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Computing Quantum Dept. of Comp. Sci. & Electrical Engineering Sci. University of Maryland Baltimore County Baltimore, MD 21250 Email: Lomonaco@UMBC.EDU WebPage: http://www.csee.umbc.edu/~lomonaco WebPage: http://www.csee.umbc.edu/~lomonaco Samuel J. Lomonaco, Jr. Overview Four Talks Elementary Lecture 1 A Rosetta Stone for Quantum Computation Three Quantum Algorithms Quantum Hidden Subgroup Algorithms An Entangled Tale of Quantum Entanglement Advanced A Rosetta Stone for Quantum Computation Dept. of Comp. Sci. & Electrical Engineering Sci. University of Maryland Baltimore County Baltimore, MD 21250 Email: Lomonaco@UMBC.EDU WebPage: http://www.csee.umbc.edu/~lomonaco WebPage: http://www.csee.umbc.edu/~lomonaco Samuel J. Lomonaco, Jr. This work is supported by: The Defense Advance Research Projects Agency (DARPA) & Air Force Research Laboratory (AFRL), Air Force Materiel Command, USAF Agreement Number F30602-01-2-0522. F30602-01- Research Institute (MSRI). The National Institute for Standards and Technology (NIST). The Mathematical Sciences L-O-O-P The L-O-O-P Fund. L- Adami, Barencol, Benioff, Bennett, Brassard, Calderbank, 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 more 1 Lomonaco, Samuel J., Jr., A Rosetta stone for quantum mechanics with an introduction to quantum computation, in AMS PSAPM/58, computation, (2002), pages 3 65. ? ? ? Why ? ? ? Quantum Computation Limits of small scale integration technology to be reached 2010-2020 2010Moore s Law, i.e., No Longer ! Moore Law, every year, double the computing power at half the price. No Longer ! A whole new industry will be built around the new & emerging quantum technology Collision Course Math Quantum Computation Comp Sci EE Multi-Disciplinary MultiPhysics The Classical World Classical Shannon Bit Decisive Individual 0 or 1 Classical Bits Can Be Copied Out Copying Machine In 2 The Quantum World Quantum Bit Introducing the Qubit Qubit ??? Indecisive Individual Can be both 0 & 1 at the same time !!! Quantum Representations of Qubits Example 1. A spin- 1 particle spin- Quantum Representations of Qubits (Cont.) Example 2. Polarization States of a Photon 2 1= , , 0= 0= or Spin Up Spin Down 1= 1 1 0 0 Where does a Qubit live ? H= Home Def. A Hilbert Space is a vector space H over together with an inner product , : H H such that 1) u +u2, v = u , v + u2, v & v, u +u = v, u + v, u 1 1 12 1 2 2) u, v = u, v 3) u, v = v , u 4) Cauchy seq u1 , u2 , in H , lim un H n The elements of H will be called kets, and kets, will be denoted by label A Qubit is a quantum system whose state is represented by a Ket lying in a 2-D Hilbert 2Space H 3 Superposition of States A typical Qubit is In de ci siv e Collapse of the Wave Function Collapse Qubit ??? 0 0 + 0 1 = Observer W ho os h = 0 0 + 1 1 2 where 0 + 1 = 1 2 !!! The above Qubit is in a Superposition of states 0 1 and It is simultaneously both 0 and 1 !!! i Kets as Column Vectors over Let Tensor Product of Hilbert Spaces The tensor product of two Hilbert spaces and K is the simplest Hilbert space such that the map ( h, k ) h k is bilinear, i.e., such that H = Pr ob |a | i2 be a 2-D Hilbert space with orthonormal basis H 0, 1 In this basis, each ket can be thought of as a column vector. For example, H K H K 1 0 = 0 and 0 1 = 1 And in general, we have = a 0 + b 1 = a + b = 1 0 b 0 1 a ( h1 + h2 ) k = h1 k + h2 k h ( k1 + k2 ) = h k1 + h k2 ( h) k = h ( k ) We define the action of on H K as ( h k ) ( h) k = h ( k ) In other words, H K is constructed in the simplest nontrivial way such that: Kronecker (Tensor) Product of Matrices ( h1 + h2 ) k = h1 k + h2 k h ( k1 + k2 ) = h k1 + h k2 ( h) k = h ( k ) ( h k ) , The Kronecker(tensor) product is defined as: a 11 A B = a 21 b1 1 b21 b11 b21 b1 2 b22 b1 2 b22 b1 1 a12 b21 b1 1 a 22 b21 b12 b22 b1 2 b22 a A = 11 a21 a12 a22 and b B = 11 b21 b12 b22 a11b11 a b = 11 21 a21b11 a b 21 21 a11b12 a11b22 a21b12 a21b22 a12b11 a12b21 a22b11 a22b21 a12b12 a12b22 a22b12 a22b22 4 So Representing Integers in Quantum Computation Let H 2 be a 2-D Hilbert space with orthonormal basis 1 0 01 = 0 1 = 0 1 = 0 1 0 0 1i 1 1 = = 0 0 0i 0 1 0,1 Then H = 0 H 2 is a 2n-D Hilbert space with induced orthonormal basis n 1 0 00 , 0 01 , 0 10 , 0 11 , 1 11 b1 b0 where we are using the convention bn 1bn 2 b1b0 = bn 1 bn 2 Representing Integers in Quantum Computation So in the Hilbert orthonormal basis 2n-D Indexing Convention for Matrices The indices of matrices start at 0, not 1. For example, in H space with induced 0 00 , 0 01 , 0 10 , 0 11 , 1 11 with binary H2 H2 H2 we represent the integer expansion n 1 m m = j = 0 m j 2 j , m j = 0 or 1, j m = mn 1 mn 2 23 = 010111 m1m0 as the ket For example, 0 index = 0 0 index = 1 0 index = 2 0 1 0 0 index = 3 5 = 101 = = 1 0 1 0 index = 4 1 index = 5 0 index = 6 0 index = 7 The Qubit Village Qubitville Kets Each in Massive Parallelism 1 , 2 , n Example. For j = 1, 2, , n , let j = H1 , H2 , n , Hn n Then 1 2 n = 0 + 1 2 j =1 0 +1 2 The Qubits in Qubit Village collectively live in H1 H2 Hn = H j=1 1 = ( 0 + 1 )( 0 + 1 ) 2 n n (0 +1) j The populace of Qubit Village is P o p u la ce = 1 2 n H j=1 n 1 = ( 00 0 + 00 1 + + 11 1 +) 2 j Other names for the populace of Qubit Village Populace = 1 2 n = 1 2 n 1 = a 2 a=0 Therefore, the n-qubit register contains all n-bit binary numbers simultaneously ! n 2n 1 5 But ! ! ! 1 2 sh oo h W ! Observer Activities in Quantum Village All activities in Quantum Village are Unitary transformations At time At time U n Pr ob =1 /2 n t=0 0 1 t=1 H T U where a unitary transformation is one such that H a U U = I = UU T Another Activity in Quantum Village Measurement Measurement Connecting Quantum Village To the Classical World Measurement Group of Friendly Physicists Another Activity in Quantum Village Measurement Observables ??? What does our observer actually observe ? Observables = Hermitian Operators H H where Group of OA Angry Physicists O A = OA T 6 Observables (Cont.) What does our observer actually observe ? ??? Observables (Cont.) What does our observer observe ? The state of an n-Qubit register can be written in the eigenket basis as ??? Let i be the eigenkets of OA, and let a i denote the corresponding eigenvalues , i.e., = i i i 2 OA i = ai i Caveat: We only consider observables whose Caveat: eigenkets form an orthonormal basis of So with probability pi = i , the observer observes the eigenvalue ai , and H i sh oo h W ! Example: Pauli Spin Matrices Consider the following observables, called the Pauli Spin matrices: 0 1 0 i 1 0 Measurement Example Consider a 2-D quantum system in state 2 = a 0 + b 1 , where a 2 + b 2 = 1 What happens if we measure First express 1 = , 2 = i 0 , 3 = 0 1 1 0 0 i i 0 *T T which can readily be checked to be Hermitian. Hermitian. E.g., 2 = 0 i 0 i = = i 0 = 2 i 0 w.r.t. observable 1 ? w.r.t. terms in of the eigenket basis of 1 The respective eigenvalues and eigenkets of these matrices are listed in the table below Thus, if a + b 0 + 1 a b 0 1 = + 2 2 2 2 Eigenvalue 1 ( 0 + 1 )/ 2 (0 1 )/ 2 2 ( 0 + i 1 )/ 2 (0 i 1 )/ 2 is observed w.r.t. , either w.r.t. 1 2 +1 3 0 1 Possibility0 Prob = a + b / 2 1 a 0 = + 1 is meas. ( 0 + 1 )/ 2 Eigenvalue or Possibility1 Prob = a b / 2 2 a1 = 1 is meas. ( 0 1 )/ 2 Eigenvalue Important Feature of Quantum Mechanics It is important to mention that: The No-Cloning Theorem NoDieks, Wootters, Zurek Dieks, Wootters, We cannot completely control the outcome of quantum measurement In Copying Machine Out 7 The No Cloning Theorem Definition. Let be a Hilbert space. Then Definition. a quantum replicator consists of an auxiliary Hilbert space H , a fixed state A # H A (called the initial state of the replicator), and a unitary transformation replicator), H The No Cloning Theorem Cloning is: # ( a 0 +b 1 ) blank @ ( a 0 + b 1 )( a 0 + b 1 ) U : HA H H HA H H such that, for some fixed state blank H , for (called the replicator state after replication of a ) depends on a . U # a blank = a a a all states a H , where a H A Cloning is NOT: NOT: # ( a 0 + b 1 ) blank @ ( a 00 + b 11 ) The No Cloning Theorem Key Idea Cloning is inherently non-linear non Quantum mechanics is inherently linear Ergo, quantum replicators do not exist Ergo, Introduction to Quantum Entanglement A Illustration of the Weirdness of Quantum Mechanics Entangled Qubits Observing Entangled Qubits Observe Only the Blue Qubit 0 0 Not Entangled Separate Unitary Transf 0 1 1 0 2 =1 /2 Pr ob U o ob P Pr 0 1 1 0 2 Entangled Not Separate ! / /2 =1 Whoosh ! 0 1 1 0 No Longer Entangled Separate Identity 8 EPR Pair EPR Pair 0 1 1 0 2 0 1 1 0 2 Einstein Podolsky Einstein Rosen Podolsky Something is Missing from Quantum Mechanics. There Must Exist Hidden Variables Hidden Variable Theory Bah ! Humbug ! vs Bah ! Humbug ! Bell Inequalities Aspect Experiment Rosen Quantum Mechanics Score So Far HVT Score = 0 QM Score = 1 Why did Einstein Podolsky Rosen Object So Vehemently ? Forces of Nature Are Local Interactions Hello ! Spacelike Distance Can t Hear Can You !! ?? All the forces of nature (i.e., gravitational, electromagnetic, weak, & strong forces) are local interactions. Spacelike Distance Mediated by another entity, e.g., gravitons, photons, etc. Propagate no faster than the speed of light c Strength drops off with distance ( x, y, z, t ) P1 Dist ( P1 , P2 ) > c T t ( X ,Y , Z , T ) P2 No signal can travel between spacelike regions of space Ergo, spacelike regions of space are physically independent, i.e., one cannot influence the other. independent, 9 Blue Qubit Meaurement of EPR Pair Red Qubit Alpha Centauri No Local Interaction ! ( 01 10 ) / Spacelike Distance 2 No force of any kind Acts instantaneously - Faster than light - Not mediated by anything Meas. Blue Qubit Instantly, Both Qubits Are Determined ! Strength does not drop off with distance - Full strength at any distance Yet, still consistent with General Relativity ! =1 /2 b ob P Pr Pr ob 2 /2 = =1 0 1 1 0 Properties of Qubits Useful for Quantum Computation Quantum Entanglement Appears to Pinpoint the Weirdness of Quantum Mechanics Properties of Quantum Properties of States Computer Data Qubits can exist in a superposition of states Qubits can be entangled Quantum States Actions onComputer Instructions Qubits collapse upon measurement collapse Qubits are transformed by unitary transformations An Application of Quantum Entanglement OxfordTeleportationDictionary Unabridged ? Teleportation: Transfering an object between Teleportation: two locations by a process of: Dissociation to obtain info Quantum Teleportation Information Transmission Reconstruction from info - Scanned to extract suff. Info. to suff. recreate original Net Effects: Effects: Destruction of original object Creation of an exact replica at the intended destination. - Exact replica is re-assembled at destination reout of locally available material 10 Asked Scotty about Teleportation Asked Scotty about Teleportation Aye, Aye, Captain ! Beam me up, Scotty ! I m just a wee bit busy. 11 More Dirac Notation Hilbert Space of morphisms from H to More Dirac Notation Let H * = Hom ( H , ) H* We call the elements of denote them as Bra s, and Bra label 12 More Dirac Notation There is a dual correspondence between t Ke Bra s as Row Vectors over Bra H * and H There exists a bilinear map defined by H H ( 1 )( 2 ) * Br a Let H be a 2-D Hilbert space with 2orthonormal basis 0 , 1 and let be the corresponding dual Hilbert space with corresponding dual basis H * = Hom ( H , 0,1 ) which we more simpy denote by Then with respect to this basis, we have 1 | 2 Bra-c-Ket Bra Bra- Ket 0 = ( 1,0 ) and 1 = ( 0,1) 0 a + 1 b = ( a, b ) Bra s & Ket s as Adjoints of One Another Bra Ket The dual correspondence If H H * is given by 1 = a 0 + b 1 2 = c 0 + d 1 then the bracket product becomes a b = a 0 + b 1 0 a + 1 b = ( a, b ) 1 | 2 = ( 0 a + 1 b )( c 0 + d 1 c = ( a , b )i = ac + bd d ) and is called the adjoint 1 2 If As a Matrix Outerproduct 1 = a 0 + b 1 2 = c 0 + d 1 Let basis H be an N-D Hilbert space with orthonormal N0 , 1 , , N 1 then 1 2 H is the linear transformation If we use the convention that matrix indices begin at 0, then the matrix of the linear transformation mk is an NxN matrix consisting of all zeroes with the exception of entry (m,k) which is 1 m,k) For example if N=4, then N=4, Entry (2,3) 1 2 H 1 2 | which, when written in matrix notation, becomes the matrix outerproduct 1 2 = i ( c , d ) = b bc bd a ac ad 0 0 2 3 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 13 Two Ways to Represent Quantum States Density Operators & Mixed Ensembles Kets & Density Operators Two Ways to Represent Quantum States Example. We have seen pure ensembles, i.e., Example. ensembles, pure states, such as Ket Prob Two Ways to Represent Quantum States Example. Consider the following state for Example. which we have incomplete knowledge, called a mixed ensemble: ensemble: Ket Prob where 1 1 p1 2 p2 k pk All unit Length & not nec. nec. Problem. Certain types of quantum states Problem. are difficult to represent in terms of kets p1 + p2 + + pk = 1 Two Ways to Represent Quantum States Ket Prob Two Ways to Represent Quantum States If for example, 1 p1 2 p2 k pk Mixed Ensemble =a 0 +b1 where Johnny von Neumann suggested that we use the following operator to represent a state: = p1 1 1 + p2 2 2 + + pk k k is called a density operator. It is a Hermitian operator. positive definite operator of trace 1. For the pure ensemble Ket a + b =1 2 2 then = (a 0 + b 1 )( 0 a + 1 b) ab 2 b 1 , Prob = 1i a2 a = (a b) = ba b 14 Two Ways to Represent Quantum States On the other hand, Quantum Mechanics from the Two Perspectives = 4 p1 3/8 3i /8 3 0 i 1 0 + i 1 1 + 1 1 = 2 2 4 3i /8 5/8 Kets Density Ops 1 1 1 p2 i 2 2 2 2 = 1 =H t i = [H, ] t Schroed. Schroed. Eq. Eq. Unitary Evolution Observation is the mixed ensemble Ket = 0 i 1 0 U 0 0 U 0U ( )/ A = | A| A =trace( A ) Prob 3 4 1 4 Density Operators We now have a more powerful way of representing quantum states. Density operators are absolutely crucial when discussing and dealing with quantum noise and quantum decoherence. decoherence. The End Quantum Computation: A Grand Mathematical Mathematical Challenge for the Twenty-First Century Century and the Millennium, Samuel J. Lomonaco, Jr. (editor), AMS PSAPM/58, (2002). Quantum Computation and Information, Samuel J. Lomonaco, Jr. and Howard E. Brandt (editors), AMS CONM/305, (2002). 15

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