Linear Algebra in Quantum Computing
Byoungwook Jang · Digital Commons - Colby (Colby College) · 2014
The immense power of computing has been continually growing, yet we still want more speed and computing capacity. If, as Moores Law states, the number of transistors on a microprocessor continues to double every other year, we will find the circuits on a microprocessor measured on an atomic scale in a couple of decades. Then, we would start to consider quantum computers, which will make use of the power of atoms to perform memory and processing tasks. The mathematics behind quantum computers will be the focus of my presentation. The information contained in qubits (quantum bits), which are analogues of classical bits, can be represented in a matrix form, namely a density matrix. It has been found that a density matrix is positive semi-definite and of trace 1. The operators acting on density matrices need to preserve the trace and positivity so that the resulting output after processing the information is still a density matrix. Therefore, these operators can be thought of as a trace-preserving positive map. In this presentation, the definitions of such maps and properties will be introduced along with the theorems that are related to completely positive maps.