Quantum Signal Processing and its generalizations via martix polynomials

Témavezető: Gilyén András Pál
Rényi Intézet
email: gilyenandras@gmail.com

Projekt leírás

In recent years, it has been revealed that the study of matrix products of the following form:

U_0 W U_1 W U_2 W ... W U_k,

where 'W' is a diagonal matrix with elements omega and omega^-1, and each 'U_i' is a 2x2 matrix over the complex numbers, makes many quantum algorithms much more understandable and easier to describe. (Here, intuitively, omega should be thought of as a variable moving on the complex unit circle.)

Characterizing matrices of this form has therefore led to a major breakthrough in the development of quantum algorithms. These matrices are exactly the 2x2 matrix-valued Laurent polynomials that have at most degree-k in omega, have even / odd parity in omega that matches 'k', and for every omega in the complex unit circle, they describe a unitary matrix.

Several open questions remain in this context. For example, given a Laurent polynomial that meets the conditions, how can we efficiently find the sequence U_0, U_1, U_2, ..., U_k? There are quite good algorithms that run in about k^3 time, but we would like to find even better ones.

An interesting question from the perspective of quantum algorithms is what happens if, for example, 'W' is a matrix with elements omega_1 and omega_2? Or what if we look at 4x4 diagonal matrices 'W' with diagonal entries omega_1, omega_1^-1, omega_2, omega_2^-1? Can we give a good characterization for the resulting matrix-valued polynomials?

Előfeltételek

Knowledge of linear algebra, polynomials, and unitary matrices. Some familiarity with complex functions.

Hivatkozások