# Quantum Information

## Hyperbolic band theory

The notions of Bloch wave, crystal momentum, and energy bands are commonly regarded as unique features of crystalline materials with commutative translation symmetries. Motivated by the recent realization of hyperbolic lattices in circuit QED, I will present a hyperbolic generalization of Bloch theory, based on ideas from Riemann surface theory and algebraic geometry. The theory is formulated despite the non-Euclidean nature of the problem and concomitant absence of commutative translation symmetries. The general theory will be illustrated by examples of explicit computations of hyperbolic Bloch wavefunctions and bandstructures.

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## Random walks and graphs in materials, biology, and quantum information science

What does mathematics, materials science, biology, and quantum

information science have in common? It turns out, there are many

connections worth exploring. I this talk, I will focus on graphs and random

walks, starting from the classical mathematical constructs and moving on to

quantum descriptions and applications. We will see how the notions of graph

entropy and KL divergence appear in the context of characterizing

polycrystalline material microstructures and predicting their performance

under mechanical deformation, while also allowing to measure adaptation in

cancer networks and entanglement of quantum states. We will discover

unified conditions under which master equations for classical random walks

exhibit nonlocal and non-diffusive behavior and see how quantum walks allow

to realize the coveted exponential speedup in quantum Hamiltonian

simulations. Recent classical and quantum breakthroughs and open questions

will be discussed.

For other events in this series see the quanTA events website.