Abstract: The Quantum Information RIT is concerned with all aspects of research at the intersection between quantum information science and mathematics. Goals for talks include: studying recent research results in quantum information from a mathematical angle; finding examples (old and new) in which existing tools from mathematics can be adapted for application in quantum information; and studying quantum algorithms for mathematical problems.
This seminar is associated with QuICS and the UMD math department, and participation from other departments is encouraged.
Abstract: In this talk, we will briefly explore the use of monoidal categories in quantum information. Namely we will introduce the ZX-Calculus which is a diagrammatic formalism that strives to enhance intuition by providing pictorial proofs and we will provide examples of how some proofs can be simplified using this framework. Later in the talk, we will go over a recent application of ZX-Calculus in fault tolerant quantum computing and how it was used to provide the intuition for this result.
References:  Coecke, Bob. "Quantum picturalism." Contemporary physics 51.1 (2010): 59-83.  de Beaudrap, Niel, Xiaoning Bian, and Quanlong Wang. "Fast and effective techniques for T-count reduction via spider nest identities." arXiv preprint arXiv:2004.05164 (2020).
Abstract: One of the most successful recent applications of quantum information has been the generation of "certified" random numbers. The unique properties of quantum devices allow the generation of numbers that are provably random, offering a potential major benefit to cryptography and other fields. Constructing the right theory to prove quantum randomness can be challenging. Quantum devices and adversaries are modeled by complex vector spaces whose dimensions may be arbitrarily large, and proving security in such a setting leads to some difficult mathematical problems. This talk will be a retrospective look at one successful approach to quantum randomness that is based on the convexity of the Schatten matrix norm.
Abstract: Aaronson and Ambainis (2009) and Chailloux (2018) showed that fully symmetric (partial) functions do not admit exponential quantum query speedups. This raises a natural question: how symmetric must a function be before it cannot exhibit a large quantum speedup?
In this work, we prove that hypergraph symmetries in the adjacency matrix model allow at most a polynomial separation between randomized and quantum query complexities. We also show that, remarkably, permutation groups constructed out of these symmetries are essentially the only permutation groups that prevent super-polynomial quantum speedups. We prove this by fully characterizing the primitive permutation groups that allow super-polynomial quantum speedups.
In contrast, in the adjacency list model for bounded-degree graphs (where graph symmetry is manifested differently), we exhibit a property testing problem that shows an exponential quantum speedup. These results resolve open questions posed by Ambainis, Childs, and Liu (2010) and Montanaro and de Wolf (2013).
Abstract: In this talk, I will present quantum query complexity of symmetric oracle problems. In particular, we will be exploring the query complexity of quantum learning problems in which the oracles form a group G of unitary matrices, and a description of the optimal success probability of a t-query quantum algorithm in terms of group characters. More generally, in the coset identification problem, for a subgroup H of group G, the task is to determine which coset the group element belongs to. The authors provide character-theoretic formulas for the optimal success probability achieved by a t-query algorithm for this problem. It generalizes a number of well-known quantum algorithms including the Bernstein-Vazirani problem, the van Dam problem and finite field polynomial interpolation.
References:  Daniel Copeland, James Pommersheim, Quantum query complexity of symmetric oracle problems, https://arxiv.org/abs/1812.09428  Orest Bucicovschi, Daniel Copeland, David A. Meyer, and James Pommersheim. Single-query quantum algorithms for symmetric problems, https://arxiv.org/abs/1503.05548
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