▁required ▁by ▁classical ▁algorithms . ▁In ▁this ▁case , ▁the ▁advantage ▁is ▁not ▁only ▁prov able ▁but ▁also ▁optimal , ▁it ▁has ▁been ▁shown ▁that ▁Gro ver ' s ▁algorithm ▁gives ▁the ▁maximal ▁possible ▁probability ▁of ▁finding ▁the ▁desired ▁element ▁for ▁any ▁number ▁of ▁oracle ▁look ups . ▁Several ▁other ▁examples ▁of ▁prov able ▁quantum ▁spe edu ps ▁for ▁query ▁problems ▁have ▁subsequently ▁been ▁discovered , ▁such ▁as ▁for ▁finding ▁coll isions ▁in ▁two - to - one ▁functions ▁and ▁evalu ating ▁N AND ▁trees . ▁ ▁Problem s ▁that ▁can ▁be ▁addressed ▁with ▁Gro ver ' s ▁algorithm ▁have ▁the ▁following ▁properties : ▁There ▁is ▁no ▁search able ▁structure ▁in ▁the ▁collection ▁of ▁possible ▁answers , ▁The ▁number ▁of ▁possible ▁answers ▁to ▁check ▁is ▁the ▁same ▁as ▁the ▁number ▁of ▁inputs ▁to ▁the ▁algorithm , ▁and ▁There ▁exists ▁a ▁boolean ▁function ▁which ▁evalu ates ▁each ▁input ▁and ▁determ ines ▁whether ▁it ▁is ▁the ▁correct ▁answer ▁ ▁For ▁problems ▁with ▁all ▁these ▁properties , ▁the ▁running ▁time ▁of ▁Gro ver ' s ▁algorithm ▁on ▁a ▁quantum ▁computer ▁will ▁scale ▁as ▁the ▁square ▁root ▁of ▁the ▁number ▁of ▁inputs ▁( or ▁elements ▁in ▁the ▁database ), ▁as ▁opposed ▁to ▁the ▁linear ▁scaling ▁of ▁classical ▁algorithms . ▁A ▁general ▁class ▁of ▁problems ▁to ▁which ▁Gro ver ' s ▁algorithm ▁can ▁be ▁applied ▁is ▁Boolean ▁satisf i ability ▁problem . ▁In ▁this ▁instance , ▁the ▁database ▁through ▁which ▁the ▁algorithm ▁is ▁iter ating ▁is ▁that ▁of ▁all ▁possible ▁answers . ▁An ▁example ▁( and ▁possible )