Models of quantum walks which admit continuous time and continuous spacetime limits have recently led to quantum simulation schemes for simulating fermions in relativistic and non relativistic regimes (Di Molfetta and Arrighi, 2020). This work continues the study of relationships between discrete time quantum walks (DTQW) and their ostensive continuum counterparts by developing a more general framework than was done in (Di Molfetta and Arrighi, 2020) to evaluate the continuous time limit of these discrete quantum systems. Under this framework, we prove two constructive theorems concerning which internal discrete transitions ("coins") admit nontrivial continuum limits in 1D+1. We additionally prove that the continuous space limit of the continuous time limit of the DTQW can only yield massless states which obey the Dirac equation. We also demonstrate that for general coins the continuous time limit of the DTQW can be identified with the canonical continuous time quantum walk (CTQW) when the coin is allowed to transition through the continuum limit process. Finally, we introduce the Plastic Quantum Walk, or a quantum walk which admits both continuous time and continuous spacetime limits and, as a novel result, we use our 1D+1 results to obtain necessary and sufficient conditions concerning which DTQWs admit plasticity in 2D+1, showing the resulting Hamiltonians. We consider coin operators as general 4 parameter unitary matrices, with parameters which are functions of the lattice step size 𝜖. This dependence on 𝜖 encapsulates all functions of 𝜖 for which a Taylor series expansion in 𝜖 is well defined, making our results very general.

För närvarande utforskas quantum walk som en potentiell metod för att analysera medicinska bilder. Med inspiration från Gradys random walk-algoritm för bildbehandling har vi utvecklat en metod som bygger på de kvantmekaniska fördelar som quantum walk innehar för att detektera och segmentera medicinska bilder. Vidare har de segmenterade bilderna utvärderats utifrån klinisk relevans. Teoretiskt sett kan quantum walk-algoritmer erbjuda en mer effektiv metod för bildanalys inom medicin jämfört med traditionella metoder för bildsegmentering som exempelvis klassisk random walk, som inte bygger på kvantmekanik. Inom området finns omfattande potential för utveckling, och det är av yttersta vikt att fortsätta utforska och förbättra metoder. För närvarande kan det konstateras att det är en lång väg att vandra innan detta är något som kan appliceras i en klinisk miljö. / Currently, quantum walk is being explored as a potential method for analyzing medical images. Taking inspiration from Grady's random walk algorithm for image processing, we have developed an approach that leverages the quantum mechanical advantages inherent in quantum walk to detect and segment medical images. Furthermore, the segmented images have been evaluated in terms of clinical relevance. Theoretically, quantum walk algorithms have the potential to offer a more efficient method for medical image analysis compared to traditional methods of image segmentation, such as classical random walk, which do not rely on quantum mechanics. Within this field, there is significant potential for development, and it is of utmost importance to continue exploring and refining these methods. However, it should be noted that there is a long way to go before this becomes something that can be applied in a clinical environment.