Basic Mathematics to Understand Caltech Quantum Computing Lecture Notes (1)

What's this page for?

This is a page for me collecting mathematical terms that often appears in quantum computing, especially to read these lecture notes smoothly without worrying much about mathematical terms. If you are interested in these fields, I would say this page will also be your help.

This page contains

Bra-ket notation
Hermite conjugate
Hermitial matrix
Unitary matrix
Qubit

Bra-ket notation (Notation)

In quantum computing (mechanics), a vector $\bm{u}$ is expressed in this way $$ |u \rang $$ where $|u \rang$ is equivalent to $$ |u \rang = \bm{u} = \begin{bmatrix} u_1 \\ \vdots \\ u_n \\ \end{bmatrix} $$ This notation $| \, \, \rang$ is called ket notation.
If we transposed $\bm{u}$ as $\bm{u}^t$, we can rewrite that $$ \lang u| = \bm{u}^t = [u_1^t \, \dots \, u_n^t] $$ and this $\lang \, \,|$ notation is called bra notation. Basically ket notation shows a column vector and bra notation shows a row vector.

We can naturally write down an inner product of these two vectors as $$ \begin{aligned} \lang u| u \rang &= \bm{u}^t \bm{u} = [u_1^t \, \dots \, u_n^t] \begin{bmatrix} u_1 \\ \vdots \\ u_n \\ \end{bmatrix} \\ \lang u| u \rang &= \sum_{i=1}^n u_{i}^t \, u_i \end{aligned} $$ We can also multiply these two vectors as follows. $$ \begin{aligned} |u \rang \lang u| = \bm{u} \, \bm{u}^t &= \begin{bmatrix} u_1 \\ \vdots \\ u_n \\ \end{bmatrix} [u_1^t \, \dots \, u_n^t] \\\\ &= \begin{bmatrix} u_1 u_1^t & \dots & u_n u_n^t\\ \vdots &\ddots& \vdots\\ u_n u_1^t & \dots & u_n u_n^t\\ \end{bmatrix} \end{aligned} $$ In Quantum Computing, this bra-ket notation is often used to express the computational basis of $\Complex$. Computational means that we use a bit (0 or 1) to express infomation. To understand this, I will take an example in $\R^3$. We can choose $\bm{e_{i (i=0, 1, 2)}} $ as an orthonormal basis for $\R^3$. $$ \bm{e_0} = \begin{bmatrix} 1 \\ 0 \\ 0 \\ \end{bmatrix} , \bm{e_1} = \begin{bmatrix} 0 \\ 1 \\ 0 \\ \end{bmatrix} , \bm{e_2} =\begin{bmatrix} 0 \\ 0 \\ 1 \\ \end{bmatrix} $$ In bra-ket world, we write those vectors as $$ | i \rang = \bm{e_i} $$ For $i=1$, $$ | 1 \rang = \bm{e_1} = \begin{bmatrix} 0 \\ 1 \\ 0 \\ \end{bmatrix} $$ $i$ refers to the index where 1 exists in the vector. Note that the norm of an orthonormal basis is always 1.

Hermite conjugate $A^\dagger$ (Operator)

A matrix $A^\dagger$ is a transposed and conjugated of the original matrix $A$. For instance, let 2 by 2 matrix $A$ as $$ A = \begin{bmatrix} i & 1 + 2i \\ -3 & 5 - 4i\\ \end{bmatrix} $$ then the daggered version $A^\dagger$ is equivalent to $$ A^\dagger = (A^T)^\ast = (A^\ast)^T = \begin{bmatrix} -i & -3 \\ 1-2i & 5 + 4i\\ \end{bmatrix} $$

Hermitial matrix (Matrix)

When $$ A^\dagger = A $$ $A$ is called Hermitial matrix.
For example, let A as $$ A = \begin{bmatrix} 1 & i \\ -i & 1\\ \end{bmatrix} $$ then the daggered $A^\dagger$ is also $$ A^\dagger = \begin{bmatrix} 1 & i \\ -i & 1\\ \end{bmatrix} $$

Unitary matrix (Matrix)

When a matrix $A$ has this property, it is called Unitary matrix. $$ A \, A^\dagger = A^\dagger \, A = I $$ Here $I$ is the identity matrix $$ I = \begin{bmatrix} 1 && \bf{0}\\ & \ddots & \\ \bf{0} && 1\\ \end{bmatrix} $$ For example, let $U_1$ $$ U_1 = \begin{bmatrix} \cos{\theta} & \sin{\theta}\\ -\sin{\theta} & \cos{\theta}\\ \end{bmatrix} \\ $$ then, apparently this $U_1$ is a unitary matrix because $$ \begin{aligned} U_1 U_1^\dagger &= \begin{bmatrix} \cos{\theta} & \sin{\theta}\\ -\sin{\theta} & \cos{\theta}\\ \end{bmatrix} \begin{bmatrix} \cos{\theta} & -\sin{\theta}\\ \sin{\theta} & \cos{\theta}\\ \end{bmatrix} \\\\ &= \begin{bmatrix} 1 &0 \\ 0 &1\\ \end{bmatrix} \\\\ &= I \end{aligned} $$ Another example is that, let $U_2$ as $$ a, b \in \Complex^2, ~~~ U_2 = \begin{bmatrix} a & b \\ -b^\ast e^{i\varphi} & a^\ast e^{i\varphi} \\ \end{bmatrix}, ~~~~ |a|^2 + |b|^2 = 1 $$ then as you can see $$ \begin{aligned} U_2^\dagger &= \begin{bmatrix} a^\ast & -b e^{-i\varphi} \\ b^\ast & a e^{-i\varphi} \\ \end{bmatrix}\\\\ U_2 U_2^\dagger &= \begin{bmatrix} a & b \\ -b^\ast e^{i\varphi} & a^\ast e^{i\varphi} \\ \end{bmatrix} \begin{bmatrix} a^\ast & -b e^{-i\varphi} \\ b^\ast & a e^{-i\varphi} \\ \end{bmatrix}\\\\ &= \begin{bmatrix} 1 &0 \\ 0 &1\\ \end{bmatrix} \\\\ &= I \end{aligned} $$ This is also a unitary matrix.

And for any unitary matrix $U$, $$ |\det{U}| = 1 $$

We can rewrite any unitary matrix $U$ using Hermite matrix $H$ as $$ U = e^{iH} $$

As you know from the first linear algebra class, when $Q$ is a orthogonal matrix $$ Q^T Q = Q Q^T = I \\ Q^T = Q^{-1} \\ |\det{Q}| = 1\\ \vdots $$ We can think of orthogonal matrices as expanded versions of unitary matrices.

Qubit

A qubit is represented as $$ | \psi \rang = \alpha |0\rang + \beta |1\rang $$ where $\alpha$ and $\beta$ are complex numbers that satisfy $$ |\alpha| ^2 + |\beta| ^2 = 1 $$ $|0\rang$ and $|1\rang$ are the computational basis of $\Complex^2$. As you already know $$ |0 \rang = \begin{bmatrix} 1 \\ 0 \\ \end{bmatrix} , |1 \rang = \begin{bmatrix} 0 \\ 1 \\ \end{bmatrix} $$ the number inside the bra-ket shows the location of 1 as an index.

References

https://cds.cern.ch/record/1522001/files/978-1-4614-6336-8_BookBackMatter.pdf
http://www.theory.caltech.edu/~preskill/ph219/ph219_2018-19
https://en.wikipedia.org/wiki/Unitary_matrix
https://en.wikipedia.org/wiki/Orthogonal_matrix

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