Geometric algebra and transition‐selective implementations of the controlled‐NOT gate
Timothy F. Havel, Shyamal S. Somaroo, Wurong Zhang, David G. Cory · Concepts in Magnetic Resonance Part A · 2004
Abstract Geometric algebra provides a complete set of simple rules for the manipulation of product operator expressions at a symbolic level, without any explicit use of matrices. This approach can be used not only to describe the state and evolution of a spin system, but also to derive the effective Hamiltonian and associated propagator in full generality. In this article, we illustrate the use of geometric algebra via a detailed analysis of transition‐selective implementations of the controlled‐NOT gate, which plays a key role in NMR‐based quantum information processing. In the appendices, we show how one can also use geometric algebra to derive tight bounds on the magnitudes of the errors associated with these implementations of the controlled‐NOT. © 2004 Wiley Periodicals, Inc. Concepts Magn Reson Part A 23A: 49–62, 2004