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.

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.

In this work, Clebsch-Gordan coefficients are studied from both a quantum mechanical and a mathematical perspective. In quantum mechanics, Clebsch-Gordan coefficients arise when two quantum systems with a certain angular momentum are combined and the total angular momentum is to be found. We start by discussing the relevant postulates of quantum mechanics. From there we move on to a discussion of quantum angular momentum, the combining of quantum systems, and Clebsch-Gordan coefficients. Finally, an algorithm for calculating the Clebsch-Gordan coefficients is developed. A firm grasp of linear algebra is required to understand this quantum mechanical perspective. The mathematical perspective is rooted in representation theory. The Clebsch-Gordan coefficients show the decomposition of the tensor product of irreducible representations of the three dimensional rotation group into the direct sum of irreducible representations. We explain the necessary theory to understand how this works. This includes a discussion of irreducible unitary representations, spherical harmonics, direct sum representations, and tensor product representations. Some experience with group theory is required to understand the mathematical perspective.

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.

As quantum computers are developing, they are beginning to become useful for practical applications, for example in the field of quantum metrology. In this work, a variational quantum algorithm is used to find an optimal probe state for measuring parameters in a noisy environment. This is achieved by optimizing a cost on a quantum computer, based on the Fisher information of the parameters to be estimated. These parameters are then estimated using maximum likelihood estimators. In a simulation, a probe state was found that performed better than the best possible state for noiseless measurements, although this could not be reproduced on an actual quantum computer.

Quantum communication has been shown to be vastly superior to classical communication in many problems. However no general statements exist which tells us how much better quantum communication is to its classical counterpart. In this thesis it was studied the minimum amount of classical bits required to exactly simulate a quantum communication process. The quantum communication process specifically studied was a quantum prepare and measurement communication problem. It has been shown that the calculation of the amount of classical bits of communication required for simulation reduces to a minimization-maximization optimization problem. Several results have been presented for for solving this optimization problem and in addition a link was made between classical simulations of quantum communication and a recent debate on the reality of the quantum state.

In this thesis we provide an elementary introduction in finite dimensional representation theory of the Lie groups SU(2) and SU(3) for undergraduate students in physics and mathematics. We will also give two application of representation theory of these two groups in physics: the spin and quark models. We begin with first discussing representation theory for finite groups to create intuition for representations. We will explain notions such as intertwining maps and complete reducibility and we will mention some application of representation theory of finite groups inquantum mechanics. Hereafter, we begin with representation theory for Lie groups and Lie algebras, especially the groups SO(3) and SU(2), as these groups will play an important role in the description of spin. One of the main results is that SU(2) is the universal cover of SO(3). Furthermore, we give a description of spin by means of representation theory of SO(3) and its Lie algebra so(3). We will show that half integer representations of the Lie algebra so(3) cannot be exponentiated to representations of the Lie group SO(3), but it can be exponentiated to its universal cover SU(2). Moreover, we study the irreducible representations of SO(3) inside the Hilbert space L2(R3). We will argue that one of the simplest quantum Hilbert spaces of a particle L2(R3), can be modified to the completion of the tensor product L2(R3) ⊗V, where, V is a finite dimensional Hilbert space that incorporates the internal degrees of freedom: spin. V carries an irreducible projective representation of SO(3). We will also discuss the addition of angular momentum of two particles in quantum mechanics. For this, we show how the tensor product of irreducible representations V and W of so(3) decomposes into SO(3) invariant subspaces of L2(R3). Hereafter, we will turn to representation theory of the Lie group SU(3) for setting up the mathematical framework for analysing the quark model. We will proof that there is a one-to-one correspondence between the irreducible representations of sl(3;C)and SU(3). We will also proof the theorem of the highest weight by which we can classify all the irreducible representations of SU(3) and sl(3;C) by their highest weight. We will also introduce the notion of the Weyl group and show that the Weyl group is a symmetry of weights of the finite dimensional representation of sl(3;C). Other properties of these representation, such as the dimension of the irreducible representations of sl(3;C) will be provided. Lastly, the quark model is discussed by means of representation theory of SU(3). We will show how this model can be used to classify two type of particles which also interact by means of the strong force: baryons and mesons. We show that we can classify the lightest mesons and baryons in so-called multiplets by the irreducible representations of SU(3). However, we will also introduce a modification of the strong force which further refines this model. A topic for further study would be how the symmetry group SU(3) describing Quantum Chromo Dynamics (QCD) can be used for the description of mesons and baryons.

In this thesis we will for a quantum Markov semi-group (Φt)t≥0 on a finite von Neumann algebra N with a trace τ , investigate the property of the semi-group being gradient-Sp for some p ∈ [1, ∞]. This property was introduced in [12] (see also [9]) and has been studied in [9, 10, 12] for quantum Markov semi-groups on compact quantum groups and on q-Gaussian algebras. Beyond these classes the property gradient-Sp has not been studied; in particular for groups and their operator algebras no (non-trivial) examples were known before this thesis. The main aim of this thesis is therefore to construct interesting examples of quantum Markov semi-groups that possess the gradient-Sp property. The reason why we are interested in constructing such semi-groups, is because they can be used to obtain properties like the Akemann-Ostrand property (AO+) and strong solidity for the underlying von Neumann algebra. Over the last decade, these properties have become a topic of interest and have been studied for several von Neumann algebras, see [3, 8, 9, 10, 12, 23, 32, 33, 37, 41]. In this thesis we shall focus on group von Neumann algebras (L(Γ), τ ) for certain discrete groups Γ that possess the Haagerup property. Namely, for such groups there exists a proper, conditionally negative definite function ψ on Γ. We can then define an unbounded operator ∆ψ on the GNS-Hilbert space L2(L(Γ), τ ) as ∆ψ(λv) = ψ(v)λv and consider the corresponding quantum Markov semi-group (e−t∆ )t≥0. For this semi-group we can investigate for what p it has the gradient-Sp property. In particular we will be considering group von Neumann algebras of Coxeter groups. Namely, a Coxeter group W possesses the Haagerup property by [4], and a proper conditionally negative function ψ on W is given by the minimal word length ψ(w) = |w| w.r.t some set of generators. We will ‘almost completely’ characterize for what types of Coxeter systems the semi-group corresponding to the word length is gradient-Sp. Moreover, in the cases that we get the gradient-S2 property, we obtain the Akemann-Ostand property (AO+) and strong solidity for L(W). Hereafter, we will also consider other quantum Markov semi-groups on L(W). We consider word lengths that arise by putting different weights on the generators, and consider the semi-groups associated to these proper, conditionally negative functions. From this we obtain (AO+) and strong solidity for L(W) for some other cases. Thereafter, we will generalize some of our results obtained for L(W) to the Hecke algebras Nq(W), which are q-deformations of L(W). For the case of group von Neumann algebras L(Γ) for general groups, we shall examine for semi-groups induced by a proper, conditionally negative function ψ, how the gradient-Sp property of the semi-group (Φt)t≥0 := (e−t∆ )t≥0 relates to the gradient-Sq property of the semi-group that is generated by the αth-root ∆α of the generator. Last, we will also show a method that allows us, for right-angled word hyperbolic Coxeter groups, to obtain (AO+) and strong solidity for L(W) without building a gradient-Sp quantum Markov semi-group, but by using a slightly different method.

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.

The first part of this thesis provides a mathematical description for bipartite quantum correlations, aiming to analyze the geometry of several sets of correlations. We explain why quantum entanglement can be used to simulate shared randomness: Cloc(Γ) ⊆ Cqd(Γ) for a sufficiently large d. The known bound for this dimension d in the literature is d ≥ dim(Cloc(Γ))+1, but we improve this by showing that the inclusion is always true for d ≥ dim(Cloc(Γ)). For the proof of this bound, we show that the set Cprivate(Γ) of correlations using private randomness is connected, which allows the use of an improved version of Carathéodory’s Theorem. In the second part of this thesis, we define and analyze a see-saw method to determine the state and measurement operators that reconstruct both the correlation itself as its entanglement dimension, by solving consecutive semidefinite programs. One of the strengths of the algorithm is its generality: it applies to different dimensions, question sets, and answer sets. Some numerical experiments demonstrated that the method can indeed reconstruct quantum correlations, although some highly entangled correlations failed to be reconstructed due to the computationallimitations. The numerical experiments motivated several new theorems, for example the fact that every correlation with |A|= 1 or |B|= 1 has entanglement dimension 1, which means that it can be written as a private randomness correlation. The proof of this result is based on the earlier described improvement for the dimension d.

The Lovász theta function, and the variants of it given by Schrijver and Szegedy are upper bounds on the independence number of a graph. These functions play an important role in several optimization problems, such as the Cohn-Elkies bound for optimal sphere packing densities. This thesis covers the properties of these functions. The main focus of this thesis is the multiplicity of Schrijver's variant. A construction for optimal solutions to the dual problem of this function for the disjunctive graph product is proposed, and its generality is studied. It is shown that for abelian Cayley graphs, the Lovász theta function, and its variants reduce a linear program. Using properties of Fourier analysis an upper bound is given on the minimal gap between the largest and least eigenvalues of optimal solutions to the dual of Schrijver's variant. This, along with numerical results, suggests that the proposed construction is more likely to hold when abelian Cayley graphs are sparse.

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.

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,{Esa},{Ftb}) = ∑a,b,s,t(P(a,b|s,t)- ψ*(Esa ⊗ Ftb)ψ)2. Here, P(a,b|s,t) is the bipartite correlation,  ψ is the state vector, and Esa and Ftb 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 Esa and Ftb 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, 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.

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.

The role of Unmanned Aerial Vehicles (UAVs), more commonly known as drones, in society continues to become more significant every day, both in everyday life and in military operations. The extent to which unmanned vehicles are used for both offensive as well as reconnaissance missions is at an all-time high. To expand the number of operational systems while managing costs, it is desirable to deploy systems that can operate fully independently. For a survey mission, this requires a planning of the complete mission before the drone leaves for enemy territory. The setting of such a mission can be stated as follows: starting from a secure base, multiple surveillance locations need to be safely reached and the acquired information has to be transmitted back to the base. There are many possible strategies for gathering this information. This report investigates how to find the strategy that maximises the expected amount of retrieved information. Specifically, such an optimal strategy tells us which route the UAV should take in enemy territory and at what moments in the mission transmissions should be made. We present a mathematical framework for formulating the problem, as well as a genetic algorithm capable of finding the optimal strategy in different scenarios.