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We explore the possible use of particle beam storage rings as quantum computers. More precisely, we consider creating an ion trap system, in which the same computational basis states can be defined as in a modern ion trap system, but in which the ions have a constant velocity and are rotating in a circular trap. The basic structures that we explore are classical and ultracold crystalline beams. What we propose is a novel method that uses the ion trap quantum computer concept, but puts the ions into a rotating frame of reference. The benefits of this approach are discussed.