▁ 2 n / 2 ▁inv oc ations ▁of ▁the ▁underlying ▁crypt ographic ▁algorithm , ▁compared ▁with ▁roughly ▁ 2 n ▁in ▁the ▁classical ▁case , ▁meaning ▁that ▁symmetric ▁key ▁lengths ▁are ▁effectively ▁hal ved : ▁A ES - 2 5 6 ▁would ▁have ▁the ▁same ▁security ▁against ▁an ▁attack ▁using ▁Gro ver ' s ▁algorithm ▁that ▁A ES - 1 2 8 ▁has ▁against ▁classical ▁br ute - force ▁search ▁( see ▁Key ▁size ). ▁ ▁Quant um ▁crypt ography ▁could ▁potentially ▁ful fill ▁some ▁of ▁the ▁functions ▁of ▁public ▁key ▁crypt ography . ▁Quant um - based ▁crypt ographic ▁systems ▁could , ▁therefore , ▁be ▁more ▁secure ▁than ▁traditional ▁systems ▁against ▁quantum ▁hack ing . ▁ ▁Quant um ▁search ▁▁ ▁Besides ▁factor ization ▁and ▁discrete ▁log arith ms , ▁quantum ▁algorithms ▁offering ▁a ▁more ▁than ▁polynomial ▁speed up ▁over ▁the ▁best ▁known ▁classical ▁algorithm ▁have ▁been ▁found ▁for ▁several ▁problems , ▁including ▁the ▁simulation ▁of ▁quantum ▁physical ▁processes ▁from ▁chem istry ▁and ▁solid ▁state ▁physics , ▁the ▁approximation ▁of ▁Jones ▁polynomials , ▁and ▁solving ▁P ell ' s ▁equation . ▁No ▁mathematical ▁proof ▁has ▁been ▁found ▁that ▁shows ▁that ▁an ▁equally ▁fast ▁classical ▁algorithm ▁cannot ▁be ▁discovered , ▁although ▁this ▁is ▁considered ▁unlikely . ▁However , ▁quantum ▁computers ▁offer ▁polynomial ▁speed up ▁for ▁some ▁problems . ▁The ▁most ▁well - known ▁example ▁of ▁this ▁is ▁quantum ▁database ▁search , ▁which ▁can ▁be ▁solved ▁by ▁Gro ver ' s ▁algorithm ▁using ▁qu adrat ically ▁fewer ▁queries ▁to ▁the ▁database ▁than ▁that ▁are