Quantum álgorithms for the combinatorial invariants of numerical semigroups

It was back in spring 2014 when the author of this doctoral dissertation was finishing its master thesis, whose main objective was the understanding of Peter W. Shor’s most praised result, a quantum algorithm capable of factoring integers in polynomial time. During the development of this master thesis, me and my yet-tobe doctoral advisor studied the main aspects of quantum computing from a purely algebraic perspective. This research eventually evolved into a sufficiently thorough canvas capable of explaining the main aspects and features of the mentioned algorithm from within an undergraduate context. Just after its conclusion, we seated down and elaborated a research plan for a future Ph.D. thesis, which would expectantly involve quantum computing but also a branch of algebra whose apparently innocent definitions hide some really hard problems from a computational perspective: the theory of numerical semigroups. As will be seen later, the definition of numerical semigroup does not involve sophisticated knowledge from any somewhat obscure and distant branch of the tree of mathematics. Nonetheless, a number of combinatorial problems associated with these numerical semigroups are extremely hard to solve, even when the size of the input is relatively small. Some examples of these problems are the calculations of the Frobenius number, the Apéry set, and the Sylvester denumerant, all of them bearing the name of legendary mathematicians. This thesis is the result of our multiple attempts to tackle those combinatorial problems with the help of a hypothetical quantum computer. First, Chapter 2 is devoted to numerical semigroups and computational complexity theory, and is divided into three sections. In Section 2.1, we give the formal definition of a numerical semigroup, along with a description of the main problems involved with them. In Section 2.2, we sketch the fundamental concepts of complexity theory, in order to understand the true significance within the inherent hardness concealed in the resolution of those problems. Finally, in Section 2.3 we prove the computational complexity of the problems we aim to solve. Chapter 3 is the result of our outline of the theory of quantum computing. We give the basic definitions and concepts needed for understanding the particular place that quantum computers occupy in the world of Turing machines, and also the main elements that compose this particular model of computation: quantum bits and quantum entanglement. We also explain the two most common models of quantum computation, namely quantum circuits and adiabatic quantum computers. For all of them we give mathematical definitions, but always having in mind the physical experiments from which they stemmed. Chapter 4 is also about quantum computing, but from an algorithmical perspective. We present the most important quantum algorithms to date in a standardized way, explaining their context, their impact and consequences, while giving a mathematical proof of their correctness and worked-out examples. We begin with the early algorithms of Deutsch, Deutsch-Jozsa, and Simon, and then proceed to explain their importance in the dawn of quantum computation. Then, we describe the major landmarks: Shor’s factoring, Grover’s search, and quantum counting. Chapter 5 is the culmination of all previously explained concepts, as it includes the description of various quantum algorithms capable of solving the main problems inside the branch of numerical semigrops. We present quantum circuit algorithms for the Sylvester denumerant and the numerical semigroup membership, and adiabatic quantum algorithms for the Ap´ery Set and the Frobenius problem. We also describe a C++ library called numsem, specially developed within the context of this doctoral thesis and which helps us to study the computational hardness of all previously explained problems from a classical perspective. This thesis is intended to be autoconclusive at least in the main branches of mathematics in which it is supported; that is to say numerical semigroups, computational complexity theory, and quantum computation. Nevertheless, for the majority of concepts explained here we give references for the interested reader that wants to delve more into them

Errors in scalable quantum Computers

A functional quantum computer potentially outperforms any classical machine exponentially in a number of important computational tasks. Therefore, its physical implementation has to scale efficiently in the number of qubits, specifically in tasks such as treatment of external error sources. Due to the intrinsic complexity and limited accessibility of quantum systems, the validation of quantum gates is fundamentally difficult. Randomized Benchmarking is a protocol to efficiently assess the average fidelity of only Clifford group gates. In this thesis we present a hybrid of Randomized Benchmarking and Monte Carlo sampling for the validation of arbitrary gates. It improves upon the efficiency of current methods while preserving error amplification and robustness against imperfect measurement, but is still exponentially hard. To achieve polynomial scaling, we introduce a symmetry benchmarking protocol that validates the conservation of inherent symmetries in quantum algorithms instead of gate fidelities. Adiabatic quantum computing is believed to be more robust against environmental effects, which we investigate in the typical regime of a scalable quantum computer using renormalization group theory. We show that a k-local Hamiltonian is in fact robust against environmental influence but multipartite entanglement is limited to combined system-bath state which we conclude to result in a more classical behavior more susceptible to thermal noise.Ein Quantencomputer wäre in einer Reihe wichtiger Berechnungen exponenziell effizienter als klassische Computer, unter Vorraussetzung einer fehlerarmen und skalierbaren Implementierung. Aufgrund der intrinsischen Komplexität und beschränkten Auslesbarkeit von Quantensystemen ist die Validierung von Quantengattern ungleich schwerer als die klassischer. Das Randomized Benchmarking Protokoll leistet dies effizient, ist jedoch beschränkt auf Cliffordgatter. In dieser Arbeit präsentieren wir ein Hybridprotokoll aus Interleaved Randomized Benchmarking und Monte Carlo Sampling zur Validierung von beliebigen Gattern. Trotz Verbesserung gegenüber vergleichbaren Protokollen skalieren die benötigten Ressourcen exponenziell. Um dies zu vermeiden entwickeln wir ein Protokoll, welches die Erhaltung von spezifischen Symmetrien von Quantenalgorithmen untersucht und dadurch Rückschlüsse auf die Fehlerrate der Quantenprozesse zulässt und demonstrieren seine Effizienz an relevanten Beispielen. Der Effekt von Umgebungseinflüssen auf adiabatische Quantencomputer wird als weit weniger gravierend angenommen als im Falle von konventionellen Systemen, ist jedoch im gleichen Maße weniger verstanden. Wir untersuchen diese Effekte mithilfe von Renormalisierungsgruppentheorie und zeigen, dass k-lokale Hamiltonoperatoren robust sind, vielfach verschränkte Zustände hingegen nur verschränkt mit der Umgebung existieren. Wir folgern daraus ein verstärkt thermisches Verhalten des Annealingprozesses.QEO/IARPA, Google, ScaleQI

Adiabatic quantum computing from an eigenvalue dynamics point of view

We investigate the effects of a generic noise source on a prototypical adiabatic quantum algorithm. We take an alternative eigenvalue dynamics viewpoint and derive a generalised, stochastic form of the Pechukas-Yukawa model. The distribution of avoided crossings in the energy spectra is then analysed in order to estimate the probability of level occupation. We find that the probability of successfully finding the system in the solution state decreases polynomially with the computation speed and that this relationship is independent of the noise amplitude. The overall regularity of the eigenvalue dynamics is shown to be relatively unaffected by noise perturbations. These results imply that adiabatic quantum computation is a relatively stable process and possesses a degree of resistance against the effects of noise. We also show that generic noise will inherently break any symmetries, and therefore remove degeneracies, in the energy spectrum that might otherwise have impeded the computation process. This suggests that the conventional stipulation that the initial and final Hamiltonians do not commute is unnecessary in realistic physical systems. We explore the effects of an artificial noise source with a specifically engineered time-dependent amplitude and show that such a scheme could provide a significant enhancement to the performance of the computation. Finally, we formulate an extended version of the Pechukas-Yukawa formalism. This provides a complete description of the dynamics of a quantum system by way of an exact mapping to a system of classical equations of motion

Notes on Quantum Computation and Information

We discuss fundamentals of quantum computing and information - quantum gates, circuits, algorithms, theorems, error correction, and provide collection of QISKIT programs and exercises for the interested reader.Comment: v2: 86 pages, 97 references. Refined the text, fixed several typos, added some text on continuous variables, and added few solved example problems. v1: 72 pages, 76 references. Suggestions, comments, and corrections are very welcome

Recipes for spin-based quantum computing

Technological growth in the electronics industry has historically been measured by the number of transistors that can be crammed onto a single microchip. Unfortunately, all good things must come to an end; spectacular growth in the number of transistors on a chip requires spectacular reduction of the transistor size. For electrons in semiconductors, the laws of quantum mechanics take over at the nanometre scale, and the conventional wisdom for progress (transistor cramming) must be abandoned. This realization has stimulated extensive research on ways to exploit the spin (in addition to the orbital) degree of freedom of the electron, giving birth to the field of spintronics. Perhaps the most ambitious goal of spintronics is to realize complete control over the quantum mechanical nature of the relevant spins. This prospect has motivated a race to design and build a spintronic device capable of complete control over its quantum mechanical state, and ultimately, performing computations: a quantum computer. In this tutorial we summarize past and very recent developments which point the way to spin-based quantum computing in the solid-state. After introducing a set of basic requirements for any quantum computer proposal, we offer a brief summary of some of the many theoretical proposals for solid-state quantum computers. We then focus on the Loss-DiVincenzo proposal for quantum computing with the spins of electrons confined to quantum dots. There are many obstacles to building such a quantum device. We address these, and survey recent theoretical, and then experimental progress in the field. To conclude the tutorial, we list some as-yet unrealized experiments, which would be crucial for the development of a quantum-dot quantum computer.Comment: 45 pages, 12 figures (low-res in preprint, high-res in journal) tutorial review for Nanotechnology; v2: references added and updated, final version to appear in journa