Most combinatorial optimization problems from theoretical computer science have a natural framing as optimization of polynomials in commuting variables. Noncommutativity is one of the defining features of quantum mechanics. So it is not surprising that noncommutative polynomial optimization plays an equally important role in quantum computer science. Our main goal here is to understand the relative hardness of commutative versus noncommutative polynomial optimization. At a first glance it might seem that noncommutative polynomial optimization must be more complex. However this is not always true and this question of relative hardness is substantially more subtle than might appear at the outset. First in this thesis we show that the general noncommutative polynomial optimization is complete for the class $\Pi_2$; this class is in the second level of the arithmetical hierarchy and strictly contains both the set of recursively enumerable languages and its complement. On the other hand, commutative polynomial optimization is decidable and belongs to $\PSPACE$. We then provide evidence that for polynomials arising from a large class of constraint satisfaction problems the situation is reversed: the noncommutative polynomial optimization is an easier computational problem compared to its commutative analogue. A second question we are interested in is about whether we could extract good commutative solutions from noncommutative solutions? This brings us to the second theme of this thesis which is about understanding the algebraic structure of the solutions of noncommutative polynomial optimization. We show that this structural insight then could shed light on the optimal commutative solutions and thereby paves the path in understanding the relationships between the commutative and noncommutative solutions. Here we first use the sum-of-squares framework to understand the algebraic relationships that are present between operators in any optimal noncommutative solution of a class of polynomial optimization problems arising from certain constraint satisfaction problems. We then show how we can design approximation algorithms for these problems so that some algebraic structures of our choosing is present. Finally we propose a rounding scheme for extracting good commutative solutions from noncommutative ones.

Lozenges are polygons constructed by gluing two equilateral triangles along an edge. We can fit lozenge pieces together to form larger polygons and given an appropriate polygon we can tile it with lozenges. Lozenge tilings of the semi-regular hexagon with sides A,B,C can be viewed as the 2D picture of a stack of cubes in a A x B x C box. In this project we investigate the typical shape of a tiling as the sides A,B,C of the box grow uniformly to infinity and we consider two cases: The uniform case where all tilings occur with equal probability and the q^Volume case where the probability of a tiling is proportional to the volume taken up by the corresponding stack of cubes. To investigate lozenge tilings we transform it into a question on families of non-intersecting paths on a corresponding graph representing the hexagon. Using the Lindström–Gessel–Viennot theorem we can define the probability of a non-intersecting path crossing a particular point in the hexagon both for the uniform and the $q$-Volume case. In each case this probability function is connected to either the Hahn or the $q$-Hahn orthogonal polynomials. The orthogonal polynomials depend on the sides of the hexagon and so we consider the asymptotic behaviour of the polynomials as the sides grow to infinity using a result due to Kuijlaars and Van Assche. This determines the density of non-intersecting paths through every point in the hexagon, which we calculate, and a ``Arctic curve" result which shows that the six corners of the hexagon are (with probability one) tiled with just one type of lozenge. / "Lozenger" är polygoner konstruerade genom att limma två liksidiga trianglar längs en kant. Vi kan montera lozengstycken ihop för att bilda större polygoner och med en lämplig polygon kan vi lozengplatta den. Lozengplattor av den semi-liksidiga hexagonen med sidorna A, B, C kan ses som 2D-bilden av en stapel kuber i en A x B x C-box. I det här projektet undersöker vi den typiska formen på en platta när sidorna A, B, C på rutan växer till oändlighet och vi tar an två fall: Det likformiga fallet där alla plattor sker med samma sannolikhet och q ^ Volymfallet då sannolikheten för en platta är proportionell mot volymen som tas upp av motsvarande kubstapel. För att undersöka plattor förvandlar vi det till en fråga om samlingar av icke-korsande vägar på en motsvarande graf som representerar hexagonen. Med hjälp av satsen Lindström – Gessel – Viennot kan vi definiera sannolikheten för att en icke-korsande väg går genom en viss punkt i hexagonen både för det enhetliga och $ q $ -volymfallet. I båda fallen är dessa sannolikhetsfunktioner relaterade till Hahn eller $ q $ -Hahn ortogonala polynomer. Dessa ortogonala polynom beror på hexagonens sidor så vi betraktar polynomens asymptotiska beteende när sidorna växer till oändlighet genom ett resultat från Kuijlaars och Van Assche. Detta bestämmer densiteten för de icke-korsande vägarna genom varje punkt i det hexagon vi beräknar. Detta bestämmer också också en '' arktisk kurva '' som visar att hexagonens sex hörn är (med sannolikhet ett) plattade med bara en typ av lozeng.

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京都大学 / 新制・課程博士 / 博士(理学) / 甲第23678号 / 理博第4768号 / 新制||理||1683(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 加藤 周, 教授 雪江 明彦, 教授 池田 保 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM