Geometric Algebra in Quantum Information Processing by Nuclear Magnetic Resonance

The relevance of information theoretic concepts to quantum mechanics has been apparent ever since it was realized that the Einstein-Podolsky-Rosen paradox does not violate special relativity because it cannot be used to transmit information faster than light [ 22 , 39 ]. Over the last few years, physicists have begun to systematically apply these concepts to quantum systems. This was initiated by the discovery, due to Benioff [ 3 ], Feynman [ 25 ] and Deutsch [ 17 ], that digital information processing and even universal computation can be performed by finite state quantum systems. Their work was originally motivated by the fact that as computers continue to grow smaller and faster, the day will come when they must be designed with quantum mechanics in mind (as Feynman put it, “there’s plenty of room at the bottom”). It has since been found, however, that quantum information processing can accomplish certain cryptographic, communication, and computational feats that are widely believed to be classically impossible [ 5 , 9 , 19 , 23 , 40 , 53 ], as shown for example by the polynomial-time quantum algorithm for integer factorization due to Shor [ 45 ]. As a result, the field has now been the subject of numerous popular accounts, including [ 1 , 11 , 37 , 60 ]. But despite these remarkable theoretical advances, one outstanding question remains: Can a fully programmable quantum computer actually be built?