In classical computers, any kind of data or information is represented in binary digits called **bits**. A bit value could be either

In this Answer, we’ll discuss the basics of a qubit.

A **qubit** (also called a quantum bit) is a quantum system’s smallest unit of data. It is a quantum analog of a classical bit. A qubit has two basis states that are usually written in Dirac notation as

A qubit can be represented in multiple ways, each providing unique insights and a different aspect of quantum computing. Let’s discuss some popular representations of a qubit.

In the **abstract state**, a qubit is described using the Dirac notation, also called **bra-ket notation**. It is a standard way of representing a quantum state. A qubit can be either a basis state or a superposition state where that state could be any linear combination of both basis states. Mathematically, a generic qubit state (

In the above equation, **complex numbers**, which means that the

According to the Born rule, the probability of measuring the qubit in the state

This equation ensures that the total probability of all possible outcomes (in this case, the qubit being in the state

In **column vector representation**, the state of a qubit is expressed as a two-dimensional complex vector. This is useful for performing matrix operations. The vector representation of the basis states is as follows:

The vector representation of the generic qubit state (

This column vector form,

For example, to apply a quantum gate to the qubit, we’ll perform a matrix-vector multiplication, which transforms the qubit’s state according to the rules defined by that gate. This is how quantum computations are carried out, making vector representation a crucial tool in quantum computing.

In quantum mechanics, a qubit’s state space, also known as its **Hilbert space**, consists of all possible quantum states, including pure and mixed ones. The Bloch sphere is a geometric representation used to visualize a qubit’s pure states within this state space.

On the Bloch sphere, each pure state corresponds to a point on the surface of the sphere, defined by angles

The angle

$\theta$ is the polar angle from the$z$ -axis of the Bloch sphere. It determines the relative weight of the basis states$\ket{0}$ and$\ket{1}$ in the quantum state. Specifically,$\theta$ controls the probability amplitudes$\alpha$ and$\beta$ in the quantum state representation.The angle

$\phi$ is the azimuthal angle around the$z$ -axis of the Bloch sphere. It represents the relative phase between the basic states$\ket{0}$ and$\ket{1}$ . The angle$\phi$ affects how the quantum state evolves and is related to the complex phase factor of the amplitude$\beta$ .

It’s important to note that the Bloch sphere does not represent the entire state space (Hilbert space), which also includes mixed states. It only represents the pure states geometrically.

The general state

$\alpha = \cos (\frac{\theta}{2}) \\ \beta = e^{\dot\iota \phi}\sin (\frac{\theta}{2})$

In this representation, the state

There are different types of qubits. Some occur naturally, and others are engineered. Let’s discuss some common types of qubits.

The **spin** qubit uses the angular momentum of particles such as electrons or nuclei. These qubits utilize the quantum property that spin can be in either the up or down direction. A spin qubit can be formed using these up and down directions, where the upward direction represents

Trapped atoms or ions are isolated using electromagnetic fields. The energy level of an electron can be used to represent the qubit state. In the natural state, the electrons hold their lowest possible energy levels. We can manipulate electrons using lasers or microwaves; e.g., lasers can excite the electron to a high energy level. We can map the low-energy state as

**Photons** are light particles that can be used to represent a qubit. We can represent the qubit state by using light properties such as polarization, path, time, or arrival. Let’s discuss all types of photon qubits.

**Polarization** is the electromagnetic field with a specific direction that a photon carries. In a polarization qubit, the polarization state represents the quantum state. The quantum states

**Time qubit** uses the time of a photon’s arrival to represent the quantum state. We can create the time qubit by splitting a photon using delay lines and beam splitters. The photon that arrives early is mapped to the

**Path qubit** uses different paths to represent the quantum state. We can use the beam splitters to put a photon in a superposition state. The top path can be represented as

**Superconductors** allow to pass the electric current without any resistance at a lower temperature. An electrical circuit of superconductors can be designed to behave like qubits. The current flow in this circuit can represent the quantum states. For example, clockwise current flow can be assumed as

Assess your understanding by attempting the following quiz.

1

What is a qubit?

A)

A classical bit used in digital computing

B)

The smallest unit of data in a quantum system

C)

A type of classical memory unit

D)

A quantum version of a digital register

Question 1 of 30 attempted

A qubit is the smallest unit of data in quantum computing, similar to a bit in classical computing but much more powerful. Unlike bits, which can be 0 or 1, qubits can be in both states simultaneously due to superposition. This makes qubits very useful for complex calculations. They can be represented in different ways, such as using math notation, vectors, or visual models like the Bloch sphere. There are also different types of qubits, including spin qubits, trapped ions, photons, and superconducting circuits, each using unique physical properties to store and process information. Understanding qubits is key to unlocking the potential of quantum computing.

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