Ph.D. Candidate: Sepehr Nezami
Research Advisor: Patrick Hayden
Date: Thursday, May 23, 2019
Time: 2:00 pm
Location: Varian 355
Title: Representation theory, Randomness, and Quantum Information Science
Often times, understanding physical systems or mathematical structures requires analyzing the generic behaviors instead of specific examples. In such cases, a common approach is identifying the degrees of freedom that are supposed to be generic and replacing them with entirely random values. This line of thinking has been fruitful in many areas of physics, such as nuclear physics, quantum chaos, quantum field theory, and quantum gravity. Within quantum information science, which is the primary focus of this thesis, multiple areas including analyzing quantum entanglement, quantum capacity of channels and quantum Shannon theory, construction of quantum codes, and benchmarking quantum systems have heavily benefited from randomization techniques. Most of this thesis is devoted to random constructions and relevant mathematical formalisms, such as representation theory and group theory, and their applications in quantum information science and quantum gravity.
In particular, we focus on the stabilizer formalism. Stabilizer formalism is the main framework for exact quantum error correction, and its power and flexibility have led to applications throughout quantum information science. We classify the symmetries of multiple copies of stabilizer states and Clifford operators, proving a stabilizer analog of the celebrated Schur-Weyl duality. A key part of this discovery is a new description and formula for the higher moments of the Clifford group in terms of discrete geometries. This work provides a framework for understanding the statistical properties of stabilizer states and the representation theory of the Clifford group. We demonstrate that using this new formalism, one can provide simple derivations for previously known results in this area, such as qubit stabilizers being 3-designs. We further discuss applications of this mathematical framework, including proving a new exponential de Finetti theorem and applications in stabilizer testing.