pling , ▁where ▁only ▁the ▁conject ure ▁that ▁the ▁average - ▁and ▁worst - case ▁complex ities ▁of ▁the ▁problem ▁are ▁the ▁same ▁is ▁needed . ▁ ▁Pro posed ▁experiments ▁ ▁The ▁following ▁are ▁propos als ▁for ▁demonstr ating ▁quantum ▁computational ▁supre m acy ▁using ▁current ▁technology , ▁often ▁called ▁N IS Q ▁devices . ▁Such ▁propos als ▁include ▁( 1 ) ▁a ▁well - defined ▁computational ▁problem , ▁( 2 ) ▁a ▁quantum ▁algorithm ▁to ▁solve ▁this ▁problem , ▁( 3 ) ▁a ▁comparison ▁best - case ▁classical ▁algorithm ▁to ▁solve ▁the ▁problem , ▁and ▁( 4 ) ▁a ▁complexity - the or etic ▁argument ▁that , ▁under ▁a ▁reasonable ▁assumption , ▁no ▁classical ▁algorithm ▁can ▁perform ▁significantly ▁better ▁than ▁current ▁algorithms ▁( so ▁the ▁quantum ▁algorithm ▁still ▁provides ▁a ▁super pol ynomial ▁speed up ). ▁ ▁Sh or ' s ▁algorithm ▁for ▁factor ing ▁integers ▁▁ ▁This ▁algorithm ▁finds ▁the ▁prime ▁factor ization ▁of ▁an ▁n - bit ▁integer ▁in ▁ ▁time ▁whereas ▁the ▁best ▁known ▁classical ▁algorithm ▁requires ▁time ▁and ▁the ▁best ▁upper ▁bound ▁for ▁the ▁complexity ▁of ▁this ▁problem ▁is ▁. ▁It ▁can ▁also ▁provide ▁a ▁speed up ▁for ▁any ▁problem ▁that ▁reduces ▁to ▁integer ▁factor ing , ▁including ▁the ▁membership ▁problem ▁for ▁matrix ▁groups ▁over ▁fields ▁of ▁odd ▁order . ▁ ▁This ▁algorithm ▁is ▁important ▁both ▁pract ically ▁and ▁histor ically ▁for ▁quantum ▁computing . ▁It ▁was ▁the ▁first ▁polynomial - time ▁quantum ▁algorithm ▁proposed ▁for ▁a ▁real - world ▁problem ▁that ▁is ▁believed ▁to ▁be ▁hard ▁for