Qubit

What is Qubit?

In classical computing, the bit is the fundamental unit of information and the basic building block of digital data. A bit can represent one of two states, typically denoted as 0 or 1. Electronic devices, such as transistors in computer circuits, are well-suited to this binary system, as they can easily build between two states-typically represented by high and low voltage. This binary nature makes bits a natural choice for encoding information in digital systems. The manipulation of bits underpins all digital computation and information processing in classical computers

In quantum computing, the quantum bit, also called qubit, is the fundamental unit of quantum information. Unlike classical bits which can only exist in a definite state of either 0 or 1, a qubit has the unique ability to exist in a superposition of both 0 and 1 simultaneously. This property can perform more complex computation than classical bits. However, When measurements happen on qubits, the superposition collapses, and the qubit assumes a definite state of either 0 or 1. The outcome is probabilistic, introducing an element of uncertainty. The probabilities assigned during the superposition influence the likelihood of the qubit collapsing into a particular state.

What is Quantum State?

In quantum information theory, a quantum state is denoted as \(|\psi>\). The quantum state corresponding to 0 is represented as |0>, and the quantum state corresponding to 1 is denoted as |1>. Unlike a classical bit which is represented by either 0 or 1, a qubit can exist in both quantum state |0> and |1>, requiring a two-dimensional vector for its representation.

How to represent a quantum state?

In this representation, the first number of the vector represents the amplitude for the state |0>, and the other represents the amplitude for the state |1>. The computational basis states, which are the most common way to represent a qubit, are given by:

\[|0> = \begin{pmatrix} 1 \\ 0 \end{pmatrix} , |1> = \begin{pmatrix} 0\\ 1 \end{pmatrix}\]

However, a qubit can also be represented using other orthogonal vector, depending on the context. For example, another representation might involve:

\[|0> = \begin{pmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{pmatrix}, |1> = \begin{pmatrix} \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \end{pmatrix}\]

This representation is more complex than the first one but is sometimes useful in specific situations. A qubit can be described as a line superposition of two elementary orthogonal states |0> and |1>, capturing all ranges of quantum states. For example, an arbitrary quantum states is given by:

\[ |\psi> = \alpha|0> + \beta|1> = \begin{pmatrix} \alpha \\ \beta \end{pmatrix} \]

The two coordinates, \(\alpha\) and \(\beta\), are known as qubit amplitudes. These amplitudes determine the probability of the qubit collapsing into a particular quantum state when measurement. Although the outcome of a measurement is not deterministic, but the total probability across all quantum states is certain to equal 1.

When a qubit is measured, the probability of it yielding the quantum state |0> is given by \(|\alpha|^{2}\), and the probability of yielding the quantum state |1> is given by \(|\beta|^{2}\). These probabilities reflect the likelihood of each outcome, with the sum of \(|\alpha|^{2}\) and \(|\beta|^{2}\) being equal to 1, ensuring that all possible outcomes are account for

\[|\alpha|^{2} + |\beta|^{2} = 1\]

In a multi-qubit system, the state of the entire system is described by the combined quantum states of each individual qubit. The state of an n-qubit system is represented as a linear combination of basis states to all possible outcomes of qubits. This system's quantum state can be described by a \(2^n\) dimensional vector, where n is the number of qubits. Each element of this vector corresponds to a possible basis state of the system, with the entire state being a superposition of these basis states. For example, an arbitrary quantum state with an n-qubit system is given by:

\[ |\psi> = \begin{pmatrix} \alpha_{0} \\ \alpha_{1} \\ \vdots \\ \alpha_{2^n - 1} \\ \end{pmatrix} \]

Here, \(\alpha_{0}\) represents the amplitude of the quantum state \(|000....000>\)(the binary representation with n bit of the number 0), \(\alpha_{2^n-1}\) represents the amplitude of the quantum state \(|111....111>\)(the binary representation with n bit of the number \(2^n-1\))

In the case of a 2-qubit system, the basis state |00>, |01>, |10>, and |11> can be denoted as a 4-dimensional vector:

These vectors form the basis for the 2-qubit system, and a quantum state of this system can be described as a linear combination of these basis vectors. For example, a qubit in a 2-qubit system can be expressed in a superposition of four basis states:

Here, \(\alpha_{00}, \alpha_{01}, \alpha_{10}\), and \(\alpha_{11}\) are quantum amplitudes of |00>, |01>, |10>, and |11>, The amplitudes of each quantum state satisfy the following equation. The probabilities of all quantum states are added to be equal to 1.

\[|\alpha_{00}|^{2} + |\alpha_{01}|^{2} + |\alpha_{10}|^{2} + |\alpha_{11}|^{2} = 1\]