Quaternionic Computing
发布时间：2021-06-08
Quaternionic Computing
QuaternionicComputing
Jos´eM.Fernandez,WilliamA.Schneeberger
arXiv:quant-ph/0307017v2 5 Nov 2004February1,2008AbstractWeintroduceamodelofcomputationbasedonquaternions,whichisinspiredonthequantumcomputingmodel.Purestatesarevectorsofasuitablelinearspaceoverthequaternions.Otheraspectsofthetheoryarethesameasinquantumcomputing:super-positionandlinearityofthestatespace,unitarityofthetransformations,andprojectivemeasurements.However,onenotableexceptionisthefactthatquaternioniccircuitsdonothaveauniquelyde nedbehaviour,unlessatotalorderingofevaluationofthegatesisde ned.Givensuchanorderingauniqueunitaryoperatorcanbeassociatedwiththequaternioniccircuitandapropersemanticsofcomputationcanbeassociatedwithit.Themainresultofthispaperconsistsinshowingthatthismodelisnomorepowerfulthanquantumcomputing,aslongassuchanorderingofgatescanbede ned.Moreconcretelyweshow,thatforallquaternioniccomputationusingnquaterbits,thebehaviourofthecircuitforeachpossiblegateorderingcanbesimulatedwithn+1qubits,andthiswithlittleornooverheadincircuitsize.Theproofofthisresultisinspiredofanewsimpli edandimprovedproofoftheequivalenceofasimilarmodelbasedonrealamplitudestoquantumcomputing,whichstatesthatanyquantumcomputationusingnqubitscanbesimulatedwithn+1rebits,andinthiswithnocircuitsizeoverhead.Beyondthispotentialcomputationalequivalence,however,weproposethismodelasasimplerframeworkinwhichtodiscussthepossibilityofaquaternionicquantummechanicsorinformationtheory.Inparticular,italreadyallowsustoillustratethattheintroduction
ofquaternionsmightviolatesomeofthe“natural”propertiesthatwehavecometoexpectfromphysicalmodels.
1Introduction
QuantumComputingrepresentsyetanotherdisconcertingpuzzletoComplexityTheory.Whatweknowtodayisthatquantumcomputingdevicescane cientlysolvecertainproblems,which,inappearance,classicalorprobabilisticcomputerscannotsolvee ciently.EventhoughwewouldliketobelievethatquantumcomputingviolatesthestrongChurch-Turingthesis,thesoretruthisthattheknownresultsdonotprovideusaproof,onlyconstituting,atbest,“strongevidence”thereof.
Yet,eventhoughwecannotprovideastrictseparationbetweenthesemodels,wedoknowcertaininclusionsbetweenvariationsofthesecomputingmodels.Perhapsthemostnatural
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