This is the first addendum, a piece of supplementary material to help you further understand topics that are important to quantum computing but not the focus of our meetings. Most of these will focus on either providing more mathematical background or looking more at the theory side of something we cover in a meeting.
This addendum will be doing the former for bra-ket notation, a type of notation widely used in quantum physics and computing but not used as often in math or computer science.
A ket is simply a column vector, i.e. a list of amplitudes in the form of a $n \times 1$ matrix. For a qubit, this list is always of size 2, where the first term represents the amplitude for $|0\rangle$ and the second term represents the amplitude for $|1\rangle$. We write a ket as $|\psi\rangle$, where $\psi$ is our quantum state variable.
A bra is simply a row vector, i.e. a list of amplitudes in the form of a $1 \times n$ matrix. Similarly to a ket, we write a bra as $\langle \psi|$. Importantly, $\langle \psi| = |\psi\rangle^\dag$ and $|\psi\rangle = \langle\psi|^\dag$.
Here, the $\dag$ symbol denotes the conjugate transpose of the vector. You may be familiar with the transpose of a matrix already; the conjugate transpose simply also flips the sign of the imaginary part of each element. For example,
$$ |\psi\rangle = \begin{bmatrix} a \\ b+ci \end{bmatrix} \qquad |\psi\rangle^\dag=\langle\psi|=\begin{bmatrix} a & b-ci \end{bmatrix} $$There is nothing special about vectors here; you can find the conjugate transpose of any matrix in a similar way.
In bra-ket notation, $\langle \psi | \psi \rangle$ represents the inner product (dot product) and $|\psi\rangle \langle \psi |$ represents the outer product. For the example above, we find that
$$ \langle \psi | \psi \rangle = \begin{bmatrix} a & b-ci \end{bmatrix} \begin{bmatrix} a \\ b+ci \end{bmatrix} = a^2+b^2+c^2 = 1 $$When the bra and the ket are the same state, this operation is the same as taking the sum of the absolute value squared of each term in $|\psi\rangle$. By our quantum state restriction, this sum must be 1. Similarly, we find that
$$ |\psi\rangle\langle\psi| = \begin{bmatrix} a \\ b+ci \end{bmatrix}\begin{bmatrix} a & b-ci \end{bmatrix} = \begin{bmatrix} a^2 & ab-aci \\ ab+aci & b^2+c^2 \end{bmatrix} $$I will use the rest of this addendum to discuss a few other properties of bra-ket notation that we will look at in more detail over the next few weeks. Don't worry if you don't understand much of this; we'll be looking at each of these properties more closely as the semester goes along! Likewise, feel free to use this page as reference anytime during the semester.
You will often see kets written next to each other, like $|\psi\rangle|\xi\rangle$. This is not matrix multiplication, since the dimensions are invalid. Instead, this is used to refer to the tensor product, sometimes denoted with the $\otimes$ symbol. For example,
$$ A = \begin{bmatrix} a \\ b \\ c \end{bmatrix}, \quad B = \begin{bmatrix} d \\ e \end{bmatrix}, \quad A \otimes B = \begin{bmatrix} ad \\ ae \\ bd \\ be \\ cd \\ ce \end{bmatrix} $$More formally, in quantum computing, the tensor product between a $m \times n$ matrix $A$ and $p \times q$ matrix $B$ is defined as
$$ A \otimes B = \begin{bmatrix} a_{11}B & \cdots & a_{1n}B \\ \vdots & \ddots & \vdots \\ a_{m1}B & \cdots & a_{mn}B \end{bmatrix} $$This matrix will have size $mp \times nq$. Tensor products are really useful in quantum computing. We'll be using them a lot throughout the semester, and I hope it becomes more clear why they're the right operation for a lot of what we're doing as the semester goes along.
The final thing I'll mention in this addendum is that you can multiply matrices with bras or kets, since they are just vectors. Often in quantum computing, this will be a square matrix with a size that corresponds to a power of 2. For example, suppose we have the matrix
$$ H = \begin{bmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ 1/\sqrt{2} & -1/\sqrt{2} \end{bmatrix} $$Let's try applying $H$ to a state, say $|0\rangle$. We would write this operation as follows:
$$ H|0\rangle = \begin{bmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ 1/\sqrt{2} & -1/\sqrt{2} \end{bmatrix}\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \end{bmatrix} = |+\rangle $$This is an interesting result! This matrix seems to convert the $|0\rangle$ state, i.e. 100% '0', into the $|+\rangle$ state, i.e. 50% '0' and 50% '1'.
There is a more subtle fact about this matrix $H$. We can observe that $HH^\dag=H^\dag H=I$, where $I$ is the identity matrix. If this property is true for some matrix $U$, then $U$ is a unitary matrix, which means that it will preserve the property that $\langle \psi | \psi \rangle=1$ for any quantum state $|\psi\rangle$ when applied. More formally, we would write $$ \langle\psi|U^\dag U|\psi\rangle = \langle\psi|\psi\rangle = 1 $$ where $\langle\psi|U^\dag$ represents the unitary $U$ being applied on the bra and $U|\psi\rangle$ represents it being applied on the ket. This unitary $H$ is extremely important; it is known as a Hadamard gate, and we will be talking about it next week.