss of information. Reducing this decoherence effect is thus vital for the function of a quantum computer. Geometric quantum computation is a method for creating error robust quantum gates by using so called geometric phases which are solely reliant on the geometry of the evolution of the system. The purpose of this project has been to develop physical schemes of geometric entangling two-qubit gates along the Schmidt sphere, a geometric construct appearing in two-qubit systems. Essentially the overall aim has been to develop new schemes for implementing robust entangling quantum gates solely by means of interactions intrinsic to the computational systems. In order to create this gate four mutually orthogonal states were defined which together spanned the two-qubit state space. Two of the states were given time dependent variables containing a total of two angles,which were used to parameterize the Schmidt sphere. By designing an evolution for these angles that traced out a cyclical evolution along geodesic lines a quantum gate with exclusively geometric phases could be created. This gate was dubbed the ”Schmidt gate” and could be shown to be entangling by analyzing a change in the concurrence of a two qubit system. Two Hamiltonians were also defined which when acted upon the predefined system of states would give rise to the aforementioned evolution on the Schmidt sphere. The project was successful in creating an entangling quantum gate which could be shown by looking at difference in the concurrence of the input and output state of a two-qubit system passing through the gate.