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We introduce and examine quantum states of a special kind, referred to as E-states, whose properties are both structurally and functionally analogous to the sieve of Eratosthenes. More broadly, the concept of an E-state is related to a certain noncommutative extension of the Dirichlet ring, also discussed here for the first time. Furthermore, we demonstrate that E-states can be implemented on a universal quantum computer and, as a particular application, we construct an algorithm whichdoi:10.4172/2090-0902.1000180 fatcat:qpvbahhpc5h6tkbdd3wrnahjau