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Unformatted text preview: A concise introduction to quantum probability, quantum mechanics, and quantum computation Greg Kuperberg UC Davis, visiting Cornell University (Dated: 2005) Quantum mechanics is one of the most interesting and surprising pillars of modern physics. Its basic precepts require only undergraduate or early grad- uate mathematics; but because quantum mechanics is surprising, it is more dicult than these prerequi- sites suggest. Moreover, the rigorous and clear rules of quantum mechanics are sometimes confused with the more dicult and less rigorous rules of quantum field theory. Many working mathematicians have an excellent intuitive grasp of two parent theories of quantum mechanics, namely classical mechanics and probabil- ity theory. The empirical interpretations of each of these theories above and beyond their mathemat- ical formalism have been a great source of ideas for mathematics proper. I believe that more mathe- maticians could and should learn quantum mechan- ics and borrow its interpretation for mathematical problems. Two subdisciplines of mathematics that have assimilated the precepts of quantum mechan- ics are mathematical physics and operator algebras. However, the prevailing intention of mathematical physics is the converse, to apply mathematics to problems in physics. The theory of operator algebras is closer to the spirit of this article; in this theory the precepts of quantum mechanics are sometimes called non-commutative probability. Recently quantum computation has entered as a new reason for both mathematicians and computer scientists to learn the precepts of quantum mechan- ics. Just as randomized algorithms can be moder- ately faster than deterministic algorithms for some computational problems (such as testing primality), some problems admit quantum algorithms that are faster (sometimes much faster) than their classical and randomized alternatives. These quantum algo- rithms can only run on a new kind of computer called a quantum computer. As of this writing, convincing quantum computers do not exist. Nonetheless, the- oretical results suggest that quantum computers are possible rather than impossible. Entirely apart from its potential as a technology, quantum computation is a beautiful subject that combines mathematics, * Electronic address: email@example.com@math. cornell.edu physics, and computer science. This article is a concise introduction to quantum probability theory, quantum mechanics, and quan- tum computation for the mathematically prepared reader. Chapters 2 and 3 depend on Section 1 but not on each other, so the reader who is interested in quantum computation can go directly from Chap- ter 1 to Chapter 3....
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- Spring '11