Gain insights into quantum computing, starting with qubits and quantum mechanics. Discover quantum gates, circuits, and algorithms like Grover’s search and Shor’s factoring. Explore potential applications.

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Quantum Computing

Jupyter Notebook

Firms like IBM, Honeywell, Zapata, Rigetti, Amazon, Google, IonQ, and others are all adopting and investing heavily in quantum computing. This is a great opportunity to get involved.

In this course, you will learn the fundamentals of quantum computing, starting with quantum bits or qubits. These are the centerpiece and most basic computational unit. 

You will then move on to quantum mechanics and the role of quantum gates, circuits, and simulating computers. From there, you will get acquainted with the many different quantum algorithms that are asymptotically faster than their classical counterparts. These include: Deutsch Jozsa, Grover’s search, and Shor’s factoring.

By the end of this course, you will have the foundations in place to start exploring more applications of quantum computing. This is just the beginning!

The Fundamentals of Quantum Computing

Previously, we have discussed how one can combine qubits by taking their tensor product to create a multi-qubit system. We will be revisiting that concept in this lesson. But before that, let’s discuss **quantum spin**. We must build some basic understanding of it, its quantization, and its measurement so we can understand **quantum entanglement**.

# Quantum spin
The **spin** of an electron or any other quantum particle is a physical property that we can measure just like **momentum** and **position**. Without going too much into the details, **spin** can be considered a quantum mechanical variant of **angular momentum**. Loosely, you can think of spin as a certain direction in space, e.g., **up** or **down**, associated with a quantum particle. 

Like other observables in quantum mechanics, spin is also quantized (discretized), meaning that it comes only in specific packets. You would recall from the previous lesson on the postulates of quantum mechanics that **measurements** on **observables** output the **eigenvalues** of the **operator**. 



# Stern-Gerlach Experiment
The **Stern-Gerlach Experiment** was the first successful experiment at measuring the spin of an electron. In it, a beam of electrons was directed through a magnetic field towards a detecting screen. The magnetic field deflected electrons in their pathway, and the screen detected two discrete positions that the electrons ended up in. This became the first experimental proof of the fact that spin is also quantized like other observables. 

For this lesson, you can think of the **Stern-Gerlach Experiment** as a black box that measures the spin of an electron, up or down. 

We now know how to measure the spin of electrons. Before moving to the crux of this lesson, let’s define another key concept in quantum mechanics, **correlation**.


# Correlated particles

In quantum mechanics, particles can combine with each other in a process that is mathematically represented as taking the tensor products of their state vectors. The correlation arises when these particles form a system in which they are so intrinsically intermixed that they cannot be thought of as independent particles anymore. This **correlation** is like the correlation you would have studied in a statistics course at college. Just like statistical correlation tells you that you can predict the value of a certain variable $Y$ with some confidence by knowing another variable $X$, correlation in quantum mechanics does the same. 

Consider two electrons that are correlated. If we were to measure one electron’s spin, we would automatically know another electron's spin, given the two electrons were highly correlated. Such electrons are called **entangled electrons**, and together they form an **entangled pair**. 


Entanglement in electrons is not limited to the discussion of spin alone. Continuing our study of qubits, entangled qubits would mean that by measuring the state of one qubit, we will automatically know the state of the second qubit. 

For illustration, let’s consider the two states $|\psi\rangle$ and $|\phi\rangle$ with definitions as follows:

$$|\psi\rangle=\frac{1}{2}|00\rangle+\frac{1}{2}|10\rangle+\frac{1}{2}|01\rangle+\frac{1}{2}|11\rangle$$ 

$$|\phi\rangle=\frac{1}{\sqrt{2}}|00\rangle+\frac{1}{\sqrt{2}}|11\rangle$$

Both states comprise two qubits in a superposition of states $|0\rangle$ and $|1\rangle$. Now, if we were to measure the state of one of these qubits, what would happen? 



The probabilities for measuring the state of the second qubit based on the measurement of the first for state $|\psi\rangle$ are described in the following table:

| | QUBIT 2: \|$0\rangle$| QUBIT 2: \|$1\rangle$     |
| :---:        |    :----:   |          :---: |
| QUBIT 1: \|$0\rangle$ | \|$\psi\rangle\rightarrow$\|$00\rangle:\frac{1}{4}$| \|$\psi\rangle\rightarrow$\|$01\rangle:\frac{1}{4}$   |
| QUBIT 1: \|$1\rangle$| \|$\psi\rangle\rightarrow$\|$10\rangle:\frac{1}{4}$| \|$\psi\rangle\rightarrow$\|$11\rangle:\frac{1}{4}$      |

This is the same as saying that the probability of measuring the first qubit in the state $|0\rangle$ is $\frac{1}{4}+\frac{1}{4}=0.5$. 

Now for something interesting. The probabilities for the second state $|\phi\rangle$ are as follows:

| | QUBIT 2: \|$0\rangle$| QUBIT 2: \|$1\rangle$     |
| :---:        |    :----:   |          :---: |
| QUBIT 1: \|$0\rangle$ | \|$\phi\rangle\rightarrow$\|$00\rangle:\frac{1}{2}$| \|$\phi\rangle\rightarrow$\|$01\rangle:0$   |
| QUBIT 1: \|$1\rangle$| \|$\phi\rangle\rightarrow$\|$10\rangle:0$| \|$\phi\rangle\rightarrow$\|$11\rangle:\frac{1}{2}$      |


Let’s learn how correlated quantum particles behave and how they are ‘inseparable’ at all distances.

Baby Steps

The Essence of Quantum Mechanics

Quantum Gates and Circuits

Simulating Quantum Computers

Getting Started with Quantum Algorithms

The Deutsch-Jozsa Algorithm

Grover's Search Algorithm

Shor's Factoring Algorithm

The Future

Quantum Entanglement