Abstract
We investigate geometric phases in terms of Heisenberg equation. We find that, equivalently to Schr\"odinger picture with a memory of its motion in terms of the geometric phase factor contained in the wave function, the observales carry with the geometric message under their evolutions in the Heisenberg picture. Such an intrinsic geometric feature may be particularly useful to implement the multi-time correlation geometric quantum gate in terms of the observables, which leads to a possible reduction in experimental errors as well as gate timing. An application is discussed for nuclear-magnetic-resonance system, where the geometric quantum gate is proposed.
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