A quantum computer is a machine that exploits quantum phenomena to store information and perform computations.
The chief goal of this article is to provide a brief but comprehensive introduction to quantum computing. It overviews some mathematical underpinnings of quantum computation for readers with only a basic knowledge of linear algebra and probability. However, it does not attempt to be exhaustive nor make an exposition of quantum physics. It further does not attempt to stay current with physical implementations of quantum computers.
After providing a brief historical background, the article introduces the notion of a quantum bit (qubit) and the linear operators (gates) that act on these. It then addresses systems of multiple qubits and their corresponding gates. Along the way, the article covers the essential concepts of separable systems, as well quantum interference and decoherence. It also describes how to represent gates as quantum circuits. To conclude, it explains the underpinnings of two simple but insightful quantum algorithms. A final section suggests further readings for those who wish to delve deeper into quantum computing.
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Pieper, Jaden K. and Lladser, Manuel E., "Quantum Computation" (2018). Applied Mathematics Faculty Contributions. 25.