Military simulation is essential for modern warfare, providing a virtual environment for training, analysis, and rehearsal of procedures. Accurate correlation between simulated entities, called tracks, and their radar detections, called tracks, is crucial for generating reliable data, vital for evaluating military operations and improving training exercises. However, factors like radar noise, communication errors, and simulation inaccuracies complicate this correlation. This thesis aims to develop a robust method for correlating simulated truth entities with corresponding radar tracks in military simulations. The proposed method tackles challenges such as radar noise and communication errors to improve the reliability and validity of simulation statistics. The research encompasses the theoretical development of three correlation algorithms, one of which serves as a benchmark for verification. The methods were evaluated across various simulated scenarios, in which the two correlation methods consistently outperformed the benchmark, particularly in scenarios with fewer data points.

The minimum vertex cover problem (MinVertexCover) is an important optimization problem in graph theory, with applications in numerous fields outside of mathematics. As MinVertexCover is an NP-hard problem, there currently exists no efficient algorithm to find an optimal solution on arbitrary graphs. We consider quantum optimization algorithms, such as the Quantum Alternating Operator Ansatz (QAOA+), to find good minimum vertex covers.

This thesis presents new 'second degree' mixing Hamiltonians for MinVertexCover in the QAOA+ framework, which allow mixing between solutions Hamming distance 2 apart. The performance of these new Hamiltonians is evaluated on a small graph. Methods to extend this idea by constructing Hamiltonians which allow mixing between solutions at Hamming distance n are also presented, along with a generalization of second degree mixing Hamiltonians to other optimization problems.

In this thesis, gate set tomography (GST) has been conducted on the nitrogen vacancy center (NV). Gate set tomography is a protocol for characterization of logic operations (gates) on quantum computing processors. The NV’s electron served as a qubit. The quantum circuits were run both experimentally as well as on an NV-simulator. GST is different from its predecessors in the sense that it estimates all aspects of the processor simultaneously, without assuming any of its parts to be ideal. However, it does assume that the gates are Markovian. This allows to analyse the gate errors more precisely via their error generators. In addition to this, the diamond norm and a measure of the amount of model violation were used to examine the results. The electron qubit can couple to the nearby nitrogen nucleus, which can cause non-Markovian dynamics. The nitrogen nucleus (𝑆 = 1) can be initialised using nuclear spin polarization. The effects of different initialisation procedures on GST’s results were
researched. Furthermore, a dynamical decoupling XY-4 echo sequence could be employed to protect the quantum state of the qubit. How the presence of the echo affected the estimated gates and their errors was probed. It was discovered that the echo was extremely good at decreasing the amount of model violation, most likely by preventing the electron from coupling to the nitrogen, which can lead to non-Markovian dynamics. Without the echo, GST’s estimates were satisfactory only if the nitrogen nucleus was initialised with a high enough fidelity, at least 0.95. Another topic for further research would be to perform GST on the electron and the nitrogen system, by modelling the nitrogen nucleus as
a qutrit.

This thesis explores the convergence of the mixing method, an iter- ative algorithm for solving diagonally constrained semidefinite programs. In this paper we first give an exposition of the convergence proof for the mixing method based on the proof by Wang, Chang, and Kolter , where we restructure some parts of the proof and provide extra de- tails. Then we construct an example where the linear convergence rate of the mixing method is close to one when near the optimal solution. The mixing method is then compared for convergence speed to a semidefinite programming solver and gradient descent on random max-cut instances. For instances of the max-cut, it is found that the mixing method outper- forms other methods.

Quantum technology is evolving faster than ever. Currently, all eyes are on the quantum computer, the promising computer that can solve problems which are unsolvable for regular computers. In order to understand this new technology, it is necessary to understand the qubit, the basic unit of quantum information. This can be done by means of the density matrix: a mathematical representation of the state of a system. The aim of this thesis is to find the density matrix of a single qubit in closed (isolated) and open quantum systems. In the case of a closed system, an alternating sequence of two processes with different Hamiltonians is considered, which both last a fixed amount of time after one another. These systems have been solved using a direct formula for a 2 × 2 matrix with distinct eigenvalues raised to the power n for any n ∈ N. In the case of an open system, dissipation is taken into account compared to a single process in a closed system. These open systems have been solved numerically using the Lindblad master equation and the spin-boson model to model the environment as a bath of bosons. However, the use of multiple approximations and assumptions questions the validity of the results for ’strong’ interactions. Suggestions for further research include investigating quantitatively when the Lindblad equation is valid to use and solve open quantum systems using different models for the environment.

In this thesis, we start with giving a mathematical description of bipartite quantum correlations and how they are built up in the Tensor model. This is needed because we want to recover the state and the operators when only the bipartite quantum correlation is known. In the literature, there are see-saw algorithms to recover the state, but they are limited to only lower dimensions. In this thesis, we explore an alternative approach, where we directly minimize the function
f(ψ,{E_s^a},{F_t^b}) = ∑_a,b,s,t(P(a,b|s,t)- ψ^*(E_s^a ⊗ F_t^b)ψ)². Here, P(a,b|s,t) is the bipartite correlation,  ψ is the state vector, and E_s^a and F_t^b are the POVMs. Furthermore, ⊗ is the Kronecker product and ^* indicates the conjugate transpose of a vector. These variables are subject to constraints and some of them can easily be transformed into penalty functions. The matrices E_s^a and F_t^b have to be Hermitian positive semidefinite, for which we parameterize them by their Cholesky decompositions. The gradient of this (now unconstrained) problem can be explicitly determined with the use of Wirtinger calculus. This offers an elegant way to determine the gradient of real-valued functions with complex variables. Also, a total description of Wirtinger calculus is also given, including a proof that the gradient indeed points towards the direction of the steepest incline. We use first-order methods like gradient descent with backtracking line search and momentum-based gradient descent to find a minimum solution of the equation. If the cost function converges towards zero, we assume that the variables converge to a correct state and measurement operators.  These methods can find large correlations of approximately 3000 separate variables in 1.5 hours and are able to find many different other correlations and states. The algorithm had some problems finding the operators and state of a family of correlations that had four inputs and two outputs. For some correlations, the algorithms found states and operators of lower dimension than the correlations were build with. 

In this thesis, we give a primal-dual interior point method specialized to clustered low-rank semidefinite programs. We introduce multivariate polynomial matrix programs, and we reduce these to clustered low-rank semidefinite programs. This extends the work of Simmons-Duffin [J. High Energ. Phys. 1506, no. 174 (2015)] from univariate to multivariate polynomial matrix programs, and to more general clustered low-rank semidefinite programs.  Clustered low-rank semidefinite programs are programs with low-rank constraint matrices where the positive semidefinite variables are only used within clusters of constraints. The free variables can be used in any constraint, and can be used to connect clusters. The solver uses this structure to speed up the computations in two ways. First, the low rank structure is used to reduce matrix products to products of the form uT M v, where M is a matrix and u and v are vectors, as already suggested by Löfberg and Parrilo in [43rd IEEE CDC (2004)]. Second, an additional block-diagonal structure is introduced due to the clusters. This gives the possibility to do operations such as the Cholesky decomposition block-wise.   We apply this to sphere packing and spherical cap packing. For sphere packing, the speed of the solver is compared to the often used arbitrary precision solver SDPA-GMP, showing the theoretical speedup in time complexity. We give a new three-point bound for the maximum density when packing spherical caps of $N$ sizes on the unit sphere.    

Large fault­tolerant universal gate quantum computers will provide a major speed­up to a variety of common computational problems. While such computers are years away, we currently have noisy intermediate­scale quantum (NISQ) computers at our disposal. In this project we present two quantum machine learning approaches that can be used to find quantum circuits suitable for specific NISQ devices. We present one gradient­based and one non­gradient based machine learning approach to optimize the created quantum circuits, to best mimic the behaviour of a given function up to measurement. We make sure that the created quantum circuits obey the restrictions of the chosen hardware, therefore the approaches can be used to find circuits perfectly suited for specific NISQ devices. This enables the user to make the best use of quantum technology in the near future. In doing this we created our own quantum simulator which can be used to simulate small quantum circuits that obey hardware restrictions. We also present the method used to implement this simulator. Finally we present the results of applying both machine learning approaches to different problem types and compare their performance.

The construction of cyclic railway timetables is an important task for Netherlands Railways (NS).This construction can be formulated as a Periodic Event Scheduling Problem (PESP). The most powerful technique for solving cyclic railway timetabling problems is constraint programming, especially via SAT solvers when PESP instances are encoded as SAT instances. SAT solvers can determine the feasibility of problem instances of NS quickly and reliably. However, in previous implementations the problem specification must explicitly indicate the track use within stations and on four-track sections. As a result, the solver also reports infeasibility if a small adjustment of the track allocation could lead to a feasible timetable. In this thesis, the Open Periodic Event Scheduling Problem (OPESP) is introduced, which is used in a new method to incorporate flexible track use in the SAT formulation. This method yields promising results that could help improve the timetabling process at NS.

Quantum coin flipping is a cryptographic primitive in which two or more parties that do not trust each other want establish a fair coin flip. These parties are not physically near each other and use quantum communication channels to interact. A quality of protocols is measured by the best possible cheating strategy, which is the solution of a complex semidefinite optimization problem. In this master thesis we show new explicit bounds in multiparty quantum coin flipping, we investigate how to explicitly formulate these problem in a standard form, we show that a fair coin flip results in the lowest possible bias and we determine more measures of the quality of a protocol. Furthermore, this master thesis presents a rigorous and detailed mathematical description of semidefinite optimization, quantum information theory and quantum coin flipping. This thesis also includes an article written together with J. Mulderij, T. Attema, I. Chiscop and F. Phillipson on distributed quantum computing. In this article, we pose new questions and formulate integer linear programs that solve to find optimal assignment of qubits to computers for a given network of quantum computers and quantum algorithm.