Synergy between Quantum Computers and Databases

Academia, industry, and societies are showing increasing interest in the possibilities of quantum computing. The research in the intersection of quantum computing and databases is still in its initial steps. This work represents several crucial data management and query processing problems that will benefit from quantum computing. We outline how quantum computing will tackle these challenges and what kind of outcomes and speed-ups we expect. We discuss the position of quantum computing in data management and raise awareness of possible security threats in encryption. We aim to be realistic and point out technical difficulties that currently restrict implementations.Peer reviewe

Reaalisten varistojen vakausindeksi: Bröckerin ja Scheidererin teoria

Tässä työssä todistan Bröckerin ja Scheidererin teorian avoimille semi-algebrallisille perusjoukoille. Teoria osoittaa, että jokaiselle reaaliselle algebralliselle varistolle on olemassa yläraja niiden polynomien lukumäärässä, joiden avulla variston osajoukkona olevia avoimia semi-algebrallisia perusjoukkoja määritellään. Tätä lukuarvoa kutsutaan reaalisen variston vakausindeksiksi. Teoria pohjaa suljetuille reaalisille kunnille, jotka yleistävät reaalilukujen kuntaa. Reaalinen algebrallinen varisto on suljetun reaalisen kunnan osajoukko, joka on määritelty polynomiyhtälöiden ratkaisujoukkona. Jokainen semi-algebrallinen joukko on määritelty Boolen yhdistelmänä äärellisestä määrästä polynomien merkkiehtoja, jotka toteuttavat tietyt yhtäsuuruudet ja epäyhtälöt. Semi-algebralliset perusjoukot ovat ne semi-algebralliset joukot, jotka toteuttavat ainoastaan annetut yhtäsuuruudet ja epäyhtälöt polynomien merkkiehdoissa. Semi-algebrallisista perusjoukoista voidaan siis rakentaa kaikki semi-algebralliset joukot ottamalla perusjoukkojen äärelliset yhdisteet, leikkaukset ja joukkoerotukset. Tämä työ pyrkii esittämään riittävät esitiedot päätuloksen todistuksen syvällistä ja yksityiskohtaista ymmärtämistä varten. Ensimmäinen luku esittelee ja motivoi tulosta yleisellä tasolla. Toinen luku käsittelee tiettyjä edistyneitä algebrallisia rakenteita, joita vaaditaan päätuloksen todistuksessa. Näitä ovat muun muassa radikaalit, alkuideaalit, assosiatiiviset algebrat, renkaan ulottuvuuden käsite sekä tietyt tekijärakenteet. Kolmas luku määrittelee suljetut reaaliset kunnat ja semi-algebralliset joukot, jotka ovat tämän työn kulmakiviä. Kolmannessa luvussa myös kehitetään neliömuotojen teoriaa. Kolmannen luvun päätulos on Wittin teoria. Neljäs luku käsittelee Pfisterin muotoja, jotka ovat tietynlaisia neliömuotoja. Työssä määritellään yleiset Pfisterin muodot kuntien yli. Tämän jälkeen kehitään niiden teoriaa rationaalifunktioiden kunnan yli. Viidennessä luvussa esitetään kaksi konstruktiivista esimerkkiä Bröckerin ja Scheidererin teorian käytöstä yhdessä ja kahdessa ulottuvuudessa. Nämä esimerkit edelleen motivoivat tulosta ja sen mahdollisia algoritmisia ominaisuuksia. Työn päätteeksi todistetaan Bröckerin ja Scheidererin teoria, joka osoittaa, että reaalisen variston vakausindeksi on olemassa ja se on äärellinen kaikille reaalisille varistoille.In this work, I prove the theorem of Bröcker and Scheiderer for basic open semi-algebraic sets. The theorem provides an upper bound for a stability index of a real variety. The theory is based on real closed fields which generalize real numbers. A real variety is a subset of a real closed field that is defined by polynomial equalities. Every semi-algebraic set is defined by a boolean combination of polynomial equations and inequalities of the sign conditions involving a finite number of polynomials. The basic semi-algebraic sets are those semi-algebraic sets that are defined solely by the sign conditions. In other words, we can construct semi-algebraic sets from the basic semi-algebraic sets by taking the finite unions, intersections, and complements of the basic semi-algebraic sets. Then the stability index of a real variety indicates the upper bound of numbers of polynomials that are required to express an arbitrary semi-algebraic subset of the variety. The theorem of Bröcker and Scheiderer shows that such upper bound exists and is finite for basic open semi-algebraic subsets of a real variety. This work aims to be detailed in the proofs and represent sufficient prerequisites and references. The first chapter introduces the topic generally and motivates to study the theorem. The second chapter provides advanced prerequisites in algebra. One of such results is the factorial theorem of a total ring of fractions. Other advanced topics include radicals, prime ideals, associative algebras, a dimension of a ring, and various quotient structures. The third chapter defines real closed fields and semi-algebraic sets that are the fundamental building blocks of the theory. The third chapter also develops the theory of quadratic forms. The main result of this chapter is Witt’s cancellation theorem. We also shortly describe the Tsen-Lang theorem. The fourth chapter is about Pfister forms. Pfister forms are special kinds of quadratic forms that we extensively use in the proof of the main theorem. First, we define general Pfister forms over fields. Then we develop their theory over the fields of rational functions. Generally, Pfister forms share multiple similar properties as quadratic forms. The fifth chapter represents one- and two-dimensional examples of the main theorem. These examples are based on research that is done on constructive approaches to the theorem of Bröcker and Scheiderer. The examples clarify and motivate the result from an algorithmic perspective. Finally, we prove the main theorem of the work. The proof is heavily based on Pfister forms

A Formal Category Theoretical Framework for Multi-model Data Transformations

Data integration and migration processes in polystores and multi-model database management systems highly benefit from data and schema transformations. Rigorous modeling of transformations is a complex problem. The data and schema transformation field is scattered with multiple different transformation frameworks, tools, and mappings. These are usually domain-specific and lack solid theoretical foundations. Our first goal is to define category theoretical foundations for relational, graph, and hierarchical data models and instances. Each data instance is represented as a category theoretical mapping called a functor. We formalize data and schema transformations as Kan lifts utilizing the functorial representation for the instances. A Kan lift is a category theoretical construction consisting of two mappings satisfying the certain universal property. In this work, the two mappings correspond to schema transformation and data transformation.Peer reviewe

MultiCategory: multi-model query processing meets category theory and functional programming

The variety of data is one of the important issues in the era of Big Data. The data are naturally organized in different formats and models, including structured data, semi-structured data, and unstructured data. Prior research has envisioned an approach to abstract multi-model data with a schema category and an instance category by using category theory. In this paper, we demonstrate a system, called MultiCategory, which processes multi-model queries based on category theory and functional programming. This demo is centered around four main scenarios to show a tangible system. First, we show how to build a schema category and an instance category by loading different models of data, including relational, XML, key-value, and graph data. Second, we show a few examples of query processing by using the functional programming language Haskell. Third, we demo the flexible outputs with different models of data for the same input query. Fourth, to better understand the category theoretical structure behind the queries, we offer a variety of graphical hooks to explore and visualize queries as graphs with respect to the schema category, as well as the query processing procedure with Haskell.Peer reviewe

Quantum Machine Learning: Foundation, New Techniques, and Opportunities for Database Research

In the last few years, the field of quantum computing has experienced remarkable progress. The prototypes of quantum computers already exist and have been made available to users through cloud services (e.g., IBM Q experience, Google quantum AI, or Xanadu quantum cloud). While fault-tolerant and large-scale quantum computers are not available yet (and may not be for a long time, if ever), the potential of this new technology is undeniable. Quantum algorithms have the proven ability to either outperform classical approaches for several tasks, or are impossible to be efficiently simulated by classical means under reasonable complexity-theoretic assumptions. Even imperfect current-day technology is speculated to exhibit computational advantages over classical systems. Recent research is using quantum computers to solve machine learning tasks. Meanwhile, the database community already successfully applied various machine learning algorithms for data management tasks, so combining the fields seems to be a promising endeavour. However, quantum machine learning is a new research field for most database researchers. In this tutorial, we provide a fundamental introduction to quantum computing and quantum machine learning and show the potential benefits and applications for database research. In addition, we demonstrate how to apply quantum machine learning to the optimization of join order problem for databases.Non peer reviewe