What is a qubit?

In classical computers, any kind of data or information is represented in binary digits called bits. A bit value could be either $0$ or $1$. Unlike the bits in classical computers, data is represented as qubits in quantum computing. Quantum mechanics allows quantum computers to process large amounts of data simultaneously. This makes them exponentially more powerful than classical computers. Learning about qubits isn’t just about understanding a new technology; it’s about stepping into the future and being part of something that could improve the world.

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

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 $\ket0$ (pronounced as “ket 0”) and $\ket1$ (pronounced as “ket 1”).

Representation

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.

Abstract state

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 ($\psi$, pronounced as “psi”) is written as follows:

In the above equation, $\alpha$ and $\beta$ both are the probability amplitudes. They are complex numbers, which means that the $\alpha$ and $\beta$ can have both real and imaginary parts. These amplitudes relate to the likelihood that the qubit will be found in a particular state when measured.

According to the Born rule, the probability of measuring the qubit in the state $\ket0$ is given by the square of the magnitude of $\alpha$, denoted as $|\alpha|^2$. Similarly, the probability of measuring the qubit in the state $\ket1$ is given by the square of the magnitude of $\beta$, denoted as $|\beta|^2$probabilities. These probabilities must add up to $1$ because the qubit can only be in one of these two states when measured. Mathematically, this is expressed as:

This equation ensures that the total probability of all possible outcomes (in this case, the qubit being in the state $\ket{0}$ or $\ket{1}$) equals 1, which is consistent with our understanding of probability in general. This constraint is crucial in quantum mechanics, ensuring the description of the qubit’s state is physically meaningful.

Column vector

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 ($\psi$) is as follows:

This column vector form, $\begin{bmatrix} \alpha \\ \beta \end{bmatrix}$, is powerful because it allows us to apply quantum gates represented as matrices. By multiplying the vector representing the qubit’s state with a matrix representing a quantum gate, we can change the qubit’s state in a controlled and predictable manner.

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.

State space

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 $\theta$ and $\phi$. This visualization helps in understanding superposition and phase.

• 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 $\beta$.

The general state$\ket \psi = \alpha \ket0 + \beta \ket1$ can be parameterized using two real numbers, $\theta$ and $\phi$, where:

In this representation, the state $\ket\psi$ is visualized as a vector pointing from the origin to the surface of the sphere, with $\theta$ and $\phi$ corresponding to the polar and azimuthal angles, respectively. The north and south poles of the Bloch sphere correspond to the basis states $\ket0$ and $\ket1$, respectively.

Qubit types

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

Spin qubit

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 $\ket0$, and the downward direction is equivalent to $\ket1$.

Trapped atoms and ions

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 $\ket0$ state and the high-energy state as $\ket1$.

Photons qubit

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 qubit

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 $\ket0$ and $\ket1$ can be represented as horizontal ($H$) and vertical ($V$) polarization, respectively. They can also be represented as diagonal ($D$) and anti-diagonal ($A$) polarizations.

Time qubit

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 $\ket0$, and the photon that arrives late is mapped to the $\ket1$.

Path qubit

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 $\ket0$ and the bottom path can be considered as $\ket1$.

Superconducting circuits

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 $\ket0$ and anticlockwise current flow can be considered as $\ket1$.

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Conclusion

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.