Members: Cihan Okay (group leader, faculty), Ho Yiu Chung (postdoc, Sep 2021 - present), Aziz Kharoof (postdoc, Oct 2021 - present), Selman Ipek (postdoc, Oct 2021 - present), Igor Sikora (postdoc, Sep 2022 - present)
Our research is supported by the US Air Force Office of Scientific Research (FA9550-21-1-0002) and the European Union (HORIZON-CL4-2021-DIGITAL-EMERGING-01).
Our research focuses on the algebraic, geometric, and topological structures responsible for the computational advantage provided by quantum computers. We are currently working on two approaches:
Shor’s factoring algorithm shows that quantum computing makes it possible to solve some computational problems efficiently, which is impossible using a conventional computer. Classical simulation algorithms provide a rigorous way to quantify this computational advantage. One way to construct such algorithms is to use quasiprobability distributions where negative values in the distribution are necessary for quantum speedup. Our most recent work introduces a new classical simulation algorithm based on polytope theory that can simulate universal quantum computation. In this algorithm, negativity completely disappears, and it is an open problem to understand quantum speedup in this simulation method. The combinatorial structure of the polytope plays a crucial role and remains to be understood. Relevant paper: $\Lambda$-simulation.
Quantum contextuality, a generalization of Bell’s non-locality, is a notion from quantum foundations, a well-established computational resource. It is a phenomenon unique to quantum theory that shows itself in measurement statistics. Our recent work introduces a topological framework to study quantum contextuality. This approach uses combinatorial objects known as simplicial sets, a more expressive framework than simplicial complexes, to represent probability distributions on spaces. The resulting objects are called simplicial distributions, and the theory subsumes earlier approaches in a unified way. Moreover, our formalism goes beyond generalizing sets of measurements and outcomes, the conventional setup for physical experiments, to spaces representing measurements and outcomes. The implications of this generalization and the mathematical properties of simplicial distributions are yet to be understood. Relevant paper: Simplicial distributions.