▁their ▁prefer ences ▁( called ▁low - rank ▁matrices ), ▁what ▁are ▁the ▁products ▁that ▁a ▁given ▁user ▁may ▁want ▁to ▁buy ? ▁A ▁common ▁linear ▁algebraic ▁classical ▁strategy ▁to ▁solve ▁this ▁problem ▁is ▁to ▁re construct ▁an ▁approximation ▁of ▁the ▁full ▁preference ▁matrix ▁and ▁use ▁it ▁to ▁predict ▁the ▁next ▁preferred ▁product . ▁Such ▁a ▁strategy ▁uses ▁at ▁least ▁a ▁time ▁polynomial ▁in ▁the ▁matrix ▁dimension . ▁In ▁ 2 0 1 6 , ▁ ▁I ord an is ▁K eren id is ▁and ▁An up am ▁P rak ash , ▁found ▁an ▁expon entially ▁faster ▁quantum ▁algorithm ; ▁this ▁algorithm ▁uses ▁the ▁H HL ▁algorithm ▁to ▁sample ▁the ▁product ▁directly ▁from ▁an ▁approximation ▁of ▁the ▁preference ▁matrix ▁without ▁re construct ing ▁the ▁matrix ▁itself , ▁thus ▁avoid ing ▁the ▁polynomial ▁limit ▁mentioned ▁above . ▁Tang ' s ▁classical ▁algorithm , ▁inspired ▁by ▁the ▁fast ▁quantum ▁algorithm ▁of ▁K eren id is ▁and ▁P rak ash , ▁is ▁able ▁to ▁perform ▁the ▁same ▁calculations ▁but ▁on ▁a ▁normal ▁computer ▁without ▁the ▁need ▁for ▁quantum ▁machine ▁learning . ▁Both ▁approaches ▁run ▁in ▁poly log arith mic ▁time ▁which ▁means ▁the ▁total ▁computation ▁time ▁scales ▁with ▁the ▁log arith m ▁of ▁the ▁problem ▁variables ▁such ▁as ▁the ▁total ▁number ▁of ▁products ▁and ▁users , ▁except ▁Tang ▁util ises ▁a ▁classical ▁rep lication ▁of ▁the ▁quantum ▁sampling ▁techniques . ▁Prior ▁to ▁Tang ' s ▁results , ▁it ▁was ▁widely ▁assumed ▁that ▁no ▁fast ▁classical ▁algorithm ▁existed ; ▁K eren id is ▁and ▁P rak ash