In Quantum Computing fundamental units of information processing are no longer classical bits (0 or 1) but quantum bits, or qubits. Qubits, unlike classical bits can exist in multiple states simultaneously due to the property of superposition (0 and 1 simulatenously), providing exponentially faster speed. Qubits in superposition means an object which is combination of multiple quantum states at the same time. Mathematically, an object in superposition can be represented by an equation that has more than one solution. Additionally in Quantum Computing, Qubits can become correlated or entangled, where pair of qubits exist in single quantum state irrespective of distance between them, allowing to perform tasks more efficiently. Like classical logic gates (AND, OR, NOT, etc) using bits, there are quantum gates with qubits (Hadamard gate, CNOT gate, etc), however they work differently then classical analogous due to use of superposition and entaglement. The behavior of quantum gates is described by unitary matrices (square matrices, with properties suitable for quantum computing), ensuring they are reversible and do not lose information. Different quantum gates have different outputs when applied to various qubits. By combining multiple gates in a defined specific order, quantum circuits can implement quantum algorithms to solve various computational problems.

In order to practically achive quantum computing, it needs new type of hardware and associated infrastructure. Currently, some of the main ways quantum computing has been studied with varying degree of success achieved, to have a commercial quantum computer include:

• Superconducting Qubits: Using electrical circuits that operate at extremely low temperatures (near absolute zero) and rely on quantization of charge and the Josephon effect to encode and manipulate quantum information. As flow of current through a superconductor is lossless (theoretically at least), so no energy is lost to the environment. This allows to build electrical circuits where current flows without dissipation or heating. A real quantum computing computing system may employ hybrid approach of combining superconducting qubits with some other types of qubits.
• Trapped Ion Qubits:These quantum bits relies on the manipulation of individual ions (charged atoms) that are trapped and controlled in electromagnetic fields. These qubits are known for their long coherence times, i.e. allow to maintain their quantum properties for relatively extended periods. Additionally, their individual control and high-fidelity gate operations make them a promising candidate for building practical quantum computers.
• Photonic Qubits: These are types of quantum qubits that use individual photons (particles of light) to encode and process quantum information. They operate based on the principles of quantum optics and find use in quantum communication and quantum computing. Photonic qubits have relatively long coherence times and the ability to travel long distances through optical fibers while preserving their quantum states.
• Quantum-Dot Qubits: They rely on the properties of quantum dots, which are nanoscale semiconductor structures, to encode and manipulate quantum information.
• Quantum Annealing: Unlike gate-based quantum computing, which uses quantum gates to perform operations, quantum annealers are designed to find the lowest energy state of a physical system that represents the solution to an optimization problem.

Development of practical quantum computer can impact many industories including finance, logistics, cryptography, optimization, material science and machine learning.

Some of the main quantum algorithms include:

Shors Algorithm

Shor's algorithm is most famous of all quantum computing algorithms, for factoring large numbers. Now, can't this not be done by using classical computations? yes, but not as efficiently. Algorithm is however not entirely quantum in nature but hybrid. It involve some classical pre-processing, then a quantum subroutine, then some classical post-processing. It is iterative and can take several rounds of the algorithm to find the desired answer.

Algorithm allows to find prime decomposition of very big numbers in O((logN)^3) time (classical part) and O(logN) space (quantum part), where N is the number to be factored. Classical algorithms on the other hand only offer exponential or sub-exponential time complexity. .

Shor's algorithm works by finding the quantum period finding step, where quantum parallelism and interference are harnessed to efficiently explore potential values of "r", where r represent the order or period of particular quantum operation. This quantum aspect of the algorithm explores multiple potential values of "r" simultaneously, leading to exponential speed up compared to classical algorithms for factoring large numbers. Once "r" is determined and postprocessing is complete, factors of "N" are calculated effectively breaking the composite number into prime components.

One of the best know encryption scheme, RSA is based on the assumption that factoring large numbers is a computationally infeasible task for classical computers, especially when the numbers are large enough (e.g., hundreds of digits). Shor's algorithm's ability to factor large numbers efficiently means that it can break RSA encryption by discovering the prime factors of "N." Once the prime factors are known, the private key from the public key can be calculated, effectively compromising the encryption. Internet security, which relies mainly on the RSA encryption scheme, can thus be compromised by Shor's algorithm.

Beyond crytography, Shor's algorithm has also far-reaching implications in the fields of mathematics, quantum computing and quantum information theory.

Grover's algorithm is designed to search an unsorted database of "N" items and find a specific item marked as the target in roughly the square root of "N" iterations. It uses quantum superposition and amplitude amplification to efficiently search through possibilities, providing a quadratic speedup compared to classical algorithms for searching an unsorted database or solving unnstructured search problems. Grover's algorithm can be used to speed up searches in unsorted databases, solve certain cryptographic problems, and improve search efficiency in data science and artificial intelligence.

This algorithm addresses the Deutsch Jose problem, which specifically involves determining whether a black-box function is constant or balanced. Black box function describe a function or process that can be used or interacted with without knowledge of its internal workings. For any inputs to the black box there are outputs, without necessarily knowing how the black box performs the computation or transformation. This quantum algorithm efficiently determines whether a given black-box function, which takes binary inputs and produces binary outputs, is constant (always 0 or always 1) or balanced (half 0s and half 1s) in just one query to the function. It demonstrates a quantum advantage over classical algorithms by providing a guaranteed answer with a single function evaluation.

This quantum algorithm addresses what is known as the Bernstein-Vazirani problem, which is a variation of the classic oracle problem in quantum computing. Suppose, given a black-box function (an oracle) that takes a binary string as input and produces a single binary output. The goal is to determine the hidden binary string that defines this function. In other words, it finds the binary string of 0s and 1s that, when combined with the hidden function (or black-box function), produces the output. It achieves this in a single query to the function by exploiting quantum parallelism and interference, providing a significant speedup over classical methods. This algorithm has primarily applications in cryptography and can be used to break classical cryptographic schemes, such as those based on secret keys represented as binary strings.

Simon's algorithm is a quantum algorithm designed to solve simon problem (oracle problem in quantum computing) i.e. finding the hidden period of a black-box function. Given an oracle that computes a function, f(x), where x is a binary string of length n and f(x) is another binary string of length n. It efficiently identifies this period using quantum superposition and interference, providing exponential speedup compared to classical algorithms for this problem.

Quantum Phase Estimation (QPE) algorithm is used to efficiently estimate the phase (angle) of a unitary operator's eigenvalues using quantum parallelism and the quantum Fourier transform. This algorithm is fundamental in quantum computing as it plays a crucial role in many quantum algorithms, particularly in solving problems related to factoring and cryptography.

TQuantum Fourier Transform (QFT) algorithm is used to efficiently transform a quantum state representing amplitudes into a new state in the frequency domain. QFT is fundamental in quantum computing and plays a pivotal role in various quantum algorithms, including Shor's algorithm. It uses quantum parallelism to compute the discrete Fourier transform much faster than classical algorithms, making it a core component in quantum computing for solving problems related to factoring and cryptography.

Quantum Random Walk (QRW) algorithm is to use quantum principles to simulate a random walk, a fundamental concept in mathematics and physics. Quantum random walks can be more complex and exhibit different behaviors compared to classical random walks. QRW algorithms leverage quantum superposition and interference to explore multiple paths simultaneously, potentially providing speedup in solving certain problems like search and optimization. They have applications in quantum algorithms, quantum simulations, and quantum machine learning.

Quantum Error Correction Codes (QECC) algorithms is to protect quantum information from errors caused by the inherent noise and imperfections in quantum hardware. These algorithms use specially designed qubit arrangements and error-correcting techniques to detect and correct errors that can occur during quantum computation, ensuring the reliability of quantum computations.

Quantum Error Correction (QEC) algorithms is used to protect quantum information from errors that can occur due to decoherence and other sources of noise in quantum systems. QEC algorithms use redundancy, encoding quantum information in a way that allows for the detection and correction of errors without directly measuring the quantum states. This enables the creation of fault-tolerant quantum computers, where quantum information can be reliably processed and stored over extended periods, advancing the practicality of quantum computing.