Quantum Mechanics in Quantum Computing
Mathew Johnson is a Ball State junior majoring in Mathematics (Option 1) with a minor in Physics. In his sophomore year, he participated in the student- faculty colloquium, where he explored quantum com- puting with several other students and faculty. Quantum mechanics is a scientific theory that seeks to describe atomic and subatomic particles (or quantum particles) as well as the interaction among them. Such particles include electrons, protons, neutrons, and photons. In classical mechanics, a particle's future state is described with certainty once its present state is known. At any given moment, its momentum and posi- tion are determined with certainty. In quantum mechanics, its future state is described probabilistically. Instead of a single momentum, a particle can assume a simultaneous range of momenta, and instead of a single position, it can take a simultaneous range of positions. Since the product of these ranges is no less than a specific positive universal constant, a decrease in the range of its position forces an increase in the range of momentum and vice versa. It is precisely this eect of quantum particles that quantum computing seeks to exploit in order to manipulate a huge amount of information simultaneously. Quantum computing is a mathematical theory currently being developed to give a theoretical basis for building a "quantum computer," which is envi- sioned to be faster than any classical computer to date. Quantum computing is based on the principles of quantum mechanics. In this article we will look at the postulates of quantum mechanics upon which quantum computing is based. A physical system consisting of one or more quantum particles will be called a quantum system. A quantum system will be called isolated if it does not interact with other quantum systems. At any given instant, a quantum system will be in a certain "state." The first postulate deals with a way of representing states of a quantum system: Postulate 1. Associated with any isolated quantum system is a complex Hilbert space. A "state" in the system is represented by a unit vector in this vector space.