Matthew Gibson-Lopez

Quantum computing promises computational speedups, but understanding when and why quantum algorithms outperform classical ones requires building the right intuition. Last semester, we explored Deutsch's algorithm—a simple quantum algorithm that solves a particular function evaluation problem with just one query, whereas classical approaches require two. This talk begins by examining the natural generalization commonly taught in textbooks: the Deutsch-Jozsa algorithm, which extends the same quantum technique to functions on n-bit inputs. We'll then push further by relaxing the strict constant-versus-balanced restriction to see what happens when we estimate the majority class size through repeated quantum measurements. Surprisingly, this provides no advantage over classical random sampling—a result that helps clarify the boundaries of quantum speedup. However, this raises a compelling question: when does repeated quantum sampling help? We'll introduce amplitude estimation, a powerful generalization of Grover's algorithm that achieves a quadratic speedup for certain estimation problems, demonstrating where quantum coherence can reduce sampling variance in ways classical approaches cannot. Throughout, we'll maintain our discrete math perspective. No background in quantum mechanics assumed.

A lattice L is an additive discrete subgroup of ℝⁿ. A set of linearly independent vectors B = {b₁, ..., b_n} ⊆ ℝⁿ is called a basis of L if L = ℤb₁ + ⋯ + ℤb_n. When n ≥ 2, L has infinitely many possible bases. Bases consisting of relatively short and nearly orthogonal vectors are considered good bases, while others are regarded as bad bases. The lattice reduction theory studies how to obtain a good basis from a given one. In this talk, we will explore several lattice basis reduction methods and algorithms, along with their applications in cryptography and number theory.

Code-based cryptography is one of the most promising candidates for Post-Quantum Cryptography. Indeed, the NIST has selected HQC as one of the standards for Post-Quantum Cryptography. In this talk, we explore some linear code constructions that have been used in this field, with emphasis on quasi-cyclic codes which are the ones implemented in HQC. The presentation will be informal and suitable for a wide range of student in STEM fields.

Solving multivariate non-linear systems of polynomial equations is a ubiquitous task in computational science, essential to fields ranging from discrete optimization and statistical inference to computer vision, systems biology, and cryptography. The standard algorithmic engine for solving these systems is Gröbner basis computation. However, the efficiency of this method is highly sensitive to the choice of monomial ordering. Despite a near continuum of possible orders, current implementations overwhelmingly rely on static, expert-guided heuristics such as GrevLex. We take a data driven approach by casting the selection of monomial orderings as a Reinforcement Learning (RL) problem over the space of admissible orderings. The trained agents outperform human expert heuristics across the test cases, sometime reducing the computational cost up to 70%. Related paper: https://arxiv.org/pdf/2602.02972 Talk will assume no background in polynomials or related algorithms, necessary concepts will be reviewed. As much as time permits, concepts from RL will also be explained.

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