Quantum computing holds great promise for solving difficult problems that would take classical computers an infinitely long time. But working out the algorithms to solve these problems efficiently remains a major hurdle. According to a Report in the 24 Feb 2006 Science, help lies in the realm of geometry. In essence, a quantum computer designer wants to figure out the shortest path from the input data of a problem to its output solution without having the number of calculations grow out of hand along the way. Using that logic, Nielsen et al., showed that finding optimal quantum circuits is essentially equivalent to finding the shortest path between two points in a certain curved geometry — a geodesic, which also represents a path that a freely falling object would take.In making this analogy, the researchers open up the possibility of using the mathematical tools of Riemannian geometry (which involves the study of curved surfaces and spaces) to suggest new and efficient quantum algorithms or to reveal limitations of the power of quantum computers. An accompanying Perspective by J. Oppenheim (sciencemag.org) highlighted the study.

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