Math 401, Fall 2025: Thesis notes, R4, Superdense coding and Quantum error correcting codes

Progress: 0/NaN=NaN% (denominator and enumerator may change)

This part may not be a part of “mathematical” research. But that’s what I initially begin with.

Tip

Or another way in quantum computing Quantum Computing and Quantum Information Section 2.3

References to begin with

Every quantum bit is composed of two orthogonal states, denoted by $|0\rangle$ and $|1\rangle$ .

Each state

\varphi=\alpha|0\rangle+\beta|1\rangle

where $\alpha$ and $\beta$ are complex numbers, and $|\alpha|^2+|\beta|^2=1$ .

All the logic gates are unitary operators in $\mathbb{C}^{2\times 2}$ .

Example: the NOT gate is represented by the following matrix:

NOT=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

Hadamard gate is represented by the following matrix:

H=\frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}

This part is intentionally left blank and may be filled near the end of the semester, by assignments given in CSE5313.