# metadata.dc.contributor.other[].value
Universitat de Barcelona. Facultat de Física

# metadata.dc.description.abstract[].value
[eng] This thesis can be classified into the research field of quantum algorithms. A typical paper in this field usually focuses on what kind of things we can do with a quantum computer, whether they are useful or advantageous compared to classical computers, and how we can do it more efficiently. Since the ideal fault-tolerant large-scale quantum computer will not be available soon, many papers focus on noisy intermediate-scale quantum (NISQ) devices, how to characterize them through various benchmarking protocols, and how to make them useful through error mitigation or variational algorithms. Most probably, anyone who ends up opening this thesis is already an expert in certain topics, but they can still find some “out of the box" elements in one chapter or another. The results presented in the thesis might not be directly useful, but hopefully, these elements can provide insights and inspiration. The main body of this thesis is organized into six independent chapters. Each chapter is complete and self-consistent. All these chapters share a similarity: They provide some mathematical ideas of quantum computing in one aspect or another. Since part of the research is conducted in collaboration, the pronoun “we" is used throughout the work. In the first chapter, we focus on actions that transform quantum states into quantum states. We begin with a more theoretical approach. These actions naturally process a group structure. And we ask, under what conditions is this group finite? Given an arbitrary finite group, how can we construct these actions? How can these actions be useful to quantum computing? The set of these actions is named "Quantum representation of finite group," and answers to these questions are provided in the main text. Then, in the second chapter, we switch to a more practical scenario. The inspiration came from a problem that appeared in post-quantum cryptography, the dihedral coset problem, which uses quantum algorithms to recover a secret of the group hidden in quantum states. We simplified this problem and designed an interactive protocol for verifying the computational capacity of a quantum device. Holding a small quantum computer, the verifier prepares quantum state samples that encode a secret and sends them one by one to the prover via a one-way quantum channel. The prover then uses his quantum device to recover the secret with a certain probability. The device’s capacity is then reflected in the accuracy of the solution. This interactive protocol can be used locally on a single device as a benchmarking protocol. In the third chapter, we explore quantum entanglement. We present a new method for visualizing entanglement during a quantum evolution, such as the execution of a quantum algorithm. We name this method "Entanglement trajectory" and show that it can provide a "fingerprint" for each quantum process. We determine the boundaries of these trajectories with Random Matrix Theory and use different examples and entanglement measurements to show that this trajectory has an intrinsic quantum meaning and provides more information about the system. In the fourth chapter, we elaborate on the entanglement presented in a succession of numbers. First, we give a numerical and analytical description of the average entropy of succession, which can provide a standard for studying any successions. Then, we simulate the example of k-almost primes and its unions for different system sizes and explore their particular feature. In the fifth chapter, we will show three different methods of using a quantum annealer to solve Multivariate Quadratic (MQ) problems. In the sixth chapter, we present a data encoder in the fixed Hamming weight subspace, which is parameter-optimal and can be extended to a sparse and binary data encoder. We will give a small summary for each chapter. At the end of this thesis, we will provide an overall conclusion for our readers as “take-home messages".

# metadata.dc.description.abstract[].value
[cat] Aquesta tesi s’emmarca en el camp de recerca dels algorismes quàntics. Explora què es pot fer amb un ordinador quàntic, si aquestes aplicacions són útils respecte als ordinadors clàssics i com es poden fer més eficients. A causa de la manca d’ordinadors quàntics tolerants a errors a gran escala, molts estudis se centren en dispositius NISQ, els seus protocols de benchmarking i tècniques d’atenuació d’errors. Aquesta tesi pot oferir elements innovadors i inspiradors per als investigadors del camp. El treball està estructurat en sis capítols independents, cadascun centrat en una idea matemàtica en computació quàntica. En el primer capítol, s’analitzen transformacions d’estats quàntics en el context de grups finits i es proposa una representació quàntica. El segon capítol desenvolupa un protocol interactiu basat en el problema del coset dièdric per verificar la capacitat de càlcul d’un dispositiu quàntic. El tercer capítol introdueix la trajectòria d’entrellaçament, una eina per analitzar l’evolució de l’entrellaçament en processos quàntics. El quart capítol estudia l’entropia de les successions numèriques i la seva relació amb estats quàntics, incloent-hi els k-almost primes. El cinquè capítol presenta tres estratègies per resoldre problemes Quadràtics Multivariats (MQ) amb quantum annealing. Finalment, el sisè capítol introdueix un codificador de dades en un subespai de pes de Hamming fix, eficient en recursos i aplicable a dades esparses i binàries. La tesi conclou amb un resum de cada capítol i una reflexió general per destacar els elements més rellevants del treball.