In 2017 Martijn Caspers, Fedor Sukochev and Dmitriy Zanin published a paper which generalises the proof of Davies' 1988 paper, and thus resolves the Nazarov-Peller conjecture. The proofs of these papers have been presented in this thesis. They have been expanded with a proof that generalises the conjecture to arbitrary Schatten classes. The optimality of the estimates in the conjecture is also studied, following the example of the 2016 paper by Coine et al. In addition, the quantum mechanical context is provided to interpret the presented results.

Quantum computing may be the key to solving many of the problems our modern world faces today. Just as regular computers boosted our computing power to scales never imagined before, so too could quantum computers allow us to solve problems that we just do not yet have the computational power for. However, a properly functioning quantum computer that outclasses current classical computers is not yet a reality. Many di_erent platforms for quantum computing are being researched. Experiments are being done with diamond vacancies (see, for example, [1]), trapped ions [2], nuclear spins in NMR [3] and silicon quantum dots [4], to name a few. One of the most promising types of qubits is based on superconducting circuits [5]. The most widely studied and most promising superconducting qubit is the Transmon qubit [6]. One of the advantages of this type of qubit is that it can be fabricated using currently available lithographic patterning equipment. However, when fabricating Transmon chips, there is a problem of non-uniformity: when using the exact same method to make two chips, they can be vastly di_erent. As such, every chip needs to be characterised after fabrication. Characterising a chip means _nding speci_c important parameters for the qubits, such as their frequency and coherence times. For use in a quantum computer, qubit frequencies will have te be targeted with more precision than is currently possible [7] [8]. As such, a technique that allows change a qubit's frequency after its fabrication is very useful. In this thesis, such a technique is presented. The characterisation process of a Transmon chip is then explained. The data of the three chips that were subjected to the aforementioned technique is also presented.