The performances of two D-Wave 2 machines (476 and 496 qubits) and of a 1097-qubit D-Wave 2X were investigated. Each chip has a Chimera interaction graph G. Problem input consists of values for the fields hj and for the two-qubit interactions Ji,j of an Ising spin-glass problem formulated on G. Output is returned in terms of a spin configuration {sj}, with sj = +1 or -1. We generated random spanning trees (RSTs) uniformly distributed over all spanning trees of G. On the 476-qubit D-Wave 2, RSTs were generated on the full chip with Ji,j = -1 and hj = 0 and solved one thousand times. The distribution of solution energies and the average magnetization of each qubit were determined. On both the 476- and 1097-qubit machines, four identical spanning trees were generated on each quadrant of the chip. The statistical independence of the these regions was investigated.

<p>Qudit, a multi-level computational unit for quantum computing, provides a larger state space for information processing, and thus can reduce the circuit complexity, simplify the experimental setup. We promote the qudit-based quantum computing by providing an overview that covers a variety of qudit topics ranging from gate universality, circuit building, algorithm design, to physical realization methods. Among all the important qudit algorithms, we perform the first experimental realization of a qudit-based phase estimation algorithm(PEA) on a photonic platform, utilizing the high dimensionality in time and frequency degrees of freedom (DoFs) in a single photon. In our scheme the controlled-unitary gates can be realized in a deterministic fashion, as the control and target registers are now represented by two DoFs in a single photon. Next we improve the PEA by introducing a new statistical and variational approach to the PEA that we called SPEA. The SPEA can determine any unknown eigenstate-eigenphase pair from a given unitary matrix  by treating the probabilistic output of an Iterative PEA (IPEA)-like circuit as an eigenstate-eigenphase proximity metric, using this metric to estimate the proximity of the input state and input phase to the nearest eigenstate-eigenphase pair and approaching this pair via a variational process on the input state and phase. The SPEA can search over the entire computational space as well as some specified given range efficiently and thus outperforms the original PEA.</p> <p> </p> <p><br></p> <p>The simulation of open quantum dynamics has attracted wide interests recently with a variety of quantum algorithms developed and demonstrated. The second half of the thesis focus on the simulation of the open quantum dynamics which is a useful application for quantum computer based on qudit as well as qubit. We perform the first quantum simulations of the  radical pair mechanism(RPM) in the avian compass with a Sz.-Nagy dilation theorem-based quantum algorithm to demonstrate the generality of the quantum algorithm and to open new opportunities for studying the avian compass with quantum computing devices. Next we apply the same quantum algorithm to simulate open quantum dynamics based on the Generalized Quantum Master Equation (GQME). This approach overcomes the limitations of the Lindblad equation by providing a rigorous derivation of the equations of motion for any subset of elements of the reduced density matrix. We validate our quantum algorithm as applied to the spin-boson benchmark model by analyzing the impact of the quantum circuit depth on the accuracy of the results when the subset is limited to the diagonal elements of the reduced density matrix.  Our findings demonstrate that our approach yields reliable results on  noisy intermediate-scale quantum (NISQ) computers.</p>

This is a project with the ambition of demonstrating the possibilities and applications of the quantum approximation optimization algorithm (QAOA). Throughout the paper discussions on the theoretical background and fundamentals of the algorithm will be done by examining the relevant nomenclature. Then a set of possible application problems will be considered where it will be discussed why this specific algorithm is of interest for each individual problem. In the fourth section these problems will concretely be tested via simulations of the QAOA and lastly an analysis of the outcomes will be done.

No description available.

In this dissertation, we investigate three different problems in the field of Quantum computation. First, we discuss the quantum complexity of evaluating the Tutte polynomial of a planar graph. Furthermore, we devise a new quantum algorithm for approximating the phase of a unitary matrix. Finally, we provide quantum tools that can be utilized to extract the structure of black-box modules and algebras. While quantum phase estimation (QPE) is at the core of many quantum algorithms known to date, its physical implementation (algorithms based on quantum Fourier transform (QFT) ) is highly constrained by the requirement of high-precision controlled phase shift operators, which remain difficult to realize. In the second part of this dissertation, we introduce an alternative approach to approximately implement QPE with arbitrary constantprecision controlled phase shift operators. The new quantum algorithm bridges the gap between QPE algorithms based on QFT and Kitaev’s original approach. For approximating the eigenphase precise to the nth bit, Kitaev’s original approach does not require any controlled phase shift operator. In contrast, QPE algorithms based on QFT or approximate QFT require controlled phase shift operators with precision of at least Pi/2n. The new approach fills the gap and requires only arbitrary constant-precision controlled phase shift operators. From a physical implementation viewpoint, the new algorithm outperforms Kitaev’s approach. iii The other problem we investigate relates to approximating the Tutte polynomial. We show that the problem of approximately evaluating the Tutte polynomial of triangular graphs at the points (q, 1/q) of the Tutte plane is BQP-complete for (most) roots of unity q. We also consider circular graphs and show that the problem of approximately evaluating the Tutte polynomial of these graphs at the point (e 2πi/5 ,e−2πi/5 ) is DQC1-complete and at points (q k , 1 + 1−q−k (q 1/2−q−1/2) 2 ) for some integer k is in BQP. To show that these problems can be solved by a quantum computer, we rely on the relation of the Tutte polynomial of a planar G graph with the Jones and HOMFLY polynomial of the alternating link D(G) given by the medial graph of G. In the case of our graphs the corresponding links are equal to the plat and trace closures of braids. It is known how to evaluate the Jones and HOMFLY polynomial for closures of braids. To establish the hardness results, we use the property that the images of the generators of the braid group under the irreducible Jones-Wenzl representations of the Hecke algebra have finite order. We show that for each braid b we can efficiently construct a braid ˜b such that the evaluation of the Jones and HOMFLY polynomials of their closures at a fixed root of unity leads to the same value and that the closures of ˜b are alternating links. The final part of the dissertation focuses on finding the structure of a black-box module or algebra. Suppose we are given black-box access to a finite module M or algebra over a finite ring R, and a list of generators for M and R. We show how to find a linear basis and structure constants for M in quantum poly(log |M|) time. This generalizes a recent quantum algorithm of Arvind et al. which finds a basis representation for rings. We then show that iv our algorithm is a useful primitive allowing quantum computers to determine the structure of a finite associative algebra as a direct sum of simple algebras. Moreover, it solves a wide variety of problems regarding finite modules and rings. Although our quantum algorithm is based on Abelian Fourier transforms, it solves problems regarding the multiplicative structure of modules and algebras, which need not be commutative. Examples include finding the intersection and quotient of two modules, finding the additive and multiplicative identities in a module, computing the order of an module, solving linear equations over modules, deciding whether an ideal is maximal, finding annihilators, and testing the injectivity and surjectivity of ring homomorphisms. These problems appear to be exponentially hard classically.

The purpose of this thesis is the theoretical study of interference and correlation effects in multimode and continuum mode quantum systems. We are concerned with interference effects in multiport devices which in a sense are generalised Mach-Zehnder interferometers. It is shown how these multimode devices can be employed for the study of negative result and interaction free measurements. Interference and coherence effects are also studied in relation to the radiation fields generated by atoms through the process of spontaneous emission. Besides first order interference, higher order coherence effects are investigated with the aid of Glauber's photodetection theory and it is found that detectors that lie in spacelike regions may display nonclassical correlations under certain conditions. It is well known that the vanishing of field commutators between regions that cannot be connected by subluminal signals reflects the locality of quantum field theory. But is it possible that these spacelike regions exhibit correlations that violate Bell type inequalities? This is the main question and principal concern of the thesis and the answer is affirmative, nonclassical correlations between spacelike regions are indeed possible. A scheme of four detectors that lie in spacelike points was also studied. In this case we do not consider the radiation field but a free scalar field in vacuum state. Nevertheless the virtual quanta of this field may induce nonclassical correlations if the intervals between the detectors are spacelike but small enough. The fundamental reason for this fact is the nonvanishing of the Feynman propagator outside the light cone. Since this propagator is decaying expotentially with the distance it is demonstrated that for large spacelike intervals field correlations obey classical inequalities. We should also note that different inertial observers will agree on the violation or not of these inequalities since the results are manifestly Lorentz invariant.

Quantum holonomies are non-Abelian Geometric Phases predominantly observed in adiabatic, non-adiabatic, or measurement-based quantum evolutions. Their significance lies in their potential utility within quantum computing due to their robustness against noise throughout the parameter path. In this report, we detail the foundational methods necessary for constructing holonomic non-Abelian gates specifically designed for tripartite states <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%7CW%3E" data-classname="equation" data-title="" />and <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%7C%5Cbar%7BW%7D%3E" data-classname="equation" data-title="" />, which serve as the logical qubits in our project. Given that the existence of a universal set of gates has already been demonstrated for each of these evolution types, our project delves into the advantages of applying these basis states across the three evolution categories. We have reformulated the Nuclear Quadrupole Resonance (NQR) Hamiltonian to be exclusively composed of two-body terms, thus rendering it more experimentally feasible. Furthermore, we have connected the W states with the remaining tripartite states to construct a four-level model system and generalized gates within this framework. Lastly, we introduce a measurement-based method that maintains its non-Abelian attributes even in the Zeno limit, where the process of projective measurement gradually approaches the adiabatic model. / Icke-Abelska geometriska faser, så kallade kvantholonomier, observeras huvudsakligen i adiabatiska, icke-adiabatiska eller mätningsbaserade manipulationer av kvanttillstånd. De har stor potential till användning inom kvantdatorberäkningar på grund av deras robusthet mot olika typer av brus. I den här rapporten beskriver vi de grundläggande metoderna som är nödvändiga för att konstruera holonoma kvantgrindar som är speciellt utformade för trekroppstillstånden <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%7CW%3E" data-classname="equation" /> och <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%7C%5Cbar%7BW%7D%3E" data-classname="equation" data-title="" />, som fungerar som de logiska kvantbitarna i projektet, givet att det är redan bevisat att alla dessa modeller kan klara kraven för universalitet. Den här rapporten fokuserar på fördelarna med att tillämpa dessa logiska kvantbitar för tre olika evolutionskategorier. Vi har omformulerat kärnkvadrupolresonans Hamiltonianen så att den uteslutande består av tvåkroppstermer, vilket gör den mer experimentellt genomförbar för att realisera adiabatiska holonoma kvantgrindar. Vidare har vi kopplat <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?W" data-classname="equation" data-title="" />-tillstånden med andra trekroppstillstånden för att konstruera ett så kallat sammanflätat <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5CLambda" data-classname="equation" data-title="" />-system och icke-adiabatiska holonoma kvantgrindar inom detta ramverk. Slutligen introducerar vi en mätningsbaserad metod som, till skillnad från tidigare resultat, bibehåller sina icke-Abelska attribut även i Zeno-gränsen, där processen med projektiv mätning gradvis närmar sig den adiabatiska kärnkvadrupolresonans-modellen.

Semiconductor quantum dot systems offer a promising platform for quantum computation. And these quantum computation candidates are normally based on spin or charge properties of electrons. In these systems, we focus on quantum computation based on electron spins since these systems has good scalability, long coherence times, and rapid gate operations. And this thesis focuses on building a theoretical description of quantum dot systems and the link between theory and experiments. In many quantum dot systems, exchange interactions are the primary mechanism used to control spins and generate entanglement. And exchange energies are normally positive, which limits control flexibility. However, recent experiments show that negative exchange interactions can arise in a linear three-dot system when a two-electron double quantum dot is exchange coupled to a larger quantum dot containing on the order of one hundred electrons. The origin of this negative exchange can be traced to the larger quantum dot exhibiting a spin triplet-like rather than singlet-like ground state. Here we show using a microscopic model based on the configuration interaction (CI) method that both triplet-like and singlet-like ground states are realized depending on the number of electrons. In the case of only four electrons, a full CI calculation reveals that triplet-like ground states occur for sufficiently large dots. These results hold for symmetric and asymmetric quantum dots in both Si and GaAs, showing that negative exchange interactions are robust in few-electron double quantum dots and do not require large numbers of electrons. Recent experiments also show the potential to utilize large quantum dots to mediate superexchange interaction and generate entanglement between distant spins. This opens up a possible mechanism for selectively coupling pairs of remote spins in a larger network of quantum dots. Taking advantage of this opportunity requires a deeper understanding of how to control superexchange interactions in these systems. Here, we consider a triple-dot system arranged in linear and triangular geometries. We use CI calculations to investigate the interplay of superexchange and nearest-neighbor exchange interactions as the location, detuning, and electron number of the mediating dot are varied. We show that superexchange processes strongly enhance and increase the range of the net spin-spin exchange as the dots approach a linear configuration. Furthermore, we show that the strength of the exchange interaction depends sensitively on the number of electrons in the mediator. Our results can be used as a guide to assist further experimental efforts towards scaling up to larger, two-dimensional quantum dot arrays. / Doctor of Philosophy / Semiconductor quantum dot systems offer a promising platform for quantum computation. And these quantum computation candidates are normally based on spin or charge properties of electrons. In these systems, we focus on quantum computation based on electron spins since these systems has good scalability, long coherence times, and rapid gate operations. And this thesis focuses on building a theoretical description of quantum dot systems and the link between theory and experiments. A key requirement for quantum computation is the ability to control individual qubits and couple them together to create entanglement. In quantum dot spin qubit systems, the exchange interaction is the primary mechanism used to accomplish these tasks. This thesis is about attaining a better understanding of exchange interactions in quantum dot spin qubit systems and how they can be manipulated by changing the configuration of the system and the number of electrons. In this thesis, we show negative exchange energy can arise in large size quantum dots. This result holds for symmetric and asymmetric shape of the large dots. And we also provide a quantitative analysis of how large quantum dots can be used to create long-distance spin-spin interactions. This capability would greatly increase the flexibility in designing quantum processors built by quantum dot spins. The interplay of these systems with different geometry can serve as a guide to assist further experiments and may hopefully be the basis to build two-dimensional quantum dot arrays.

While optimization problems are ubiquitous in all domains of engineering, they are of critical importance to power systems engineers. A safe and economical operation of the power systems entails solving many optimization problems such as security-constrained unit commitment, economic dispatch, optimal power flow, optimal planning, etc. Although traditional optimization solvers and software have been successful so far in solving these problems, there is a growing need to accelerate the solution process. This need arises on account of several aspects of grid modernization, such as distributed energy resources, renewable energy, smart inverters, batteries, etc, that increase the number of decision variables involved. Moreover, the technologies entail faster dynamics and unpredictability, further demanding a solution speedup. Yet another concern is the growing communication overhead that accompanies this large-scale, high-speed, decision-making process. This thesis explores three different directions to address such concerns. The first part of the thesis explores the learning-to-optimize paradigm whereby instead of solving the optimization problems, machine learning (ML) models such as deep neural networks (DNNs) are trained to predict the solution of the optimization problems. The second part of the thesis also employs deep learning, but in a different manner. DNNs are utilized to model the dynamics of IEEE 1547.8 standard-based local Volt/VAR control rules, and then leverage efficient deep learning libraries to solve the resulting optimization problem. The last part of the thesis dives into the evolving field of quantum computing and develops a general strategy for solving stochastic binary optimization problems using variational quantum eigensolvers (VQE). / Doctor of Philosophy / A reliable and economical operation of power systems entails solving large-scale decision-making mathematical problems, termed as optimization problems. Modern additions to power systems demand an acceleration of this decision-making process while managing the accompanying communication overheads efficiently. This thesis explores the application of two recent advancements in computer science -- machine learning (ML) and quantum computing (QC), to address the above needs. The research presented in this thesis can be divided into three parts. The first part proposes replacing conventional mathematical solvers for optimization problems, with ML models that can predict the solutions to these solvers. Colloquially referred to as learning-to-optimize, this paradigm learns from a historical dataset of good solutions and extrapolates them to make new decisions in a fast manner, while requiring potentially limited data. The second part of the thesis also uses ML models, but differently. ML models are used to represent the underlying physical dynamics, and convert an originally challenging optimization problem into a simpler one. The new problem can be solved efficiently using popular ML toolkits. The third and final part of the thesis aims at accelerating the process of finding optimal binary decisions under constraints, using QC.

Ce mémoire étudie le problème ancestral 1 de déterminer la puissance relative de l’intrication préalable en complexité de la communication comparée à la communication quantique. L’idée maîtresse du mémoire est d’opérer un rapprochement entre la complexité de la communication et la théorie des jeux non-locaux. Spécifiquement, nous contemplons une variété de manières de convertir des jeux non-locaux pour lesquels il est su que beaucoup d’intrication est requise en problèmes de complexité de la communication. Ce faisant, nous obtenons les problèmes de communications affichant les plus grands écarts connus à ce jour entre les deux modèles pour des problèmes fonctionnels. / Ce mémoire étudie le problème ancestral 1 de déterminer la puissance relative de l’intrication préalable en complexité de la communication comparée à la communication quantique. L’idée maîtresse du mémoire est d’opérer un rapprochement entre la complexité de la communication et la théorie des jeux non-locaux. Spécifiquement, nous contemplons une variété de manières de convertir des jeux non-locaux pour lesquels il est su que beaucoup d’intrication est requise en problèmes de complexité de la communication. Ce faisant, nous obtenons les problèmes de communications affichant les plus grands écarts connus à ce jour entre les deux modèles pour des problèmes fonctionnels.