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

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

This characteristic becomes obvious within a basic addition algorithm. Because the number of steps, and inherently the execution time is directly determined by the size of the number inputs, the algorithm scales linearly in time. Constant and linear time algorithms generally scale to practical execution times in common use cases, however, one category of algorithm in particular suffers from the characteristic of quickly becoming impractical as it grows. These are known as an exponential time algorithm and they pose a huge problem for traditional computers as the execution time can quickly grow to an impractical level as 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. Quantum computing by contrast, is probabilistic. It is the manipulation of these probabilities as they move between qubits that form the basis quantum computing. Qubits are physically represented by quantum phenomena.

HOW QUANTUM PROCESSING WORKS
A qubit possesses an inherent phase component, and with this characteristic of a wave, a qubit’s phase can interfere either constructively or destructively to modify its probability magnitudes within an interaction.

BLOCH SPHERE
A Bloch sphere visualizes a qubit’s magnitude and phase using a vector within a sphere. In this representation, the two, classical bit states are located at the top and bottom poles where the probabilities become a certainty, while the remaining surface represent probabilistic quantum states, with the equator being a pure qubit state where either classical bit state is possible. When a measurement is made on a qubit, it decoheres to one of the polar definitive state levels based on its probability magnitude.

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

HADAMARD GATES
Some quantum gates have no classic digital analogs. The Hadamard gate, or H gate is one of the most important unary quantum gates, and it exhibits this quantum uniqueness. Take a qubit at state level 1 for example. If a measurement is made in between two H gates, the collapsing of the first H gate’s superposition would destroy this information, making the second H gate’s effect only applicable to the collapsed state of the measurement.

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

CONTROL GATES
Control gates trigger a correlated change to 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 digital NOT gate, when the control qubit is at state level of 1. Because the control qubit is placed in a superposition by the H gate, the correlation created by entanglement through the CNOT gate, also places the target qubit into a superposition.

When the control or target qubit state is collapsed by measurement the other qubits' state is always guaranteed to be correlated by the CNOT operation. CNOT gates are used to create other composite control gates such as the CCNOT gate or Toffoli gate which requires two control qubits at a 1 state to invert the target qubit, the SWAP gate which swaps two qubit states, and the CZ gates which performs a phase flip. When combined with the fact that a qubit is continuous by nature and has infinite states, this quickly scales up to a magnitude of information processing that rapidly surpasses traditional computing.

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