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In this thesis, we develop algorithms similar to the Gaussian elimination algorithm in symplectic and split orthogonal similitude groups. As an application to this algorithm, we compute the spinor norm for split orthogonal groups. Also, we…
The discrete logarithm problem in a finite group is the basis for many protocols in cryptography. The best general algorithms which solve this problem have time complexity of $\mathcal{O}(\sqrt{N}\log N)$, and a space complexity of…
Quantum algorithms are able to solve particular problems exponentially faster than conventional algorithms, when implemented on a quantum computer. However, all demonstrations to date have required already knowing the answer to construct…
We create a variety of new quantum algorithms that use Grover's algorithm and similar techniques to give polynomial speedups over their classical counterparts. We begin by introducing a set of tools that carefully minimize the impact of…
The discrete logarithm problem in Jacobians of curves of high genus $g$ over finite fields $\FF_q$ is known to be computable with subexponential complexity $L_{q^g}(1/2, O(1))$. We present an algorithm for a family of plane curves whose…
An instance of a group testing problem is a set of objects $\cO$ and an unknown subset $P$ of $\cO$. The task is to determine $P$ by using queries of the type ``does $P$ intersect $Q$'', where $Q$ is a subset of $\cO$. This problem occurs…
The exact evaluation of the molecular ground state in quantum chemistry requires an exponentially increasing computational cost. Quantum computation is a promising way to overcome the exponential problem using polynomial-time quantum…
Vector set orthogonal normalization and matrix QR decomposition are fundamental problems in matrix analysis with important applications in many fields. We know that Gram-Schmidt process is a widely used method to solve these two problems.…
Quadratic Unconstrained Binary Optimization (QUBO) is a broad class of optimization problems with many practical applications. To solve its hard instances in an exact way, known classical algorithms require exponential time and several…
The first quantum algorithm to offer an exponential speedup (in the query complexity setting) over classical algorithms was Simon's algorithm for identifying a hidden exclusive-or mask. Here we observe how part of Simon's algorithm can be…
Quantum algorithms require a universal set of gates that can be implemented in a physical system. For these, an optimal decomposition into a sequence of available operations is desired. Here, we present a method to find such sequences for a…
In the past decade quantum algorithms have been found which outperform the best classical solutions known for certain classical problems as well as the best classical methods known for simulation of certain quantum systems. This suggests…
We study the computational complexity of quantum state isomorphism problems under group actions: given two quantum circuits that prepare pure or mixed states, decide whether the two states are related by a group action. This can be seen as…
We propose a quantum algorithm which uses the number of qubits in an optimal way and efficiently simulates a physical model with rich and complex dynamics described by the quantum sawtooth map. The numerical study of the effect of static…
The ultimate objective of this paper is to create a stepping stone to the development of new quantum algorithms. The strategy chosen is to begin by focusing on the class of abelian quantum hidden subgroup algorithms, i.e., the class of…
We introduce a quantum algorithm to perform the Laplace transform on quantum computers. Already, the quantum Fourier transform (QFT) is the cornerstone of many quantum algorithms, but the Laplace transform or its discrete version has not…
Amongst the most remarkable successes of quantum computation are Shor's efficient quantum algorithms for the computational tasks of integer factorisation and the evaluation of discrete logarithms. In this article we review the essential…
To address the issue of excessive quantum resource requirements in Kuperberg's algorithm for the dihedral hidden subgroup problem, this paper proposes a distributed algorithm based on the function decomposition. By splitting the original…
Gate model quantum computers with too many qubits to be simulated by available classical computers are about to arrive. We present a strategy for programming these devices without error correction or compilation. This means that the number…
Assuming the polynomial hierarchy is infinite, we prove a sufficient condition for determining if uniform and polynomial size quantum circuits over a non-universal gate set are not efficiently classically simulable in the weak…