Abstract: The aim of this paper is to build quantum circuits that implement discrete-time quantum walks having an arbitrary position-dependent coin operator. The position of the walker is encoded in base 2: with n wires, each corresponding to one qubit, we encode 2(n) position states. The data necessary to define an arbitrary position-dependent coin operator is therefore exponential in n. Hence, the exponentiality will necessarily appear somewhere in our circuits. We first propose a circuit implementing the position-dependent coin operator, that is naive, in the sense that it has exponential depth and implements sequentially all appropriate position-dependent coin operators. We then propose a circuit that “transfers” all the depth into ancillae, yielding a final depth that is linear in n at the cost of an exponential number of ancillae. Themain idea of this linear-depth circuit is to implement in parallel all coin operators at the different positions. Reducing the depth exponentially at the cost of having an exponential number of ancillae is a goal which has already been achieved for the problem of loading classical data on a quantum circuit (Araujo in Sci Rep 11:6329, 2021) (notice that such a circuit can be used to load the initial state of the walker). Here, we achieve this goal for the problem of applying a position-dependent coin operator in a discrete-time quantum walk. Finally, we extend the result of Welch (New J Phys 16:033040, 2014) from position-dependent unitaries which are diagonal in the position basis to position-dependent 2 x 2-block-diagonal unitaries: indeed, we show that for a position dependence of the coin operator (the block-diagonal unitary) which is smooth enough, one can find an efficient quantum-circuit implementation approximating the coin operator up to an error epsilon (in terms of the spectral norm), the depth and size of which scale as O(1/epsilon). A typical application of the efficient implementation would be the quantum simulation of a relativistic spin-1/2 particle on a lattice, coupled to a smooth external gauge field; notice that recently, quantum spatial-search schemes have been developed which use gauge fields as the oracle, to mark the vertex to be found (Zylberman in Entropy 23:1441, 2021), (Fredon arXiv:2210.13920). A typical application of the linear-depth circuit would be when there is spatial noise on the coin operator (and hence a non-smooth dependence in the position).

Abstract: We present a novel method of inferring the dark matter (DM) content and spatial distribution within galaxies, using convolutional neural networks (CNNs) trained within state-of-the-art hydrodynamical simulations (Illustris-TNG100). Within the controlled environment of the simulation, the framework we have developed is capable of inferring the DM mass distribution within galaxies of mass similar to 10(11)-10(13)M(circle dot) from the gravitationally baryon-dominated internal regions to the DM-rich, baryon-depleted outskirts of the galaxies, with a mean absolute error always below approximate to 0.25 when using photometrical and spectroscopic information. With respect to traditional methods, the one presented here also possesses the advantages of not relying on a pre-assigned shape for the DM distribution, to be applicable to galaxies not necessarily in isolation, and to perform very well even in the absence of spectroscopic observations.

Abstract: Purpose: The aim of this work is to investigate the feasibility of the Jagiellonian Positron Emission Tomography (J -PET) scanner for intra-treatment proton beam range monitoring. Methods: The Monte Carlo simulation studies with GATE and PET image reconstruction with CASToR were performed in order to compare six J -PET scanner geometries. We simulated proton irradiation of a PMMA phantom with a Single Pencil Beam (SPB) and Spread -Out Bragg Peak (SOBP) of various ranges. The sensitivity and precision of each scanner were calculated, and considering the setup's cost-effectiveness, we indicated potentially optimal geometries for the J -PET scanner prototype dedicated to the proton beam range assessment. Results: The investigations indicate that the double -layer cylindrical and triple -layer double -head configurations are the most promising for clinical application. We found that the scanner sensitivity is of the order of 10-5 coincidences per primary proton, while the precision of the range assessment for both SPB and SOBP irradiation plans was found below 1 mm. Among the scanners with the same number of detector modules, the best results are found for the triple -layer dual -head geometry. The results indicate that the double -layer cylindrical and triple -layer double -head configurations are the most promising for the clinical application, Conclusions: We performed simulation studies demonstrating that the feasibility of the J -PET detector for PET -based proton beam therapy range monitoring is possible with reasonable sensitivity and precision enabling its pre -clinical tests in the clinical proton therapy environment. Considering the sensitivity, precision and cost-effectiveness, the double -layer cylindrical and triple -layer dual -head J -PET geometry configurations seem promising for future clinical application.