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Introduction to Quantum Information Processing CS 467 / CS 667 Phys 667 / Phys 767 C&O 481 / C&O 681 Lecture 1 (2005) Richard Cleve DC 3524 [email protected] 1 Overview 2 Moore’s Law 109 number of transistors 108 107 106 105 year 104 1975 1980 1985 1990 1995 2000 2005 Following trend … atomic scale in 15-20 years Quantum mechanical effects occur at this scale: • Measuring a state (e.g. position) disturbs it • Quantum systems sometimes seem to behave as if they are in several states at once • Different evolutions can interfere with each other 3 Quantum mechanical effects Additional nuisances to overcome? or New types of behavior to make use of? [Shor, 1994]: polynomial-time algorithm for factoring integers on a quantum computer This could be used to break most of the existing public-key cryptosystems on the internet, such as RSA 4 Quantum algorithms Classical deterministic: START FINISH Classical probabilistic: p1 p2 START p3 FINISH Quantum: α1 α2 START α3 FINISH POSITIVE NEGATIVE 5 Also with quantum information: • Faster algorithms for combinatorial search [Grover ’96] • Unbreakable codes with short keys [Bennett, Brassard ’84] • Communication savings in distributed systems [C, Buhrman ’97] • More efficient “proof systems” [Watrous ’99] … and an extensive quantum information theory arises, which generalizes classical information theory For example: a theory of quantum error-correcting codes quantum information theory classical information theory 6 This course covers the basics of quantum information processing Topics include: • Quantum algorithms and complexity theory • Quantum information theory • Quantum error-correcting codes* • Physical implementations* • Quantum cryptography • Quantum nonlocality and communication complexity * Jonathan Baugh 7 General course information Background: • classical algorithms and complexity • linear algebra • probability theory Evaluation: • 3 assignments (15% each) • midterm exam (20%) • written project (35%) Recommended text: “Quantum Computation and Quantum Information” by Nielsen and Chuang (available at the UW Bookstore) 8 Basic framework of quantum information 9 Types of information is quantum information digital or analog? probabilistic digital: 1 1 analog: 1 r 0 0 1 0 p q • Probabilities p, q 0, p + q = 1 • Cannot explicitly extract p and q (only statistical inference) • In any concrete setting, explicit state is 0 or 1 • Issue of precision (imperfect ok) 0 r [0,1] • Can explicitly extract r • Issue of precision for setting & reading state • Precision need not be perfect to be useful 10 Quantum (digital) information 0 1 1 0 0 1 • Amplitudes , C, ||2 + ||2 = 1 • Explicit state is α β • Cannot explicitly extract and (only statistical inference) • Issue of precision (imperfect ok) 11 Dirac bra/ket notation Ket: ψ always denotes a column vector, e.g. 1 Convention: 0 0 0 1 1 1 2 d Bra: ψ always denotes a row vector that is the conjugate transpose of ψ, e.g. [ *1 *2 *d ] Bracket: φψ denotes φψ, the inner product of φ and ψ 12 Basic operations on qubits (I) (0) Initialize qubit to |0 or to |1 † (1) Apply a unitary operation U (U U = I ) Examples: cos sin Rotation: sin cos 1 Hadamard: H 2 1 1 1 1 0 1 NOT (bit flip): x X 1 0 1 0 Phase flip: z Z 0 1 13 Basic operations on qubits (II) (3) Apply a “standard” measurement: 0 + 1 0 with prob 2 1 with prob 2 ψ′ 1 ψ ||2 0 ||2 … and the quantum state collapses () There exist other quantum operations, but they can all be “simulated” by the aforementioned types Example: measurement with respect to a different orthonormal basis {ψ, ψ′} 14 Distinguishing between two states Let be in state 1 2 0 1 or 1 2 0 1 Question 1: can we distinguish between the two cases? Distinguishing procedure: 1. apply H 2. measure This works because H + = 0 and H − = 1 Question 2: can we distinguish between 0 and +? Since they’re not orthogonal, they cannot be perfectly distinguished … 15 n-qubit systems Probabilistic states: p000 p 001 x, px 0 p010 p 011 px 1 p100 x p101 p 110 p111 Quantum states: x, αx C α x 2 x 1 α000 α 001 α010 α 011 α100 α101 α 110 α111 Dirac notation: |000, |001, |010, …, |111 are basis vectors, so ψ αx x x 16 Operations on n-qubit states Unitary operations: † (U U = I ) Measurements: α x x x αx x x α 000 α 001 α111 U α x x x ? 000 with prob α000 2 with prob α001 2 001 111 2 with prob α111 … and the quantum state collapses 17 Entanglement Product state (tensor/Kronecker product): α 0 β 1 α' 0 β' 1 αα' 00 αβ ' 01 βα' 10 ββ' 11 Example of an entangled state: 1 2 00 1 2 11 … can exhibit interesting “nonlocal” correlations: 18 Structure among subsystems qubits: #1 time U W #2 V #3 #4 unitary operations measurements 19 Quantum computations Quantum circuits: 0 1 1 0 1 1 0 0 1 1 0 1 “Feasible” if circuit-size scales polynomially 20 Example of a one-qubit gate applied to a two-qubit system (do nothing) u00 u01 U u u 10 11 U The resulting 4x4 matrix is Maps basis states as: 00 0U0 01 0U1 10 1U0 11 1U1 0 u00 u01 0 u u 0 0 I U 10 11 0 0 u00 u01 0 u10 u11 0 21 Controlled-U gates U u00 u01 U u u 10 11 Resulting 4x4 matrix is controlled-U = Maps basis states as: 00 00 01 01 10 1U0 11 1U1 1 0 0 0 0 0 0 1 0 0 0 u00 u01 0 u10 u11 22 Controlled-NOT (CNOT) a a b ab ≡ X Note: “control” qubit may change on some input states 0 + 1 0 − 1 0 − 1 0 − 1 23 24