Quaternionic Computing(3)
发布时间：2021-06-08
Quaternionic Computing
Transformations.Generallyspeaking,thetransformationsthatareallowedarelinearmap-
pings.Inaddition,inorderforthequantitiesabovetobeproperprobabilities,thesetransformationsmustpreserveL2-norm.TheonlyrelevantlinearandL2-normpreserv-ingoperationsareunitarytransformations2.Theseareusuallyrepresentedinthematrixforminwhichthecolumnvectorsaretheimagesofthecanonicalbasisunderthegiventransformation,listedinlexicographicalorder.
Circuits.Thecomputationaldeviceismodelledasacircuit,which,withoutlossofgenerality,
canbeassumedtohavethefollowingcharacteristics
Theinputtothecircuitisanypurestate. Thecircuitisanarrayofelementaryuniversalgates.
Forexample,wecanchoosethe2-qubitCNOTgateandarbitrary1-qubitrotationsasauniversalset.Furthermore,weallowgatestooperateonanytwoarbitrarywires,notnecessarilycontiguous3.
Algorithms.AquantumalgorithmcanbeformallydescribedasaclassicalTuringMachine,
whichgivenaclassicalstringxwillgeneratea(classical)descriptionforaquantumcircuit.Thequantumcomputercanthenproduceananswerbasedontheresultofmeasurementsoftheoutputwiresofthequantumcircuit.Withoutlossofgenerality,wecanassumethatthecircuitistobeevaluatedwiththegroundstate(theallzerocomputationalbasevector)asitsinitialstate.Thealgorithmissaidtobee cientifthecorrespondingTMrunsintimepolynomialonthesizeoftheinputx,whichinturnsimpliesthatcircuitsizeisalsopolynomial.
Fromapurelyabstractpointofview,itcanbeinferredthattheonlyrequirementsofthismodelisthatthestatespacehasalinearstructureandapropernorm-inducinginnerproduct,sothatthemeasurementruleisalwayssound.Traditionally,quantumcomputinghasbeendescribedintermsofcomplexHilbertspaces,butinprinciple,aswejustdiscussed,asoundmodelofcomputationcanbede nedonanyotherHilbertspace.Inparticular,inthispaperwestudymodelsofrealcomputingandquaternioniccomputing,basedonthe2n-dimensionalvectorspacesontherealsandthequaternions,respectively.
3
3.1RealComputingDe nitions
Intuitively,therealcomputingmodelisde nedasarestrictedversionofquantumcomputing,whereallamplitudesinthestatevectorsarerequiredtoberealnumbers.Conjugationis
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