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INTRO
In modern digital computers, these instructions boil down to the manipulation of information represented by distinct binary states. These bits can be represented abstractly by various physical phenomena, such as mechanical, optical, magnetic or electrical methods, and the process by which this binary information is manipulated is also versatile, with semiconductors being the most prolific medium for these machines. Basically, a binary computer moves individual bits of data through a handful of types of logic gates.

LIMITATIONS OF ALGORITHMS
In digital computing, binary information passes through a processing machine in discrete time steps. This is called the complexity of an algorithm. An example of such an algorithm would be one that determines whether a number is even or odd. These are known as linear temporal algorithms and they run at a speed directly correlated to the size of the algorithm's input.

This feature becomes evident in a basic addition algorithm. Since the number of steps, and inherently the execution time, is directly determined by the size of the number of inputs, the algorithm scales linearly over time. Constant-time and linear algorithms generally scale to practical execution times in common use cases. However, one category of algorithms in particular suffers from the characteristic of quickly becoming impractical as it grows. These are known as exponential-time algorithms and pose a huge problem for traditional computers because the execution time can quickly reach an impractical level as the input size increases.

QUBIT
Much like how digital systems use bits to express their fundamental unit of information, quantum computers use an analog called a qubit. In contrast, quantum computing is probabilistic. It is the manipulation of these probabilities as they move between qubits that forms the basis of quantum computing. Qubits are physically represented by quantum phenomena.

HOW QUANTUM TREATMENT WORKS
A qubit has an inherent phase component, and with this characteristic of a wave, the phase of a qubit can interfere constructively or destructively to change its probability amplitudes within an interaction.

BLOCH SPHERE
A Bloch sphere visualizes the magnitude and phase of a qubit using a vector in a sphere. In this representation, the two classical binary states are located at the upper and lower poles where probabilities become a certainty, while the remaining surface represents probabilistic quantum states, the equator being a pure qubit state where either Classic binary state is possible. When a measurement is made on a qubit, it decoheres towards one of the final polar state levels according to its probability magnitude.

PAULI DOORS
The Pauli gates rotate the vector, which represents the probability amplitude and phase of the qubit, by 180 degrees around the respective x, y and z axes of its Bloch sphere. For the X and Y gates, this effectively inverts the probability magnitude of the qubit while the Z gate only inverts its phase component.

HADAMARD DOORS
Some quantum gates have no classical digital analogues. The Hadamard gate, or H gate, is one of the most important unary quantum gates and exhibits this quantum uniqueness. Consider for example a qubit at state level 1. If a measurement is made between two H gates, the collapse of the superposition of the first H gate would destroy this information, making the effect of the second H gate only applicable to the reduced state of the measurement.

OTHER UNARY DOORS
Besides the Pauli gates and the Hadamard gate, two other fundamental gates, known as the S gate and the T gate, are common to most quantum computing models.

CONTROL DOORS
Control gates trigger a correlated change of a target qubit when a state condition of the control qubit is met. A CNOT gate causes a state flip of the target qubit, much like a NON-digital gate, when the control qubit is at state level 1. Since the control qubit is placed in superposition by the H gate, the correlation created by entanglement via the CNOT gate also places the target qubit in a superposition.

When the state of the control or target qubit is reduced by measurement, it is still guaranteed that the state of the other qubits is correlated by the CNOT operation. CNOT gates are used to create other composite control gates such as the CCNOT gate or the Toffoli gate which requires two control qubits at state 1 to invert the target qubit, the SWAP gate which swaps two qubit states and the CZ gates which carry out a phase. return. Combined with the fact that a qubit is continuous in nature and has infinite states, this allows for a scale of information processing that is rapidly surpassing traditional computing.

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