Quantum Topology and its Applications

Quantum cryptography is known for enabling functionalities that are unattainable using classical information alone. Recently, Secure Software Leasing (SSL) has emerged as one of these areas of interest. Given a circuit ? from a circuit class, SSL produces an encoding of ? that enables a recipient to evaluate ? and also enables the originator of the software to later verify that the software has been returned, meaning that the recipient has relinquished the possibility to further use the software. Such a functionality is unachievable using classical information alone, since it is impossible to prevent a user from keeping a copy of the software. Recent results have shown the achievability of SSL using quantum information for compute-and-compare functions (a generalization of point functions). However, these prior works all make use of setup or computational assumptions. We show that SSL is achievable for compute-and-compare circuits without any assumptions.
We proceed by studying quantum copy-protection, which is a notion related to SSL, but where the encoding procedure inherently prevents a would-be quantum software pirate from splitting a single copy of an encoding for ? into two parts each allowing a user to evaluate ?. Using quantum message authentication codes, we show that point functions can be copy-protected without any assumptions against one honest and one malicious evaluator. We then show that a generic honest-malicious copy-protection scheme implies SSL. By prior work, this yields SSL for compute-and-compare functions.

This is joint work with Anne Broadbent, Stacey Jeffery, Supartha Podder, and Aarthi Sundaram.

The entanglement of free fermions on Hamming graphs will be discussed. This will be used to showcase how tools of algebraic combinatorics such as the Terwilliger algebra are well suited for this analysis. The usefulness of a Heun operator generalization will also be stressed and extensions to other association schemes will be mentioned.

I will describe a way to compute anomalies in general (2+1)D fermionic topological phases. First, a mathematical characterization of symmetry fractionalization for (2+1)D fermionic topological phases is presented, and then this data will be used to define a (3+1)D state sum for a topologically invariant path integral that depends on a generalized spin structure and G bundle on a 4-manifold. This path integral is a cobordism invariant and describes a (3+1)D fermion symmetry-protected topological state (SPT). The special case of time-reversal symmetry with ?2=−1? gives a ℤ16 invariant of the 4D Pin+ smooth bordism group, and gives an example of a state sum that can distinguish exotic smooth structure.

Please note, the last 3 minutes of the talk are missing from the video

Topological orders have a mathematical axiomatization in terms of their higher fusion categories of extended operators; the characterizing property of these higher fusion categories is that they are satisfy a nondegeneracy condition. After overviewing some of the higher category theory that goes into this axiomatization, I will describe what we do and don't know about the classification of topological orders in various dimensions.

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.