Quantum Computing from Scratch: Understanding States, Logic, and Quantum Algorithms

Introduction

Quantum computing may seem like an abstract and futuristic concept, but it is already shaping the present. While classical computers face limitations in solving highly complex problems, quantum computers promise a revolution by leveraging the laws of quantum mechanics to process information exponentially more efficiently.

But what exactly differentiates a quantum computer from a traditional one? Instead of using classical bits, which can be either 0 or 1, quantum computing works with qubits, which can exist in multiple states simultaneously due to quantum superposition. Additionally, the phenomenon of entanglement allows qubits to communicate instantaneously, unlocking new possibilities for ultra-fast calculations, digital security, and complex simulations.

Despite its revolutionary potential, this technology still faces significant challenges, such as quantum error correction, qubit stabilization, and scalability. However, tech giants like IBM, Google, and Intel are heavily investing in developing functional quantum computers, and the first signs of practical applications are emerging in fields like cryptography, molecular modeling, and artificial intelligence.

🔬💡 Quantum Computing: The Technology Revolution Has Begun! – Explore the fundamentals of quantum computing and discover how qubits, superposition, and entanglement are shaping the future of computers. 🚀

In this article, you will learn the fundamentals of quantum computing, including its states, logic, and key algorithms, as well as how this technology could impact the future. If you've never studied the subject before, don't worry—this guide is designed for beginners who want to understand this revolution from the basic principles to real-world applications. 🚀

1.0 What is Quantum Computing?

1.1 Fundamental Differences Between Classical and Quantum Computing

Classical computing and quantum computing are distinct computational paradigms. While classical computing relies on bits, which can only take the values 0 or 1, quantum computing operates with qubits, which can exist in multiple states simultaneously due to the principles of quantum mechanics.

📌 The Classical Model vs. The Quantum Model

Feature	Classical Computing	Quantic Computing
Information Unit	Bit (0 or 1)	Qubit (0, 1 or a superposition of both)
Processing	Sequential or parallel	Massively parallel (via superposition and entanglement)
Computing Logic	Based on traditional logic gates (AND, OR, NOT)	Based on Quantic Gates (Hadamard, CNOT, Toffoli)
States	One state at a time	Multiple states simultaneously
Examples	Spreadsheets, web browsing, conventional mathematical calculations	Molecular simulation, complex optimization, advanced cryptography

📌 Por Que Isso é Revolucionário?

The main advantage of quantum computing is its ability to perform calculations exponentially more efficiently than classical computers. This enables the solution of problems that would be infeasible even for the world's most powerful supercomputers.

✔ Example 1 – Prime Factorization and Cryptography

• RSA encryption, widely used to secure banking transactions and sensitive data, relies on the difficulty of factoring very large prime numbers.

• Shor’s Algorithm, when executed on a sufficiently powerful quantum computer, could break RSA encryption quickly, necessitating the development of new cryptographic systems.

✔ Example 2 – Molecular Simulation for Drug Discovery

• Modeling the behavior of complex molecules to develop new pharmaceuticals requires enormous computational power.

• Quantum computers can simulate chemical interactions with unparalleled precision, accelerating scientific discoveries in medicine and chemistry.

✔ Example 3 – Optimization in Logistics and Finance

• Problems such as vehicle routing, resource allocation, and complex financial calculations can be solved much faster using quantum algorithms, such as Grover’s Algorithm.

These are just a few examples of how quantum computing can revolutionize multiple industries. 🚀plos de como a computação quântica pode transformar diversas indústrias.

1.2 Why Do More Logical States Amplify Quantum Computing’s Computational Power?

To understand this in a simple way, let’s use the following analogy: imagine you need to find a lost key inside a massive building with 1,000 rooms.

How Does a Classical Computer Solve This Problem?

A classical computer works like a person who opens the doors one by one, checking each room until they find the key. If the key is in the last room, the search requires checking all 1,000 rooms.

• If each attempt takes 1 second, in the worst case, the computer would need 1,000 seconds to find the key.

• Even with parallel processing (multiple classical computers searching different rooms simultaneously), the process still requires a substantial amount of resources.

How Does a Quantum Computer Solve This Problem?

Now, imagine that instead of opening the doors one by one, you could enter all 1,000 rooms at the same time and instantly know where the key is.

This is possible because qubits can exist in multiple states simultaneously due to quantum superposition.

• A quantum computer with 10 qubits can represent 2¹⁰ = 1,024 states at the same time.

• A quantum computer with 20 qubits can represent 2²⁰ = 1,048,576 states simultaneously.

This means that instead of testing each possibility one by one, a quantum computer can process all possibilities at once and find the answer with significantly fewer operations.

Why Does This Amplify Computational Power?

While a classical bit can store only one value at a time (either 0 or 1), a qubit can be in a combination of 0 and 1 simultaneously.

This allows a relatively small number of qubits to process an enormous amount of information in parallel, making quantum computing exponentially more powerful than its classical counterpart for specific problems.

Classical Computing	Quantic Computing
Uses bits (0 ou 1)	Works with qubits (0 and 1 simultaneously)
Test one possibility at the time	Analyze multiple possibilities at the same time
Scale linearly	Scale exponentially

In practice, this means that a quantum computer can solve complex problems – such as chemical simulations, optimization and cryptography – in much less time than a classical computer ever could.🚀

1.3 The Quantum Mechanics Basis for Computing

Quantum computing didn’t just come out of nowhere – it’s based on principles of quantum mechanics, one of the most fascinating and counterintuitive fields of physics. To understand how quantum computers work, it’s essential to understand three key concepts: superposition, entanglement and interference.

📌 Superposition – The Magic of Qubits

In classical computing, a bit can be either a 0 or a 1. In quantum computing, a qubit can be both a 0 and a 1 at the same time thanks to superposition.

Classical State	Quantic State
0	|0⟩
1	|1⟩
Doesn't exist	α|0⟩+β|1⟩, such α e β represent probability values

🔹 Example: Imagine you are tossing a coin. Before it lands, it can either be heads or tails – a qubit works in a similar way. Until it is measured, it is in a state of superposition, carrying both possibilities simultaneously.

✅ Application of Superposition in Quantum Computing

• Quantum computers can explore multiple solutions simultaneously, which makes them ideal for solving optimization and search problems in large databases.

📌 Entanglement – Instant Connection Between Qubits

Entanglement is one of the most fascinating phenomena in quantum physics. When two qubits become entangled, the state of one instantly affects the state of the other, regardless of the distance between them.

🔹 Example: If two entangled qubits are measured, the result of one automatically determines the result of the other.

✅ Applications of Entanglement in Quantum Computing

• Quantum communication can leverage entanglement for ultra-secure information exchange (such as in quantum cryptography).

• Entanglement enables extremely fast operations, improving the efficiency of complex calculations.

  
  

📌 Interference – Controlling Probabilities

Quantum interference allows certain quantum states to reinforce or cancel each other out, increasing the probability of obtaining the correct answer in a computation.

🔹 Example: In Grover’s Algorithm, which is used to accelerate database searches, quantum interference helps eliminate incorrect solutions and reinforce the correct ones.

✅ Applications of Interference in Quantum Computing

• Enables the design of quantum algorithms that enhance desirable states and eliminate errors more precisely.

• Plays a crucial role in optimizing quantum calculations, leading to faster and more efficient problem-solving. 🚀

2.0 Estados Quânticos e a Representação Matemática

Quantum computing relies on qubits, which are fundamentally different from classical bits, enabling much greater computational power through properties such as superposition and entanglement. In this section, we will explore how qubits are represented mathematically and how they can be manipulated.

2.1 The Concept of a Qubit

While a classical bit can only assume the values ​​0 or 1, a qubit can exist in a superposition of these states, which drastically expands the space of computational possibilities.

📌 Mathematical Representation of a Qubit

A qubit can be expressed mathematically as a linear combination of the classical states ∣0⟩ and ∣1⟩:

∣ψ⟩=α∣0⟩+β∣1⟩

where:

• α e β are complex coefficients called probability amplitudes.

• ∣α∣^2+∣β∣^2 = 1 → The sum of the squares of the probabilities must be 1 (unitary norm).

• When the qubit is measured, it collapses to ∣0⟩ with probability ∣α∣^2 or to ∣1⟩ with probability ∣β∣^2.

📌 Visualizing the Qubit: The Bloch Sphere

To better understand a qubit, we use the Bloch Sphere, which represents its state in a three-dimensional space:

∣ψ⟩=cos(θ/2​)∣0⟩+(e^(iϕ)).sin(θ/2​)∣1⟩

• θ (theta) → Defines the position of the qubit on the sphere.

• ϕ (phi) → Represents the relative phase between the states ∣0⟩ and ∣1⟩.

🔹 Why does this matter?

• While a classical bit can only be at the poles of the sphere (representing either 0 or 1), a qubit can occupy any point on the surface of the Bloch Sphere, allowing for much more efficient calculations.

Classic Bit vs. Qubit: representation goes from one-dimensional to three-dimensional

🔹 Exemplo: Example: If a qubit is prepared in the ∣+⟩ state (which is a balanced superposition between 0 and 1), in the Bloch Sphere it will be located halfway between ∣0⟩ and ∣1⟩.

2.2 Operations and Measurements on Qubits

Qubits are manipulated by quantum gates, which play a similar role to logic gates in classical computing.

📌 Quantum Gates and Qubit Manipulation

Quantum gates change the states of qubits without destroying them, enabling quantum information processing. Follow some of the main ones:

Ports (column 1), Diagrammatic representation (column 2), Transformation matrix (column 3)

X Gate (Pauli-X Gate / Quantum NOT Gate)

The X gate, also known as the Pauli-X gate or the Quantum NOT gate, is the quantum equivalent of the classical NOT gate. It flips the state of a qubit, transforming ∣0⟩ into ∣1⟩ and vice versa.

Y Gate (Pauli-Y Gate)

The Y gate, also called the Pauli-Y gate, is a fundamental quantum gate that performs a rotation around the Y-axis of the Bloch Sphere. This transformation combines state inversion and phase rotation, resulting in a more complex quantum evolution.

Z Gate (Pauli-Z Gate)

The Z gate, or Pauli-Z gate, executes a rotation around the Z-axis of the Bloch Sphere. Unlike the X and Y gates, it does not swap ∣0⟩ and ∣1⟩, but instead introduces a phase shift to the ∣1⟩ state, which plays a crucial role in quantum interference.

Hadamard Gate (H Gate)

The Hadamard gate creates superposition by transforming a qubit into a state where it has an equal probability of being measured as ∣0⟩ or ∣1⟩. When applied to ∣0⟩, the qubit enters a state where there is 50% probability of collapsing into either 0 or 1 upon measurement.

CNOT Gate (Controlled-NOT Gate)

The CNOT gate, or Controlled-NOT gate, is one of the most essential two-qubit quantum gates. It operates conditionally:

• If the control qubit is in ∣0⟩, the target qubit remains unchanged.

• If the control qubit is in ∣1⟩, the target qubit is flipped (∣0⟩ becomes ∣1⟩, and vice versa).

• This gate is fundamental for generating entanglement, a key property of quantum computation.

SWAP Gate

The SWAP gate is a quantum operation that exchanges the states of two qubits. If one qubit is in state ∣α⟩ and the other in ∣β⟩, after applying the SWAP gate:

• The first qubit will be in ∣β⟩.

• The second qubit will be in ∣α⟩.

• This gate is crucial for rearranging quantum states within a quantum circuit.

Toffoli Gate (CCNOT – Controlled-Controlled-NOT Gate)

The Toffoli gate, or CCNOT gate, extends the logic of the CNOT gate by incorporating two control qubits and one target qubit. It functions as follows:

• The target qubit is flipped (∣0⟩ ↔ ∣1⟩) only if both control qubits are in state ∣1⟩.

• Otherwise, the target qubit remains unchanged.

• This gate is particularly significant because it enables the construction of reversible classical logic within a quantum framework, making it useful for quantum error correction and quantum arithmetic operations. 🚀

📌 The Measurement Process and the Collapse of the State

Measuring a qubit forces its superposition to collapse into one of two classical states, 0 or 1, according to the probabilities defined by ∣α∣^2 and ∣β∣^2.

If we have a qubit in the state ( ∣0⟩+∣1⟩ ) / sqrt(2)​ (perfect superposition), when we measure it, it collapses with a 50% chance to ∣0⟩ and a 50% chance to ∣1⟩. It is as if the result (reality) is not defined until it is measured. This differs from the common sense, that if we have 1+1 we know that a calculator will return 2 before we perform the operation. The value is already defined before we know it, but in quantum computing, there really is no 'expected value'.

📌 Quantum Coherence and Decoherence

In practice, quantum computers face challenges due to decoherence, which occurs when a quantum system interacts with the environment and loses its superposition.

• Quantum Coherence: When qubits remain in superposition and entanglement states, maintaining quantum processing capacity.

• Quantum Decoherence: When external noise causes the system to collapse prematurely, losing its quantum properties.

🔹 Solution?

Quantum Error Correction: Algorithms and physical methods to minimize decoherence and preserve quantum information for longer.

3.0 Applications

Quantum algorithms leverage principles such as superposition, interference, and entanglement to process information more efficiently than classical algorithms. In specific cases, like database searches and prime number factorization, quantum computing provides exponential acceleration in problem-solving.

3.1 Key Quantum Algorithms and Their Applications

✅ Grover’s Algorithm (Fast Search and Optimization)

• Solves unstructured search problems with quantum efficiency.

• While a classical algorithm needs to test N entries, Grover’s algorithm reduces this to √N, significantly speeding up searches.

• Applications: Database searching, logistics optimization, artificial intelligence.

✅ Shor’s Algorithm (Prime Factorization and Cryptography Breaking)

• Performs prime factorization exponentially faster than the best classical algorithms.

• Classical computers make breaking large RSA encryption keys impractical, while Shor’s algorithm can break them in feasible time given enough qubits.

• Applications: Cybersecurity, quantum cryptography, blockchain security.

  
  

✅ Molecular Simulations and Material Discovery

• Quantum computers can model highly complex chemical and physical systems that would be impossible for classical supercomputers.

• Enables prediction of chemical reactions, new material structures, and molecular interactions.

• Applications: Pharmaceutical research, drug discovery, superconductors, and advanced materials.

  
  

These algorithms demonstrate how quantum logic can solve previously intractable problems, paving the way for a new era of high-performance computing.

3.2 Entanglement and Its Importance

Quantum entanglement is one of the most intriguing phenomena of quantum mechanics and plays a fundamental role in quantum computing.

📌 What is Entanglement?

• When two qubits are entangled, measuring one instantly determines the state of the other, regardless of distance.

• This defies classical intuition and enables instantaneous correlations across vast distances.

🌌✨ Quantum Entanglement: The Mystery of Instant Connection – Discover how entangled qubits defy classical physics and pave the way for a new era in quantum computing and communication. 🚀

📌 Why Is Entanglement Important?

✅ Quantum Cryptography and Secure Communication

• Entanglement enables Quantum Key Distribution (QKD), ensuring 100% secure communications.

• If a third party attempts to intercept the key, the act of measurement itself alters the quantum state, making eavesdropping impossible.

✅ Parallel Processing and Distributed Quantum Computing

• Entanglement synchronizes qubits across different locations, enabling simultaneous operations at a scale far beyond traditional parallel computing.

• This capability is crucial for solving massive computational problems in seconds instead of thousands of years.

📌 Example:

If we create a pair of entangled qubits in the state: ( ∣00⟩+∣11⟩ ) / sqrt(2)

• Measuring the first qubit and obtaining 0 instantly collapses the second qubit to 0.

• If the first qubit is 1, the second qubit immediately becomes 1.

• This happens instantly, regardless of the distance between the qubits!

📌 Applications of Entanglement

• Quantum Communication Networks → Enables instantaneous quantum-secure communication, revolutionizing cybersecurity.

• Quantum Sensors and Radars → Ultra-sensitive detection systems capable of measuring subtle environmental changes with extreme precision.

• Large-Scale Information Transfer → Transmitting quantum states instead of classical data, eliminating the need for physical data transmission.

With these advancements, quantum entanglement is unlocking a new computational paradigm, redefining both security and computing power, and challenging the limits of traditional technology. 🚀

4.0 The Future of Quantum Computing

4.1 Quantum Computers Today – State of the Technology

Quantum computing is no longer a purely theoretical concept and is being actively developed by some of the world’s largest technology companies. Today, several approaches are being explored to build functional and scalable quantum computers.

📌 Main Companies and Technologies

✅ IBM → Develops superconducting quantum processors and offers access via IBM Quantum Experience.

✅ Google → Created Sycamore, a quantum processor that achieved quantum supremacy in 2019.

✅ D-Wave → Focused on adiabatic quantum computing, applicable to optimization problems.

✅ Intel → Works on silicon-based qubits, aiming at integration with existing semiconductor technology.

✅ Rigetti Computing → Competes with IBM and Google in the development of superconducting processors.

✅ PsiQuantum and Xanadu → Invest in photon-based quantum computing, a promising alternative.

📌 Different Quantum Hardware Technologies

🔹 Superconductors (IBM, Google, Rigetti) → Qubits formed by superconducting circuits operating at extremely low temperatures.

🔹 Trapped ions (IonQ, Honeywell) → Uses charged atoms manipulated by lasers.

🔹 Photons (PsiQuantum, Xanadu) → Uses light particles to process information in a scalable way.

🔹 Silicon qubits (Intel, UNSW Sydney) → Attempts to take advantage of technologies already used in the manufacture of conventional chips.

Advances in these different methods indicate that there is no single path to quantum computing — each approach has its advantages and challenges, and the future may involve a convergence of several of them.

4.2 Challenges to Making Quantum Computing Feasible

Quantum computing promises a technological revolution, but there are still major scientific and technological barriers that need to be overcome before its widespread adoption.

📌 Key Challenges in Quantum Computing

✅ Quantum Error Correction

• Qubits are highly sensitive to environmental disturbances and experience decoherence, losing their quantum properties rapidly.

• Quantum error correction techniques are being developed to ensure reliable calculations even with imperfect hardware.

✅ Scalability

• The largest quantum processors today contain dozens to a few hundred qubits, but millions of qubits are estimated to be necessary for advanced applications.

• Researchers are exploring ways to interconnect qubits stably without increasing error rates.

✅ Accessibility and Democratization

• Quantum computers require extremely expensive and complex infrastructure, including cryogenic cooling.

• Cloud-based quantum simulators (e.g., IBM Quantum Experience) already allow students, researchers, and businesses to experiment with quantum computing without needing dedicated hardware.

✅ Finding Real-World and Practical Applications

• Some fields have demonstrated quantum advantages, but many applications are still experimental.

• The sectors that stand to benefit the most from quantum computing include:

  • Cryptography and cybersecurity

  • Molecular modeling and drug discovery

  • Logistics and supply chain optimization

  • Artificial Intelligence and machine learning

Condor, IBM's most powerful quantum processor officially launched in 2023.

As quantum technology advances, researchers and industry leaders continue working to overcome these challenges and unlock the full potential of quantum computing. 🚀

4.3 How to Study and Delve Deeper into the Subject

If you want to learn more about quantum computing, there are several free and paid resources available to explore this universe.

📌 Recommended Online Courses

• IBM Quantum Computing for Developers (IBM) → Free course with access to a real simulator.

• Quantum Computing for the Determined (Michael Nielsen) → Free series on fundamental concepts.

• Quantum Mechanics and Quantum Computation (Coursera - UC Berkeley) → Introductory course on how quantum computing works.

• MIT Quantum Computing Curriculum → MIT resources on quantum algorithms and hardware.

📌 Recommended Books

📖 Quantum Computation and Quantum Information – Michael Nielsen e Isaac Chuang (Classic).

📖 Quantum Computing for Everyone – Chris Bernhardt (for Beginners).

📖 Dancing with Qubits – Robert Sutor (Practical introduction to quantum programming).

📌 Tools for Practicing Quantum Computing

• IBM Quantum Experience →Allows you to run real quantum algorithms in the cloud.

• Qiskit (IBM) → Open source library for programming on quantum computers.

• Cirq (Google) → Tool for simulating quantum algorithms.

• QuTiP → Software for simulating quantum mechanics.

Studying quantum computing may seem intimidating at first, but with the right resources, anyone can start exploring this groundbreaking field. The future of technology could be quantum, and those who start now will have a head start on the revolution to come. 🚀

Conclusion

Quantum computing is no longer just a distant theoretical concept. Companies like IBM, Google, and Intel are actively developing functional quantum hardware, and recent breakthroughs indicate that we are moving closer to practical applications that could revolutionize fields such as cybersecurity, artificial intelligence, scientific discoveries, and process optimization.

However, significant challenges remain. Quantum error correction, qubit scalability, and necessary infrastructure are barriers that must be overcome before quantum computing becomes widely accessible to businesses and developers. Today, most experiments occur in highly controlled environments, and the operational costs are still prohibitive for many everyday applications.

Even so, we are already seeing tangible signs of how this technology could profoundly impact society. Shor's and Grover's algorithms demonstrate computational acceleration far beyond any classical system. Quantum entanglement and superposition enable new computing architectures, and industries such as pharmaceuticals, finance, and logistics could be transformed by quantum modeling and optimization.

📌 What Can We Expect in the Coming Years?

🔹 Expanded Access via Cloud-Based Quantum Computing → Companies are already allowing users to experiment with quantum computing online, making learning and application development more accessible.

🔹 Advancements in Qubit Stability → Technologies such as quantum error correction and new materials for qubits promise greater reliability and scalability.

🔹 First Real-World Practical Applications → In the coming years, companies will begin applying quantum computing to solve problems that are currently infeasible for classical computers.

If you want to understand and explore the future of computing, now is the time to start. Knowledge of quantum logic, algorithms, and quantum hardware will become increasingly valuable. While quantum computing may take time to reach its full potential, being prepared now could position you at the forefront of this revolution. 🚀

Enjoyed this article? Share it on social media and continue to follow us to stay tuned on the latest in AI, breakthroughs and emerging technologies.

Thanks for your time!😉