Related papers: A 3-Stranded Quantum Algorithm for the Jones Polyn…
We study an efficient algorithm to hash any single qubit gate (or unitary matrix) into a braid of Fibonacci anyons represented by a product of icosahedral group elements. By representing the group elements by braid segments of different…
We initiate a systematic study of pseudo-deterministic quantum algorithms. These are quantum algorithms that, for any input, output a canonical solution with high probability. Focusing on the query complexity model, our main contributions…
We study the deterministic query complexity of Boolean functions on slices of the hypercube. The $k^{th}$ slice $\binom{[n]}{k}$ of the hypercube $\{0,1\}^n$ is the set of all $n$-bit strings with Hamming weight $k$. We show that there…
The k-means method is a widely used clustering algorithm. One of its distinguished features is its speed in practice. Its worst-case running-time, however, is exponential, leaving a gap between practical and theoretical performance. Arthur…
The Bayesian network structure learning (BNSL) problem asks for a directed acyclic graph that maximizes a given score function. For networks with $n$ nodes, the fastest known algorithms run in time $O(2^n n^2)$ in the worst case, with no…
We consider online algorithms for the $k$-server problem on trees. Chrobak and Larmore proposed a $k$-competitive algorithm for this problem that has the optimal competitive ratio. However, a naive implementation of their algorithm has…
In this thesis, we prove several results concerning field-theoretic invariants of knots and 3-manifolds. In Chapter 2, for any knot $K$ in a closed, oriented 3-manifold $M$, we use $SU(2)$ representation spaces and the Lagrangian field…
The fastest known algorithm for factoring a degree $n$ univariate polynomial over a finite field $\mathbb{F}_q$ runs in time $O(n^{3/2 + o(1)}\text{polylog } q)$, and there is a reason to believe that the $3/2$ exponent represents a…
We continue our study of the degree of the colored Jones polynomial under knot cabling started in "Knot Cabling and the Degree of the Colored Jones Polynomial" (arXiv:1501.01574). Under certain hypothesis on this degree, we determine how…
In this manuscript we define Vassiliev measures of complexity for open curves in 3-space. These are related to the coefficients of the enhanced Jones polynomial of open curves in 3-space. These Vassiliev measures are continuous functions of…
It has recently been shown that starting with a classical query algorithm (decision tree) and a guessing algorithm that tries to predict the query answers, we can design a quantum algorithm with query complexity $O(\sqrt{GT})$ where $T$ is…
Our goal is to compute the minimal-order recurrence of the colored Jones polynomial of the 7_4 knot, as well as for the first four double twist knots. As a corollary, we verify the AJ Conjecture for the simplest knot 7_4 with reducible…
Let $k$ be a finite field, and $L$ be a $q$-linearized polynomial defined over $k$ of $q$-degree $r$ ($L=\sum^r_{i=0}a_iZ^{q^i}$, with $a_i\in k$). This paper provides an algorithm to compute a characteristic polynomial of $L$ over a large…
Quantum signal processing (QSP) and quantum singular value transformation (QSVT) have provided a unified framework for understanding many quantum algorithms, including factorization, matrix inversion, and Hamiltonian simulation. As a…
In this work we propose a single rounding algorithm for the fractional solutions of the standard LP relaxation for $k$-clustering. As a starting point, we obtain an iterative rounding $(\frac{3^p + 1}{2})$-Lagrangian Multiplier-Perserving…
In this master thesis, I present a new family of knots in the solid torus called lassos, and their properties. Given a knot $K$ with Alexander polynomial $\Delta_K(t)$, I then use these lassos as patterns to construct families of satellite…
We present subquadratic algorithms in the algebraic decision-tree model for several \textsc{3Sum}-hard geometric problems, all of which can be reduced to the following question: Given two sets $A$, $B$, each consisting of $n$ pairwise…
The $K$-medoids problem is a challenging combinatorial clustering task, widely used in data analysis applications. While numerous algorithms have been proposed to solve this problem, none of these are able to obtain an exact (globally…
Quantum signal processing and quantum singular value transformation are powerful tools to implement polynomial transformations of block-encoded matrices on quantum computers, and has achieved asymptotically optimal complexity in many…
In this paper we study the complexity of quantum query algorithms computing the value of Boolean function and its relation to the degree of algebraic polynomial representing this function. We pay special attention to Boolean functions with…