Discrete Analysis Seminar

Quantum locally testable codes provide a means for encoding quantum data that protects against errors in such a way that the errors can be detected using only local tests on the encoded data. This notion of quantum local testability (for a certain class of codes, namely CSS codes) can be expressed as a form of high-dimensional expansion, which generalizes graph expansion to chain complexes. We will describe the state-of-the-art construction of such codes, which is based on the hemicube, a cellular complex that is topologically equivalent to the real projective plane. No prior background in quantum codes will be assumed.

In a continuation of last week's talk, we will first finish the analysis of the hemicubic code. While the length-N hemicubic code has order sort(N) distance and 1/log(N) soundness, making it essentially the state-of-the-art quantum locally testable code, it remains an open question to improve these parameters to linear distance and constant soundness (as well as to improve the rate and locality parameters). We will discuss this open problem in more depth, and describe some recent constructions that improve certain parameters at the cost of others

We will introduce the multivariate quantum eigenvalue transform, and discuss an application to matrix functions of normal matrices. Joint work with Yonah Borns-Weil and Tahsin Saffat.