Quantum Computing and Entanglement for Mathematicians

Joseph M. Landsberg

F. J. Duarte

W. S. Curran

As strange as it may seem, developments in quantum physics and computer science promise to have important consequences for mathematicians, and for mathematics in general. The idea of what counts as a proof is at issue, as is the future role of human mathematicians. Additionally, some famous intractable problems may be solvable, based on the weirdness of quantum particles.

F.J. Duarte

Benoît Vermersch

Douglas J. Matzke, Cyrus D. Cantrell, Michael J. Manthey

Quantum computing concepts are described using geometric algebra, without using complex numbers or matrices. This novel approach was developed in the first author’s Ph.D. dissertation in Electrical Engineering at University of Texas at Dallas (May 2002) and enables the expression of the principle ideas of quantum computation without requiring an advanced degree in mathematics or physics. Using a topologically derived algebraic notation that relies only on addition and the anticommutative geometric product, this paper describes the following quantum computing concepts: bits, vectors, states, orthogonality, qubits, classical states, superposition states, spinor, reversibility, unitary operator, singular, entanglement, ebits, separability, information erasure, destructive interference and measurement. These central quantum concepts can be described simply in geometric algebra, thereby facilitating the understanding of quantum computing concepts by non-physicists and non-mathematicians.