
In honor of the International Year of Quantum Technology, I plan to write as many articles as possible about different aspects of the quantum field. However, to discuss deeper and more technical topics, I first need to explain the basics well enough so everyone can catch up later.
Quantum computers are systems that utilize the powers of quantum Physics and mechanics to execute computations. In order for me (or anyone, really) to explain how quantum computers can be better at solving some problems or how they would solve these problems, we first need to discuss how they work.
Quantum computers are powerful because they can utilize interference, superposition, and entanglement. If you already know about these concepts, great; if not, keep an eye out for my next article, which will dive deep into them.
Let’s take a quick detour first and discuss why ML, AI, and software engineers may benefit from learning some Quantum Computing basics. For ML and AI engineers, Quantum Machine learning is a topic at the intersection of quantum mechanics and machine learning. Because of superposition, quantum can "enhance" ML models by allowing them to address different environmental inputs more efficiently. As for software engineers, as we approach larger, more capable quantum computers, we will need software engineers to develop applications for those computers. The good news is you don’t really need to know all the low-level quantum physics and mechanics to develop this application; you only need a high-level understanding of how things work! That is what I am here to do!
So, today, I will discuss how quantum computers use these phenomena, using qubits as the fundamental building block for computations. The article’s title already gives away what we will discuss, but let me expand on what I will cover here.
Qubits are very important, and we can discuss them in various ways. In this article, we will discuss physical qubits (how qubits are "made"), how we address them mathematically, and different approaches to visualizing them.
Without any further ado, let’s jump right in.
A qubit, short for "quantum bit," is the fundamental unit of quantum information and the basic building block of quantum computers. You can think of qubits as the quantum analog of a classical bit.
Section 1: Qubits Physically
Theoretically speaking, any quantum system that has two distinguishable states can be used as a qubit. For example, if I have an electron, that electron will have different energy levels. Then, I can use two energy levels to describe a qubit. So, the ground can be used to define the 0 state of the qubit, while the first excited energy level can be used to define state 1.
Another example is the polarization of a photon, where we can use the horizontal polarization as state 0 and the vertical polarization as state 0.
Following that, scientists use various approaches to construct physical qubits with different accuracy and properties. Let me introduce you to 6 types of qubits used in research and industry today.
- Superconducting qubits are based on superconducting circuits, where electrical current can flow without resistance. The direction of the current in these tiny circuits indicates whether the qubit is in state 0 or 1. We use electromagnetic fields to control and manipulate these circuits to perform computations. Companies like IBM, Google, and Rigetti Computing have built quantum processors using superconducting qubits.
- Trapped ion qubits: In this type of qubits, individual ions are trapped using electromagnetic fields and manipulated using lasers or microwave radiation. The internal energy levels of the ions are used to represent the qubit states. Companies such as IonQ and Quantinuum are developing trapped ion quantum computers.
- Topological qubits: To create topological qubits, rely on anyon particles, which have more freedom to acquire any phase when they exist in two-dimensional systems. In other words, when two anyons interact somehow, the interaction gives them special topological properties that we use to construct qubits. Microsoft is one of the companies researching this type of qubits.
- Photonic qubits: These qubits utilize the quantum properties of photons, such as their polarization or path, to encode quantum information. Photonic qubits can be manipulated using optical elements, like beam splitters, and we measure them using single-photon detectors. Xanadu and PsiQuantum are companies developing photonic quantum computing platforms.
- Quantum dots are semiconductor nanostructures that can trap individual electrons. The electron spin is then used to represent the 0 and 1 states. Quantum dots can be manipulated using electric and magnetic fields.
- Spin qubits: These are based on atoms’ nuclear or spin states, often in solid-state systems like diamond or silicon. Researchers can control and read out the spin states by manipulating the magnetic environment and applying microwave pulses.
The reason there is more than one way (and counting) to construct qubits is because each one of these approaches has its pros and cons. Some are more resilient to error, some allow for longer decoherence time (which allows for longer computation times), and some are easier to produce or require less maintenance. That is why I can’t say which one of those is the best approach or will be the future of quantum; it is also the case that we don’t know what new approaches to constructing qubits researchers are currently working on.
Though all these techniques are used to construct qubits physically, the resultant qubits are what we call "physical qubits." However, we talk about algorithms, and when we say qubits, we often mean "logical qubits."
So, what is the difference between those two?
Physical qubits are implemented using a specific physical system using one of the approaches above. Physical qubits store and manipulate quantum information and are subject to noise and errors. The performance and reliability of a quantum computer depend on the quality of the physical qubits, such as their coherence time, gate fidelity, and error rates.
Logical qubits are an abstract representation of qubits in fault-tolerant quantum computing. They are constructed by encoding the quantum information across multiple physical qubits using quantum error correction (QEC). By distributing the information redundantly, QEC protects the quantum information from errors and noise, which allows us to detect and correct the errors in the qubits without measuring them.
The ratio of physical qubits to logical qubits often varies depending on the physical qubit approach, the specific QEC code, and the selected error threshold. Generally speaking, the more physical qubits used to encode a single logical qubit, the more robust the quantum computer will be against errors.
Section 2: Qubits Mathematically
When discussing algorithms or applications of quantum computing, we don’t often address how qubits are made. We usually use a punch of math equations to explain an algorithm. So, it is important to know how we can address qubits mathematically.
Mathematically, qubits are represented as vectors, and we use the Dirac notation or the bra-ket to address them. This notation was introduced by the physicist Paul Dirac to differentiate the state of a qubit from that of the classical binary 0 or 1. So, when speaking of qubits, we say the qubit state is either |0⟩ or |1⟩ (and we read as ket 0 and ket 1, respectively).
Ket vectors are column vectors in a complex vector space. So:
1- |0⟩ represents the quantum state corresponding to the classical bit 0 as a column vector.

2- |1⟩ represents the quantum state corresponding to the classical bit 1 as a column vector.

3- |ψ⟩ represents an arbitrary quantum state as a column vector

where α and β are complex numbers that must satisfy the condition:

Bra vectors are the complex conjugate transposes of ket vectors and are row vectors. The bras for the jets we just mentioned are:
1- ⟨0| represents the bra vector corresponding to the quantum state |0⟩ and can be represented as a row vector.

2- ⟨1| represents the bra vector corresponding to the quantum state |1⟩ and can be represented as a row vector.

3- ⟨ψ| represents the bra vector corresponding to the quantum state |ψ⟩ and can be represented as a row vector:

where α^ and β^ are the complex conjugates of α and β.
We can also use the bra-ket notation to represent qubits’ inner and outer products.
The result of the inner product of |0⟩ and |1⟩ is zero because they are orthogonal states. In quantum computing, we refer to them as the computational basis.
We use the bra-ket notation to describe a qubit’s arbitrary state as a linear combination of the computational basis.

Here, α and β are complex numbers determining the probability of the qubit being in state 0 or 1.
The probabilities of measuring the qubit in either state are given by the squared magnitudes of the coefficients, i.e., |α|² and |β|². The sum of these probabilities is always 1, ensuring that the qubit will be either 0 or 1 when measured.
Section 3: Qubits Graphically
To better understand qubits, we needed to create a visual way to do so. In this article, we will discuss two approaches to visualizing qubits.
1- Bloch Sphere
The first one we will talk about is Bloch spheres. A Bloch sphere is a geometrical representation of the state of a single qubit in quantum mechanics. It is a three-dimensional sphere with a radius of 1 on which the states of a qubit can be visualized as points on the sphere’s surface.
As we just discussed, a qubit can be represented as a combination of the states 0 and 1.
A Bloch sphere represents a qubit’s state using coordinates (θ, ψ) as follows:

Here, θ ranges from 0 to π, and ϕ ranges from 0 to 2π. The north pole of the Bloch sphere corresponds to the |0⟩ state, and the south pole corresponds to the |1⟩ state. The superposition states lie on the surface between these poles, with the angles θ and ϕ determining the specific state.

Though Bloch spheres help visualize the state of one qubit, they fail to visualize multi-qubit systems. This is a problem because we need to visualize multi-qubit systems to understand practical quantum systems. To overcome this issue, IBM introduced a different version of the Bloch sphere called the Q-sphere.
Q-Sphere

The Q-sphere is an approach used to represent the state of a quantum system with one or more qubits. Each node’s radius is proportional to the probability of its basis state, while its color indicates its phase.
2- The Dimensional Circle Notation (DCN)
The second approach we will discuss is the circle notation. This visualization technique simplifies understanding complex quantum states by depicting them more intuitively.
The dimensional circle notation (DCN) graphically represents quantum states using circles to depict complex numbers. These circles visualize the complex numbers that describe the quantum state’s amplitude and phase. The magnitude of the amplitude is shown as the filled area within a circle. At the same time, the phase is indicated by the angle of a radial line within the circle relative to a vertical line.

While the circle notation makes the visualization of quantum states more accessible, it still faces some limitations. Some aspects of quantum algorithms are not always intuitive and may require additional effort to understand and visualize. Another limitation of the circle notation is the ability to visualize the state of a large number of qubits effectively.
Final Thoughts
Before we get into the deep and mysterious technical aspects of quantum computing, I wanted to spotlight the field’s fundamental building blocks. By far, the core building block of quantum computing is qubits.
Qubits are essential, allowing us to harvest the superpowers of quantum mechanics, entanglement, and superposition. Without resilient, high-quality qubits, we will not be able to build more practical quantum computers capable of running life-size applications. So, it is important to know where we are when it comes to academia and industry constructing and using physical qubits.
In this article, I went through everything you need to know about qubits, from how they are constructed to how we address them mathematically and visually.
Hopefully, now you have a better understanding of qubits and what scientists mean when they use the word in different contexts. In future articles, we will dive deeper into more quantum topics, so stay tuned!