Life with Physics

More about Orthonormal Basis First of all, consider any vector $|\psi\rangle$ expanded within an orthonormal basis $|\psi\rangle=\sum_{i=1}^n\psi_i|\epsilon_i\rangle$. Because of the orthonormal property \begin{equation}\langle \epsilon_i |\epsilon_j\rangle=\delta_{ij}\,,\tag{1}\end{equation} by applying $\langle\epsilon_i|$ on both sides of the expansion , we obtain $\psi_i = \langle \epsilon_i |\psi\rangle$. Since the relation \begin{equation}|\psi\rangle =\sum_{i=1}^n\psi_i|\epsilon_i\rangle = \sum_{i=1}^n \langle \epsilon_i |\psi\rangle\,|\epsilon_i\rangle = \sum_{i=1}^n |\epsilon_i\rangle\langle\epsilon_i |\psi\rangle\end{equation} holds for any $|\psi\rangle$, we obtain the completeness property \begin{equation} \sum_{i=1}^n |\epsilon_i\rangle\langle\epsilon_i |= \cal{I}\,,\tag{2}\end{equation} where $\cal{I}$ is the identity  transformation  that leaves any vector unchanged.    Secondly, let's write an operator within the orthonormal basis \beg...

Inner Product  Given a  vector space  $V$ over the field $\mathbb{C}$, we define an inner product, i.e, a map $(\,,\,): \, V\times V \rightarrow \mathbb{C}$ that satisfies the following axioms: $\Big(|\psi\rangle\,,\,  |\phi\rangle\Big)=\Big(|\phi\rangle\,,\,  |\psi\rangle\Big)^*$ $\Big(|\psi\rangle\,,\,  |\phi\rangle+|\chi\rangle\Big) = \Big(|\psi\rangle\,,\,  |\phi\rangle\Big)+\Big(|\psi\rangle\,,\,  |\chi\rangle\Big)$ $\Big(|\psi\rangle\,,\,  c|\phi\rangle\Big)=c\Big(|\psi\rangle\,,\, |\phi\rangle\Big)$ for any $c\in\mathbb{C}$ $\Big(|\psi\rangle\,,\, |\psi\rangle\Big)\rangle \geq 0$, equality holds iff $|\psi\rangle=\mathbf{0}$ Note: In the  dual vector space , $\langle\, | \,\rangle$ is bilinear. In contrast, here the inner product $(,)$ is linear in its second argument and antilinear in its first argument.  The definition of inner product leads to the  projection theo...

Dual Vector Space Given a  vector space  $V$, its dual $V^*$ is a set of linear maps $f:\,V\rightarrow \mathbb{C}$. Following the  Dirac notation , we denote such map $f$ by $\langle f|$. When acting on a vector $|\psi\rangle$, it leads to a complex number denoted by $\langle f|\psi\rangle$. The map is linear such that \begin{equation}\langle f|\,\Big(c_1|\psi\rangle + c_2|\phi\rangle\Big)=c_1\,\langle f|\psi\rangle+c_2\langle f|\phi\rangle\,.\end{equation} $V^*$ becomes a vector space when the linear maps are equipped with the addition and scaling operations: \begin{eqnarray}\langle f+g| \psi\rangle&=& \langle f| \psi\rangle + \langle g| \psi\rangle\,,\\ \langle c\,f|\psi\rangle&=& c \langle f|\psi\rangle\,,\end{eqnarray} for all $f, g \in V^*$, $|\psi\rangle\in V$ and $c\in\mathbb{C}$. Note: $\langle f|\psi\rangle$ is bilinear in each of the two arguments. We will use Greek letters for vectors in the vector space and use English letters for maps in the d...

Basis A set of vectors $|\alpha_1\rangle, \cdots, |\alpha_n\rangle$ is a basis of a  vector space  if $|\alpha_1\rangle, \cdots, |\alpha_n\rangle$ are linearly independent. That is, the only solution of the equation \begin{equation}x_1 |\alpha_1\rangle+x_2 |\alpha_2\rangle+\cdots + x_n |\alpha_n\rangle =\mathbf{0}\end{equation} is $x_1=\cdots=x_n=0$. Any vector $|\psi\rangle$ in the vector space is a linear combination of $|\alpha_1\rangle, \cdots, |\alpha_n\rangle$, i.e. \begin{equation}|\psi\rangle =c_1 |\alpha_1\rangle+c_2 |\alpha_2\rangle+\cdots + c_n |\alpha_n\rangle\end{equation} for $c_1,\cdots, c_n\in \mathbb{C}$. [Exercise 1] If there is vector $|\psi\rangle$ that can not be a linear combination of linearly independent vectors $|\alpha_1\rangle, \cdots, |\alpha_n\rangle$, then $|\alpha_1\rangle, \cdots, |\alpha_n\rangle, |\psi\rangle$ are linearly independent. [Solution] Consider the equation $x_1 |\alpha_1\rangle+\cdots + x_n |\alpha_n\rangle +d|\psi\rangle =\ma...

This series of posts is just a refresh of linear algebra basics in the language of quantum mechanics. One of the motivations is that in many books like Prof. Georgi's "Lie Algebra in Particle Physics" , linear algebra is directly presented in quantum mechanics notations. Vector Space The fundamental concept in linear algebra is the so-called vector space (or linear space). It is a set of elements called vectors that can be added together or multiplied by scalars  in the field $\mathbb{F}$. When the scalars are complex numbers, we can denote the vectors by  kets  $|\,\rangle$. So in the language of quantum mechanics, the basic definition of vector space over $\mathbb{C}$ can be formulated as follows: $|\psi\rangle + |\phi\rangle = |\phi\rangle + |\psi\rangle$. $\Big(|\psi\rangle + |\phi\rangle\Big)+|\chi\rangle=|\psi\rangle+\Big(|\phi\rangle + |\chi\rangle\Big)$. There exists an $\mathbf{0}$ such that $|\psi\rangle + \mathbf{0}=|\psi\rangle$ for every $|\psi\rangle$....

I occasionally browsed a youtube video on "2023 MIT Integration Bee". I thought it is a competition between students taking the course of calculus. I took a try and it turns out that I was not able to even solve the first problem in the allowed 4 mins.  The first problem is to solve \begin{equation} I \equiv \int_0^{\infty} \frac{\sqrt[3]{\tan x}}{(\sin x + \cos x)^2}dx\,.\tag{1}\end{equation} It is obvious that \begin{eqnarray}I &=&  \int_0^{\infty} \frac{\sqrt[3]{\tan x}}{(\tan x + 1)^2} \frac{dx}{\cos^2 x} =  \int_0^{\infty} \frac{\sqrt[3]{x}}{(x+1)^2}dx\\&=&-\left. \frac{\sqrt[3]{x}}{x+1}\right|_0^{\infty} +  \int_0^{\infty} \frac{d\,\sqrt[3]{x}}{x+1}\\&=&  \int_0^{\infty}\frac{dx}{x^3+1}\,.\tag{2}\end{eqnarray} I know how to solve the indefinite integral $\int\frac{dx}{x^3+1}$, but I was not able to accomplish the entire calculation within 4 mins. Here is another youtube video  that provides step-by-step calculat...

In this post, we will show how the wave optics becomes the geometrical optics in the limit of zero wavelength. When in the university, I only took one optics course at the level of college physics, which only mentioned the concept of Fermat's principle without too much discussions. Fermat's principle In geometrical optics, the Fermat's principle states that the path $\vec{r}$ of a light ray from the point A to B always minimizes the quantity: \begin{equation}\mathbb{S}[\vec{r}] = \int_A^B n(\vec{r})\,ds\,, \tag{1}\end{equation} where $n(\vec{r})$ is the distribution of reflective index of the medium in space and $ds$ is the differential arc length along the path \begin{equation}ds =\sqrt{d\vec{r}\cdot d\vec{r}}\,.\end{equation} The Fermat's principle provides an explicit connection between the geometrical optics and the classical mechanics in the form of principle of least action. Warm up: Snell's law Before deriving the general equation of motion from the Fermat...

The sum of all natural numbers One may see the following mysterious identity that the sum of all natural numbers equals to $-\frac{1}{12}$:   \begin{equation} 1 + 2 + 3 + \cdots = -\frac{1}{12}\,.\tag{1}\end{equation} The rigorous mathematical interpretation relies on the so-called Riemann zeta function $\zeta(s)$ and its analytic continuation. This is not the focus of this post, so we just highlight the main conclusions: \begin{equation}\zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^s}\tag{2}\end{equation} is only valid when $\mathfrak{Re} (s) > 1$. After analytic continuation, we can compute $\zeta(-1)=-\frac{1}{12}$. Therefore, mathematically, Eq.(1) should be interpreted in the language of analytic continuation. In this post, we explain Eq. (1) by applying a scheme that is commonly used in physics when treating a divergent series $\sum_{n=1}^{\infty}a_n$:  Multiply each term $a_n$ by some regulator such as $e^{-n\epsilon}$ to make the series converge.  Compute the r...

We introduced the HHL algorithm and its subroute quantum phase estimation (QPE) algorithm in this blog . Now we introduce two quantum versions of machine learning algorithms: quantum principal component analysis (qPCA) and quantum support vector machine (qSVM), which replies on QPE and HHL, respectively. Data Preparation Suppose we have $n$ data samples and each data sample has $p$ features, i.e., $\mathbf{x}^{(i)}\equiv[x_1^{(i)}, \cdots, x_p^{(i)}]^T$ for $i=1,\cdots, n$. To apply quantum algorithms, we encode each data sample $\mathbf{x}^{(i)}$ in a quantum state \begin{eqnarray}\mathbf{x}^{(i)}\equiv[x_1^{(i)}, \cdots, x_p^{(i)}]^T \quad\xrightarrow{p=2^m} \quad |\psi_i\rangle\equiv \frac{1}{\left|\left| \mathbf{x}^{(i)}\right|\right|_2}\sum_{\tau\in \{0,1\}^m}x_{\tau}^{(i)}|\tau\rangle\,.\tag{1}\end{eqnarray} To encode the entire dataset, we entangle the data state $|\psi_i\rangle$ with an auxiliary state $|i\rangle$ and form a pure state \begin{equation}|\psi\rangle\equiv\sum_{i=...

Background Quantum computing can bring exponential speedup over some classical algorithms. Most of the early-age algorithms are designed as proof of concept and thus lack of practical interests. Shor's algorithm for integer factorization is the first quantum algorithm that will bring practical applications: breaking the current RSA encryption  on the internet. However, implementing Shor's algorithm requires the manipulation of thousands of qubits, which is far beyond today's engineering capability. The breakthrough starts with the appearance of Harrow-Hassidim-Lloyd (HHL) algorithm in 2009, which solves linear equations in logarithmic time. Solving linear equations is of great importance in almost all the scientific and engineering disciplines. It is also an indispensable subroute in many machine learning algorithms. Besides its wide applications, implementing HHL algorithm only requires tens of qubits and thus seems feasible in the near future. Along with the rise of deep...

Qubits  A qubit is a well-defined quantum two-level system with states $|0\rangle$ and $|1\rangle$. Unlike a classical bit that is either 0 or 1, a qubit can be in a superposition of both $|0\rangle$ and $|1\rangle$ as \begin{equation}|\text{qubit}\rangle\equiv\cos\frac{\theta}{2} |0\rangle +\sin\frac{\theta}{2} e^{i\phi}|1\rangle\,.\end{equation} Notes: Quantum states are the same up to an overall phase factor. To remove the ambiguity of the overall phase factor, we use the conversion that the coefficient in front of $|0\rangle$ is always a real positive number. $\phi$  is the relative phase difference between $|0\rangle$ and $|1\rangle$. Each qubit state can be one-to-one mapped to a point on the Bloch sphere . Note that the polar angle $\theta$ in the spherical coordinate system takes the value in $[0, \pi]$. To ensure the coefficient in front of $|0\rangle$ is always a real positive number, we parameterize the qubit state as $\frac{\theta}{2}$ rather than $\theta$. As...