Distributive Quantum Computing - Department Of Computer Science And .
QuantumComputing ?Quantum ComputingSamuel J. Lomonaco, Jr.Dept. of Comp. Sci.Sci. & Electrical EngineeringUniversity of Maryland Baltimore CountyBaltimore, MD 21250Email:Lomonaco@UMBC.EDUWebPage:WebPage: http://www.csee.umbc.edu/ lomonacohttp://www.csee.umbc.edu/ lomonacoL-O-O-PThis work is supported by:The Defense Advance Research ProjectsAgency (DARPA) & Air Force ResearchLaboratory (AFRL), Air Force Materiel Command,USAF Agreement Number F30602-01-2-0522. The National Institute for Standardsand Technology (NIST) The Mathematical Sciences ResearchInstitute (MSRI). L-O-O-PThe L-O-O-P Fund.The Institute of Scientific InterchangeThese talksand othersare available at:http://www.csee.umbc.edu/ lomonacohttp://www.csee.umbc.edu/ lomonacoOverviewFour Talks A Rosetta Stone for Quantum Computation Quantum Algorithms & Beyond Distributive Quantum Computing Topological quantum Computing and theJones Polynomial A Quantum Computing Knot Theoretic Mystery-- Can be found on my webpage.OverviewFour Talks A Rosetta Stone for Quantum Computation Quantum Algorithms & Beyond Distributive Quantum Computing Topological quantum Computing and theJones Polynomial1
ARosetta StoneforQuantum ComputationSamuel J. Lomonaco, Jr.Dept. of Comp. Sci.Sci. & Electrical EngineeringUniversity of Maryland Baltimore CountyBaltimore, MD 21250Email:Lomonaco@UMBC.EDUWebPage:WebPage: http://www.csee.umbc.edu/ lomonacohttp://www.csee.umbc.edu/ lomonacoL-O-O-PLomonaco, Samuel J., Jr., A Rosetta stonefor quantum mechanics with an introduction toquantum computation,computation, in AMS PSAPM/58,(2002), pages 3 – 65.Adami, Barencol, Benioff, Bennett, Brassard,Calderbank, Chen, Crepeau, Deutsch,DiVincenzo, Ekert, Einstein,Feynman, Grover, Heisenberg,Jozsa, Knill, Laflamme,Lloyd, Lomonaco,Peres,Popescu,Preskill, Podolsky,Rosen, Schumacher,Shannon, Shor,Simon, Sloane,Schrodinger, Townsend, Unruh,von Neumann, Vazirani,Wootters, Yao, Zeh,Zurek& many moreQuantum Computation and Information,Information, Samuel J.Lomonaco, Jr. and Howard E. Brandt (editors), AMSCONM/305, (2002).? ? ? Why ? ? ?QuantumComputationCollision CourseMathQuantumComputation Limits of small scale integrationtechnology to be reached 20102010-2020Moore’ No Longer !Moore’s Law,Law, i.e.,every year, double the computing powerat half the price. No Longer ! A whole new industry will be built aroundthe new & emerging quantum technologyPhysicsCompSciEEMultiMulti-Disciplinary2
dual0 or 1Classical Bits Can Be CopiedOutInCopyingMachineQuantumBitIntroducingthe QubitQubit? ? ?TheQuantumWorldQuantum Representationsof QubitsExample 1. A spinspin- 1 particle2IndecisiveIndividualCan be both 0 & 1at the same time !!!Spin Up11Spin Down003
Quantum Representationsof Qubits (Cont.)Example 2. Polarization States of a Photon1 ,0 1 ,0 orH Where does a Qubit live ?Def. A Hilbert Space is a vectorspace H over together with an innerproduct , : H H such thatHomeu1 u2, v u1, v u2, v & v,u1 u2 v, u1 v,u22) u, λ v λ u, v3) u, v v , u4) Cauchy seq u1 , u2 , in H , lim un Hn 1)The elements of H will be called kets,kets, andwill be denoted by labelSuperposition of StatesHA typical Qubit is?IndecisiveA Qubit is a quantumsystem whose state isrepresented by a Ketlying in a 22-D HilbertSpacewhere2 α 0 0 α1 12α 0 α1 1The above Qubit is in a Superposition of states01andIt is simultaneously both“Collapse”Collapse” of the Wave FunctionH!!! Prob ai 2Whooshi1!!!be a 2-D Hilbert space with orthonormal basis0,α 0 0 α1 1 ObserverandKets as Column Vectors overLetQubit01In this basis, each ket can be thought of as acolumn vector. For example, 1 0 0 0 1 1 andAnd in general, we have 0 1 a ψ a 0 b 1 a b 10b4
Tensor Product of Hilbert SpacesThe tensor product of two Hilbert spacesand K is the “simplest” Hilbert space suchthat the map ( h, k )h kHH K H Kis bilinear, i.e., such that ( h1 h2 ) k h1 k h2 k h ( k1 k2 ) h k1 h k2 ( λ h) k h (λk ) We define the action ofon H K asλ ( h k ) ( λ h) k h ( λ k )Kronecker (Tensor) Product of Matricesa12 a22 aA 11 a21 bB 11 b21andIn other words,H K isconstructed in the simplest nontrivial way such that: ( h1 h2 ) k h1 k h2 k h ( k1 k2 ) h k1 h k2 ( λh) k h ( λk ) λ ( h k ) , λ So b12 b22 1 0 01 0 1 0 1 0 1 0 0 1i 1 1 0 0 0i 0 1 The Kronecker(tensor) product is defined as: a11 A B a 21 b11b21b1 1b21b12 b 2 2 b1 2 b 2 2 a11b11 a11b12 a ba11b22 11 21 a21b11 a21b12 a b 21 21 a21b22 a 12 a 22 a12b11a12b21a22b11a22b21b11b21b1 1b21b12 b 2 2 b1 2 b 2 2 a12b12 a12b22 a22b12 a22b22 Representing Integers in Quantum ComputationRepresenting Integers in Quantum ComputationLet H2 be a 2-D Hilbert space with orthonormalbasisSo in the 2n-D Hilbertorthonormal basis0,1n 1Then H 0 H2 is a 2n-D Hilbert spacewith induced orthonormal basis000 , 001 , 0 10 , 011 , 1 11where we are using the conventionbn 1bn 2b1b0 bn 1 bn 2 000 , 0space with induced01 , 0 10 , 0 11 , 1 11we represent the integerexpansionmwith binarym j 0 m j 2 j , m j 0 or 1, jn 1as the ket b1 b0Hm m n 1 m n 2For example,m1m023 0101115
The Qubit VillageIndexing Convention for MatricesQubitvilleThe indices of matrices start at 0, not 1.H2 H2 H2 0 index 0 0 index 1 0 index 2 0 1 0 0 index 35 101 1 0 1 0 index 4 1 index 5 0 index 6 0 index 7Massive ParallelismExample. ForThenj 1,2, , n ,let Ψ j 0 12n 0 1 Ψ1Ψ2 Ψn 2 j 1 n 1 ( 0 1 )( 0 1 ) 2 Ψ1 , Ψ 2 ,ΨnEach inH1 ,, HnH2 , The Qubits in Qubit Village collectively live inH1 H 2 Hn n Hj 1j The populace of Qubit Village isP o p u la ce Ψ 1 Ψ 2 Ψn n Hj 1j Other names for the populace of Qubit VillagePopulace Ψ1 Ψ 2Ψ n Ψ 1Ψ 2ΨnBut ! ! !Ψ1Ψ2(0 1)ΨnhosohWn 1 ( 00 0 00 1 11 1 ) 2 n 2n 1 1 a 2 a 0Therefore, the n-qubit register contains alln-bit binary numbers simultaneously !Kets!Prob 1/2 nFor example, inObserveraActivities in Quantum VillageAll activities in Quantum Village are UnitarytransformationsAt timeAt timeUΨ1Ψ0t 0HUt 1Hwhere a unitary transformation is one such thatTU U I UUTMeasurementConnectingQuantum Villageto theClassical World6
Another Activity in Quantum Village:Another Activity in Quantum Village:MeasurementMeasurementMeasurementGroup ofGroup of Friendly Physicists?ObservablesWhat does our observeractually observe ?OALet ϕ i be the eigenkets of OA, and let aidenote the corresponding eigenvalues , i.e.,OA ϕ i ai ϕ iwhereCaveat:Caveat: We only consider observables whoseTeigenkets form an orthonormal basis ofOA OAObservables (Cont.)What does our observer observe ?The state of an n-Qubit register canbe written in the eigenket basis asΨ i αi ϕi2So with probability pi α i , the observerobserves the eigenvalue a i , andϕi?Observables (Cont.)H H!PhysicistsWhat does our observer actuallyobserve ?Observables Hermitian OperatorshosohWAngry?HExample: Pauli Spin MatricesConsider the following observables, called the Pauli Spinmatrices:matrices:0 10 i1 0 , σ 2 i 0 , σ 3 0 1 1 0 σ1 which can readily be checked to be Hermitian.Hermitian. 0 i i 0 E.g.,σ 2† *TT 0 i 0 i i 0 σ2 i 0 The respective eigenvalues and eigenkets of these matrices arelisted in the table belowσ1σ2 1(0 1 )/ 2(0 i 1 )/ 2σ30 1(0 1 )/ 2(0 i 1 )/ 21Eigenvalue7
Measurement ExampleImportant Feature ofQuantum MechanicsConsider a 22-D quantum system in stateψ a 0 b 1 , where a 2 b 2 1What happens if we measure ψFirst expressψ Thus, ifψ ψis observed w.r.t.w.r.t. σ , either1Possibility02Eigenvaluea 0 1 is meas.(0σ1a b 0 1 a b 0 1 2 2 2 2 Prob a b / 2ψw.r.t.w.r.t. observable σ 1 ?in terms of the eigenket basis ofor 1 )/ 2Possibility12Prob a b / 2EigenvalueIt is important to mention that:We cannot completelycontrol the outcome ofquantum measurementa1 1 is meas.ψ( 0 1 )/ 2The NoNo-Cloning TheoremDieks,Dieks, Wootters,Wootters, ZurekInCopyingMachineOutThe No Cloning TheoremHDefinition.be a Hilbert space. ThenDefinition. Leta quantum replicator consists of anauxiliary Hilbert space H , a fixed stateAψ # H A (called the initial state ofthe replicator),replicator), and a unitary transformationU : H A H H HA H Hsuch that, for some fixed state blank H ,U ψ # a blank ψ a a afor all states a H , where ψ a H A(called the replicator state after replication ofa ) depends on a .The No Cloning Theorem Cloning is:ψ# ( a 0 b 1 ) blankKey Ideaψ@ ( a 0 b 1 )( a 0 b 1 ) Cloning is NOT:NOT:ψ# ( a 0 b 1 ) blankThe No Cloning Theoremψ@ ( a 00 b 11 ) Cloning is inherently nonnon-linear Quantum mechanics is inherently linear Ergo,Ergo, quantum replicators do not exist8
Entangled QubitsIntroductiontoQuantum EntanglementA Illustration of the Weirdnessof Quantum MechanicsUnitaryTransf U0 0 Not Entangled Entangled Separate Not Separate !EPR PairObserving Entangled QubitsObserve Onlythe Blue QubitWhoosh !0 1/2 1ProbobPr 1/20 1 1 021 0 No Longer Entangled Separate IdentityEPR PairPodolsky0 1 1 02EinsteinPodolskyRosenBah ! Humbug !Something is Missing from Quantum Mechanics.There Must Exist Hidden VariablesHidden Variable TheoryvsQuantum MechanicsScore So Far0 1 1 02Einstein0 1 1 02RosenSomething is Missing from Quantum Mechanics.Bell InequalitiesThere Must Exist Hidden Variables HVT Score 0 QM Score 1Hidden VariableTheoryvs Quantum MechanicsAspectExperiment9
Forces of Nature Are Local InteractionsWhy didEinsteinPodolskyRosenAll the forces of nature (i.e., gravitational,electromagnetic, weak, & strong forces) arelocal interactions. Mediated by another entity, e.g., gravitons,photons, etc. Propagate no faster than the speed of light c Strength drops off with distanceObject SoVehemently ?Spacelike DistanceHello !Can’Can’t HearYou !! ?( x, y, z, t )P2Dist ( P1 , P2 ) c T t( X ,Y , Z , T )No signal can travel between spacelike regions of spaceErgo, spacelike regions of space are physicallyindependent, i.e., one cannot influence the other.Blue QubitMeaurement of EPR PairEarth( 01 10 ) / The forces of nature are localinteractionsSpacelike DistanceP1The EPR PerspectiveRed QubitAlpha Centauri2 Spacelike regions of space arephysically independentAll perfectlyreasonableassumptions !No Local Interaction ! No force of any kind- Not mediated by anything Acts instantaneouslySpacelike DistanceProb0 1Instantly,Both QubitsAre Determined !/2 1Meas. BlueQubit Strength does not drop off with distanceobPr 1/2- Faster than light- Full strength at any distanceYet, still consistent with General Relativity !1 010
Properties of Qubits Usefulfor Quantum ComputationQuantum EntanglementAppears to Pinpointthe Weirdness ofQuantum MechanicsProperties ofof StatesQuantum Computer Data Properties Qubits can exist in a superposition ofstates Qubits can be entangled QuantumonComputerActionsStates Instructions Qubits “collapse”collapse” upon measurement Qubits are transformed by unitarytransformationsAn Application of Quantum nUnabridged Dictionary?Teleportation:Teleportation: Transfering an object betweentwo locations by a process of: Dissociation to obtain info- Scanned to extract suff.suff. Info. torecreate original Information Transmission Reconstruction from info- Exact replica is rere-assembled at destinationout of locally available materialNet Effects:Effects: Destruction of original object Creation of an exact replica at theintended destination.Asked Scotty about TeleportationAye, Aye, Captain !I’mBeamjustmea weeup, bitScottybusy.!11
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Skip to More Dirac Notation and theDensity OperatorSkip to the DeutschDeutsch-JozsaAlgorithmD-J n.Definition. A coin is fair (or balanced)balanced) if it hasheads on one side and tails on the other side. It isunfair (or constant)constant) if either it has tails on bothsides, or heads on both sides.HTSide1Side2Fair (Balanced)TTSide1Side2Unfair (Constant)THHHSide1Side2Side1Side2Fair (Balanced)Unfair (Constant)ObservationObservation.Observation. In the classical world, we needto observe both sides of the coin to determinewhether or not it is fair ?But what about inthe quantum world ?We represent a coin mathematically as aBoolean function:f : { 0, 1} { 0, 1 }Side1Side2HT13
The Unitary Implementation of fLetUfbe the unitary transformationUfHx y Hx f ( x) ythenx0 12Uf( 1) f ( x) x0 12Moreover,0H1H ( 1) f (0) ( 1) f (1) ( 1) f (0) ( 1) f (1) 0 122 HUfHCase 1.fCase 1.f1is fair,fair, i.e., balanced ( 1) f (0) ( 1) f (1) Output 0i 0 1 1 1 1 12 is unfair,unfair, i.e., constant ( 1) f (0) ( 1) f (1) Output 0 1 0i 1 1 0 12 So If we only make one observation, i.e., if weobserve the left register, then we candetermine whether or not f is fair orunfair.MoreDiracNotationSkip Dirac Notation and the DensityOperatorMore Dirac NotationLetH * Hom ( H ,We call the elements ofdenote them aslabel)H*More Dirac NotationHilbert Spaceof morphismsfrom H toThere is a dual correspondence betweentKeψ† ψH * and HBraH* H ( ψ 1 )( ψ 2 ) There exists a bilinear mapdefined byBra’Bra’s, andwhich we more simpy denote byψ 1 ψ 2BraBra-c-KetKetBra14
Bra’Bra’s as Row Vectors overBra’Bra’s & Ket’Ket’s as Adjoints of One AnotherLet H be a 22-D Hilbert space withorthonormal basis 0 , 1and letH * Hom ( H ,The dual correspondence†)be the corresponding dual Hilbert space withcorresponding dual basisH H *is given by0, 1 a b a 0 b 1 Then with respect to this basis, we have0 ( 1,0 ) and 1 ( 0,1)and is called the adjoint0 a 1 b ( a, b )ψ1 ψ 2IfIf ψ1 a 0 b 1 ψ 2 c 0 d 1thenψ 1 ψ 2 ( 0 a 1 b ) ( c 0 d 1) c ( a , b )i ac bd d as a Matrix Outerproduct ψ1 a 0 b 1 ψ 2 c 0 d 1ψ1 ψ 2Hthen the bracket product becomesLetbasis† 0 a 1 b ( a, b )ψis the linear transformationψ1 ψ 2 Hψ 1 ψ 2 ψwhich, when written in matrix notation, becomes thematrix outerproduct a ac ad ψ 1 ψ 2 i( c , d ) b bc bd H be an NN-D Hilbert space with orthonormal0 , 1 , , N 1If we use the convention that matrix indices beginat 0, then the matrix of the linear transformationm kis an NxN matrix consisting of all zeroes with theexception of entry (m,k)m,k) which is 1For example if N 4,N 4, then 0 02 3 0 0 0 0 0 0 0 0 0 0 1 0 0 0 Entry (2,3)Density Operators&Mixed Ensembles15
Two Ways to Represent Quantum StatesKetsDensity Operators&ψTwo Ways to Represent Quantum StatesExample.Example. We have seen pure ensembles,ensembles, i.e.,pure states, such asKetρProbψ1Problem.Problem. Certain types of quantum statesare difficult to represent in terms of ketsTwo Ways to Represent Quantum StatesExample.Example. Consider the following state forwhich we have incomplete knowledge, called amixed ensemble:ensemble:KetProbwhereψ1ψ2ψkp1p2pkAll unit Length& not nec.nec. Two Ways to Represent Quantum StatesKetψ1ψ2ψkProbp1p2pkJohnny von Neumann suggested that we use thefollowing operator to represent a state:ρ p1 ψ1 ψ1 p2 ψ2 ψ2 pk ψk ψkp1 p2 pk 1ρ is called a density operator.operator. It is a Hermitianpositive semisemi-definite operator of trace 1.ψFor the pure ensemble KetProbTwo Ways to Represent Quantum States1,ρ 1i ψ ψTwo Ways to Represent Quantum StatesOn the other hand,If for example,3 0 i 1 0 i 1 1 3/8 3i /8 1 1 2 2 4 3i /8 5/8 ρ 4ψ a 0 b1where2p12a b 1thenMixedEnsembleρ (a 0 b 1 ) ( 0 a 1 b) a a ( a b) ba b 2ab 2b ψ1p2ψ1is the mixed ensembleKet ψ 0 i 11Prob(34)/ψ2 ψ22 ψ2 11416
Density OperatorsQuantum Mechanics from the Two PerspectivesKetsρψi ψ Hψ tψUψA ψ A ψ We now have a more powerful way ofDensity Opsiρ ρ [H,ρ] tU ρU †A trace( Aρ)representing quantum states.Schroed.Schroed.Eq.Eq. Density operators are absolutelyUnitaryEvolutionObservationWeird !crucial when discussing and dealingwith quantum noise and quantumdecoherence.Measurement PhilosopherTurfψBlackBoxQ. Sys.Statewhere O QuantumWorld jProb ψ Pj ψOutψjEigenvalue Pj ψψ Pj ψQ. Sys.Stateλ j P j Spectral Decomposition17
Distributive Quantum ComputingDistributive Quantum Computing . for quantum mechanics with an introduction to quantum computation, in AMS PSAPM/58, (2002), pages 3 - 65. Quantum Computation and InformationQuantum Computation and Information,Samuel J.
How did the Distributive Property help you solve the problems? Lesson 5.3 Enrich Apply the Distributive Property Use the Distributive Property to help solve each problem. Possible answer: I used the Distributive Property
1. Quantum bits In quantum computing, a qubit or quantum bit is the basic unit of quantum information—the quantum version of the classical binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics.
Quantum computing is a subfield of quantum information science— including quantum networking, quantum sensing, and quantum simulation—which harnesses the ability to generate and use quantum bits, or qubits. Quantum computers have the potential to solve certain problems much more quickly t
Topics History of Computing What are classical computers made of? Moore's Law (Is it ending?) High Performance computing Quantum Computing Quantum Computers Quantum Advantage The Qubit (Information in superposition) Information storage Different kinds of quantum computers Superconducting vs Ion Traps Annealing vs Universal/Gate quantum computers
Quantum computing with photons: introduction to the circuit model, the one-way quantum computer, and the fundamental principles of photonic experiments . quantum computing are presented and the quantum circuit model as well as measurement-based models of quantum computing are introduced. Furthermore, it is shown how these concepts can
Keywords—machine learning, quantum computing, deep learning, quantum machine intelligence I. INTRODUCTION There is currently considerable interest in quantum computing. In the U.S. in January 2019, the National Quantum Initiative Act authorized 1.2 billion investment over the next 5-10 years (Rep Smith, 2019). A quantum computing race is ongoing
According to the quantum model, an electron can be given a name with the use of quantum numbers. Four types of quantum numbers are used in this; Principle quantum number, n Angular momentum quantum number, I Magnetic quantum number, m l Spin quantum number, m s The principle quantum
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