Quantum Computing Quantum Computing be found at: be found at: - PDF document

Quantum Quantum Computing ? Computing ? Other PowerPoint talks on QC are can Other PowerPoint talks on QC are can Quantum Computing Quantum Computing be found at: be found at: Samuel J. Lomonaco, Jr. Samuel J. Lomonaco, Jr. Dept. of Comp. Sci. & Electrical Engineering Dept. of Comp. Sci. & Electrical Engineering University of Maryland Baltimore County University of Maryland Baltimore County http://www.csee.umbc.edu/~lomonaco/Lectures.html http://www.csee.umbc.edu/~lomonaco/Lectures.html Baltimore, MD 21250 Baltimore, MD 21250 Email: Email: Lomonaco@UMBC.EDU Lomonaco@UMBC.EDU WebPage: http://www.csee.umbc.edu/~lomonaco WebPage: http://www.csee.umbc.edu/~lomonaco L- -O O- -O O- -P P A A Adami, Barencol, Benioff, Bennett, Brassard, Calderbank, Chen, Crepeau, Deutsch, Rosetta Stone Rosetta Stone DiVincenzo, Ekert, Einstein, Feynman, Grover, for for Heisenberg, Jozsa, Knill, Laflamme, Lomonaco, Quantum Computation Quantum Computation Lloyd, Peres, Samuel J. Lomonaco, Jr. Samuel J. Lomonaco, Jr. Popescu, Preskill, Podolsky,Rosen, Schumacher, Shannon, Dept. of Comp. Sci. & Electrical Engineering Dept. of Comp. Sci. & Electrical Engineering Shor,Simon, Sloane, University of Maryland Baltimore County University of Maryland Baltimore County Baltimore, MD 21250 Baltimore, MD 21250 Schrodinger, Townsend, Unruh, Email: Email: Lomonaco@UMBC.EDU Lomonaco@UMBC.EDU WebPage: WebPage: http://www.csee.umbc.edu/~lomonaco http://www.csee.umbc.edu/~lomonaco L- -O O- -O O- -P P von Neumann, Vazirani, Wootters, Yao, Zeh, & many more Zurek 1

Quantum Computation and Information, Samuel J. Quantum Computation and Information Samuel J. Lomonaco, Samuel J., Jr., Lomonaco, Samuel J., Jr., A Rosetta stone A Rosetta stone Lomonaco, Jr. and Howard E. Brandt (editors), Lomonaco, Jr. and Howard E. Brandt (editors), AMS AMS for quantum mechanics with an introduction to for quantum mechanics with an introduction to CONM/305, (2002). CONM/305, (2002). quantum computation quantum computation, in AMS PSAPM/58, , in AMS PSAPM/58, (2002), pages 3 (2002), pages 3 – – 65. 65. ? ? ? Why ? ? ? ? ? ? Why ? ? ? Collision Collision Course Course Quantum Quantum Math Physics Computation Computation Quantum Quantum • Limits of small scale integration Limits of small scale integration Computation Computation technology to be reached 2010- technology to be reached 2010 -2020 2020 • No Longer ! No Longer ! Moore’s Law, i.e., Moore’s Law , i.e., Comp EE every year, double the computing power every year, double the computing power Sci at half the price. No Longer ! at half the price. No Longer ! Multi Multi- -Disciplinary Disciplinary • A whole new industry will be built around A whole new industry will be built around the new & emerging quantum technology the new & emerging quantum technology Classical Classical Shannon Shannon The The Bit Bit Classical Classical Decisive Individual World World 0 or 1 2

Classical Bits Classical Bits Can Can Be Be Copied Copied The The Out Quantum Quantum In Copying World World Machine Quantum Representations Quantum Representations Quantum Bit Quantum Bit Introducing Introducing of Qubits of Qubits Qubit Qubit the Qubit the Qubit 1 Example 1. A spin Example A spin- - particle particle ? ? ? ? ? ? 2 Indecisive Individual Spin Up Spin Down Can be both 0 & 1 1 0 at the same time !!! 1 0 H = Quantum Representations Quantum Representations Where does a Qubit live ? Where does a Qubit live ? of Qubits (Cont.) of Qubits (Cont.) Def. A Hilbert Space is a vector space over together with an inner H � Home product such that , : H H H H � − − − − × → Example 2. Polarization States of a Photon Example Polarization States of a Photon 1) u u v , u v , u v , & vu u , v u , vu , + + = = + + + = = + 1 2 1 2 1 2 1 2 2) u , v u v , λ λ = λ λ 1 = , 0 = 3) u v , v u , = or lim u ∈ H 4) ∀ Cauchy seq in , u u … , , H n 1 2 , n →∞ →∞ 1 = � 0 = ↔ = ↔ The elements of will be called The elements of will be called kets kets, and , and H will be denoted by label will be denoted by 3

Superposition of States Superposition of States Amplitudes A A Qubit Qubit is a is a quantum quantum A typical Qubit is ??? system system whose whose state state is is 0 1 = = α α + + α α represented by a Ket represented by a Ket 0 1 lying in a 2- lying in a 2 -D Hilbert D Hilbert 2 2 where 1 α α + + α α = = Space H 0 1 Space The above Qubit is in a Superposition Superposition of states 0 1 and It is simultaneously both and !!! 0 1 Kets as Column Vectors over � “Collapse” of the Wave Function “Collapse” of the Wave Function Kets as Column Vectors over H Let be a 2-D Hilbert space with orthonormal basis Qubit 0 , 1 0 1 α α + + α α = = 0 1 In this basis, each ket can be thought of as a Observer column vector. For example,   1   0 =   and =   0 1 0 1     And in general, we have   0   1  a  i a 0 b 1 a b ψ = = + + = + =       1 0 b       Tensor Product of Hilbert Spaces Tensor Product of Hilbert Spaces In other words, In other words, The tensor product of two Hilbert spaces H H H K K is constructed in the simplest non- ⊗ and is the “simplest” Hilbert space such K trivial way such that: that the map ( ( h k , ) ) � h k ⊗ H H K K H K × × → → ⊗  ( ( h h ) ) k h k h k + + ⊗ ⊗ = ⊗ + ⊗ is bilinear, i.e., such that 1 2 1 2  ⊗  ( ( h h ) ) k h k h k h ( ( k k ) ) h k h k + + ⊗ ⊗ = ⊗ + ⊗  ⊗ + + = ⊗ + ⊗ 1 2 1 2 1 2 1 2  ⊗  h ( ( k k ) ) h k h k ( ( ) ) ( ) � ( ) , �  h k h k h k ⊗ + + = ⊗ + ⊗  λ λ ⊗ ⊗ = = ⊗ λ λ λ ⊗ ∀ ∈ λ 1 2 1 2  ( ( ) ) ( ) h k h k  λ λ ⊗ ⊗ = = ⊗ λ λ We define the action of on as � H H K K ⊗ ( ( ) ) ( ) ( ) h k � h k h k λ λ ⊗ ⊗ λ λ ⊗ ⊗ = ⊗ λ 4