Introduction to Quantum Computing. Lecture 1

Transcription

1 Introduction to Quantum Computing Lecture 1 1

2 OUTLINE Why Quantum Computing? What is Quantum Computing? History Quantum Weirdness Quantum Properties Quantum Computation 2

3 Why Quantum Computing? 3

4 Transistors per chip Transistor Density? 10 7 Pentium Pro Pentium Year 4

5 Transistor Size Electrons per device 10 4 (4M) (16M) (64M) (256M) (1G) (4G) (Transistors per chip) (16G) ? 1 electron/transistor Year 5

6 Why Quantum Computing? By 2020 we will hit natural limits on the size of transistors Max out on the number of transistors per chip Reach the minimum size for transistors Reach the limit of speed for devices Eventually, all computing will be done using some sort of alternative structure DNA Cellular Automaton Quantum 6

7 What is Quantum Computing? 7

8 Introduction The common characteristic of any digital computer is that it stores bits Bits represent the state of some physical system Electronic computers use voltage levels to represent bits Quantum systems possess properties that allow the encoding of bits as physical states Direction of spin of an electron The direction of polarization of a photon The energy level of an excited atom 8

9 Spin States An electron is always in one of two spin states spin up the spin is parallel to the particle axis spin down the spin is antiparallel to the particle axis Notation: Spin up: Spin down: 9

10 qubit A qubit is a bit represented by a quantum system By convention: A qubit state 0 is the spin up state A qubit state 1 is the spin down state

11 Definitions A qubit is governed by the laws of quantum physics While a quantum system can be in one of a discrete set of states, it can also be in a blend of states called a superposition That is a qubit can be in: 0 1 c c 1 1 c c 1 2 = 1 11

12 Measurement If a qubit is realized by the spin of an electron, it is possible to measure the qubit value by passing the electron through a magnetic field If the qubit encodes a 0> then it will be deflected upward If the qubit encodes a 1> then it will be deflected downward 12

13 Superposition Measurement If the qubit is in a superposition state it cannot be determine if it will deflect up or down However, the probability of each possible deflection can be found Probability of 0 c 0 2 c c 1 1 Probability of 1 c

14 Quantum Computing History 14

15 History In the 1970 s s Fredkin, Toffoli, Bennett and others began to look into the possibility of reversible computation to avoid power loss. Since quantum mechanics is reversible, a possible link between computing and quantum devices was suggested Some early work on quantum computation occurred in the 80 s Benioff 1980,1982 explored a connection between quantum systems and a Turing machine Feynman 1982, 1986 suggested that quantum systems could simulate reversible digital circuits Deutsch 1985 defined a quantum level XOR mechanism 15

16 Existing Quantum Computers liquid NMR quantum computers with 2 12 qubit registers. Ion Trap method have achieved a single CONTROLLED NOT and 4 qubit entangled states linear optics, Superconductive Device 16

17 Quantum Weirdness 17

18 Weird Measurement One of the unusual features of Quantum Mechanics is the interaction between an event and its measurement Measurement changes the state of a quantum system Measurement of the superposition state of a qubit forces it into one of the qubit states in an unpredictable manner 18

19 Comparison I Compare qubits to classical bits Assumption Classical Quantum A bit always has a definite value True False, a qubit need not have a definite value until the moment after it is observed A bit can only be 0 or 1 True False, a qubit can be in a superposition of 0 and 1 simultaneously A bit can be copied without affecting its value A bit can be read without affecting its value True True False, a qubit in an unknown state cannot be copied without disrupting its state False, reading a qubit that is initially in a superposition will change the value of the qubit 19

20 Comparison II Assumption Classical Quantum Reading one bit has no effect on another unread bit True False, if the qubit being read is entangled with another qubit reading one will affect the other 20

21 Quantum Phenomena 21

22 Quantum Phenomena There are five quantum phenomena that make quantum computing weird Superposition Interference Entanglement Non-determinism Non-clonability 22

23 Superposition The Principal of Superposition states if a quantum system can be measured to be in one of a number of states then it can also exist in a blend of all its states simultaneously RESULT: An n-bit n qubit register can be in all 2 n states at once Massively parallel operations 23

24 Interference We see interference patterns when light shines through multiple slits This is a quantum phenomena which is also present in quantum computers A quantum computer can operate on several inputs at once, the results interfere with each other producing a collective result 24

25 Entanglement If two or more qubits are made to interact, they can emerge from the interaction in a joint quantum state which is different from any combination of the individual quantum states RESULT: If two entangled qubits are separated by any distance and one of them is measured then the other, at the same instant, enters a predictable state 25

26 Non-Determinism Quantum non-determinism refers to the condition of unpredictability If a quantum system is in a superposition state and then measured, the measured state can not be predicted. 26

27 Non-Clonability It is impossible to copy an unknown quantum state exactly If you asked a friend to prepare a qubit in a superposition state without telling you which superposition state, then you could not make a perfect copy of the qubit Useful in quantum cryptology 27

28 Quantum Computation 28

29 Quantum Computation Changes to a quantum state can be described using the language of quantum computation Single Qubit Gates Classical Not Gate - Truth table 0 1 and 1 0 Quantum Not Gate - Truth table 0 1 and

30 Quantum Computation Superposition of states? Not without further knowledge of the properties of quantum gates The quantum NOT gate acts LINEARLY α 0 + β 1 α 1 + β 0 Linear behaviour is a general property of quantum mechanics Non-linear behaviour can lead to apparent paradoxes - Time Travel - Faster than light communication - Violates the 2 nd Law of Thermodynamics 30

31 Quantum Computation NOT gate representation X we get for any α α 0 + β 1 β α 0 1 α β X or β 0 α 1 β = = β α to summarize α 0 + β 1 α 1 + β 0 31

32 Quantum Computation Are there any constraints on what matrices may be used as quantum gates? Of course! We require the normalization condition α β = 1 ψ = α 0 + β 1 for and the result ψ ' = α' 0 + β'1 after the gate has acted The appropriate condition for this (of course) is that the matrix representing the gate is UNITARY UU= I where U is the adjoint of That's it!!! Anything else is a valid quantum gate. U 32

33 Quantum Computation Two more important gates Z gate Z leaves 0 unchanged flips the sign of 1 to - 1 Hadamard Gate H ( + ) ( ) turns 0 into turns 1 into Note: Applying H twice to a state does nothing to it. H 2 = I 33

34 Quantum Computation Hadamard Gate: A most useful gate indeed! if H = 1 ( X + Z) and ψ = α 0 + β 1 then 2 H ψ = 1 ( X ψ + Z ψ ) 2 = α 1 0 α 1 β α 1 α + β β + = + = 0 1 β 2 α β 2 α β for for H H 1 0 α=1, β=0 H 0 = ( ) α = 0, β = 1 H 1 = ( ) 34

35 Quantum Computation Review: Important single-qubit gates α 0 + β 1 α 0 + β 1 α 0 + β 1 X Z H β 0 + α 1 α 0 β α + β

36 Quantum Computation Arbitrary Single Qubit Quantum Gate - complete set from properties of a much smaller set U β δ γ γ i 2 cos sin i 2 i e α e 0 = e β δ i γ γ i 2 sin cos 2 0 e 0 e 2 2 Global Phase Factor Rotation about z Rotation Scaling Constant α, βγ, and δare all real valued 36

37 Quantum Computation Classical Universal Gates (example) - The NAND gate is a classical Universal Gate. Why? NOT gate using NAND AND gate using NAND OR gate using NAND Universal Quantum Gates - An arbitrary quantum Computation on n qubits can be generated by a finite set of gates that are UNIVERSAL for quantum computation * Need to introduce some multiple quibit quantum gates 37

38 Multiple Qubit Gates Controlled-NOT (CNOT) Gate - two input qubits: : control and target A A B B A if control is 0 target left alone or else control is 1 target qubit is flipped or In General AB, AB, A 38

39 CNOT quantum gate A A U CN = B B A B0 A0 = 1 B 1 if A = 0 then we get B0 AB A = 0 0 A0 B 1 AB A B = B AB A1 B AB A0 = 0 0 if A = 1 then we get A1 = 1 B 0 B1 Any multiple qubit qubit logic gate may be composed from CNOT and Single Qubit Gates 39

40 Other Computational Bases Measurements ( ) ( 0 1 ) - In terms of + =, = basis states α + β α β ψ = α 0 + β 1 = α + β = Generally any basis state can represent an arbitrary qubit state ψ = α a + β b - If orthonormal then we can perform a measurement in keeping with probability interpretation 40

41 Quantum Circuits Elements of a Quantum Circuit - each line in a circuit represents a "wire" * passage of time * photon moving from one location to another - assume the state input is a computational basis state - input is usually the state consisting of all 0 s - no loops allowed ie: : acyclic - No FANIN(not reversible therefore not Unitary) - FANOUT (can't copy a qubit) 41

42 Quantum Circuits Quantum Qubit Swap Circuit ab, aa, b ( ) ( ) a a b, a b = b, a b b, a b b = b, a a b aa, b ba, b ba, x x 42

43 Controlled-U U Gate Quantum Circuits - A Controlled-U U Gate has one control qubit and n target qubits - where U is any unitary matrix acting on n qubits U 43

44 Quantum Circuits Measurement Operation - Converts a single qubit state into a probabilistic classical bit M ψ M 44

45 Quantum Circuits Can we make a Qubit Copying Circuit? - Copying a classical bit can be done with the Classical CNOT gate bit to be copied x x x x original bit 0 y x y x scratch-pad initialized to zero copied bit 45

46 Quantum Circuits Can we make a Qubit Copying Circuit? - How about copying a qubit in an unknown state using a controlled-cnot gate? ψ = a 0 + b 1 bit to be copied a 0 + b 1 Output State a 00 + b 10 0 a 00 + b 11 scratch-pad initialized to zero 46

47 Quantum Circuits Can we make a Qubit Copying Circuit? - Does ψ ψ = a 00 + b 11? ( )( ) 2 2 ψ ψ = a 0 + b 1 a 0 + b 1 = a 00 + ab 01 + ab 10 + b 11 - Unless ab = 0this does not copy the quantum state input 2 2 a 00 + ab 01 + ab 10 + b 11 a 00 + b 11 - It is impossible to make a copy of the unknown quantum state - NO CLONING THEOREM - 47

48 Quantum Circuits Bell States, EPR States, EPR Pairs x y H β xy In Out ( ) 2 β00 ( ) 2 β01 ( ) 2 β00 ( ) 2 β

49 Quantum Algorithms Initial State ( ) xy, xy, f x Final State Data Register x U f x y y f ( x) ψ Target Register xy, = 2 ψ 0, 0 f 0 + 1, 0 f 1 0, f 0 + 1, f 1 = = 2 2 ( ) ( ) ( ) ( ) 49

50 Quantum Algorithms Eureka!!!! Both values of the function show up in the final state solution. ψ = 0, f 0 + 1, f 1 ( ) ( ) 2 This can be generalized to functions on arbitrary number of bits using the HADAMARD TRANSFORM or WALSH-HADAMARD HADAMARD TRANSFORM 50

51 Quantum Algorithms Deutsch's Algorithm Circuit - Combines quantum parallelism and interference 0 H x x H 1 H U f y y f ( x) ψ 0 ψ 1 ψ 2 ψ 3 51

52 Quantum Algorithms Deutsch's Algorithm Calculations - Combines quantum parallelism and interference ψ 0 = 01 ψ ψ ψ = ± if f 0 = f ψ = ± if f 0 f ( ) ( ) ( ) ( ) 52

53 Quantum Algorithms Deutsch's Algorithm Conclusion ψ 0 1 ± 0 if f 0 = f 1 2 ψ = 0 1 ± 1 if f 0 f ( ) ( ) ( ) ( ) realizing f ( 0) f ( 1) is 0 if f ( 0) = f ( 1) and 1 otherwise ψ 3 f 0 f =± ( ) ( ) 2 measuring the 1 st qubit gives f ( 0) f ( 1) 53

54 Quantum Algorithms Deutsch's Algorithm Results - The quantum circuit has given us the ability to determine a GLOBAL PROPERTY of f ( x) namely f ( 0) f ( 1) using only ONE evaluation of f ( x) - A classical computer would require at least two evaluations! - Difference between quantum parallelism and classical randomized algorithms ( ) ( ) * One might think the state 0 f f 1 corresponds to probabilistic classical computer that evaluates f ( 0) with probability 1/2 or f () 1 with probability ½. These are classically mutually exclusive. * Quantum mechanically these two alternatives can INTERFERE to yield some global property of the function f and by using a Hadamard gate can recombine the different alternatives 54

55 Quantum Algorithms Deutsch-Jozsa Algorithm - A simple case of a more general algorithm - Application is called Deutsch's Problem x is a number from 0 to 2 n -1 Alice x n bits each time Bob f ( x) Constant for all values of x Balanced: 1 for 1/ 2 the values of x or 0 otherwise - Classically Alice can only send one value of x each time - Best classical algorithm requires up to 2 n /2+ 1queries n 2 /2 0 's and one 1 Balanced 55

56 ψ 0 ψ 1 ψ 2 ψ 3 Quantum Algorithms Deutsch-Jozsa Algorithm - If Bob and Alice were able to exchange qubits instead of classical bits and if Bob calculated f(x) using a unitary transform U f then Alice could determine the function in one query. - Alice has an n qubit register and a single qubit register which she gives to Bob - Prepares query and answer register in a superposition state - Bob evaluates f(x) ) and puts result into answer register - Alice interferes the states in the superposition using a hadamard transform on the query register 56

57 Quantum Algorithms Deutsch-Jozsa Algorithm Circuit 0 n n H x x n H U f 1 H y y f ( x) ψ 0 ψ 1 ψ 2 ψ 3 ψ n x = 0 1 ψ = n n x { 0,1} ψ 1 2 { 0,1} f ( ) ( x ) 1 x 0 1 ψ = n n x 2 2 Bob's function evaluation is stored in the amplitude 57

58 Hadamard transform: helps to calculate effect on a state x By checking the cases x=0 and x=1 separately for a single qubit thus Quantum Algorithms Deutsch-Jozsa Algorithm - detour H x = ( 1) where is the bitwise inner product of x and z, modulo 2 z xz z 2 ( ) 1 1 n H x,..., x = 1 x z xz x z 1 n n n n z1 zn n 1,..., H n x = z 1 ( ) x z z 2 n z,..., 2 z 58

59 Quantum Algorithms Deutsch-Jozsa Algorithm Circuit ψ amplitude for is ( ) 1 z 0 1 ψ = n z x n Case 1: If f is constant the amplitude for is +1 or -1 depending on the constant value f(x) ) takes. Since ψ 3 is unit length then all other amplitudes must be zero. - An observation will yield 0s for all qubits in the register ( ) x z+ f x query register f ( ) ( x ) x 1 2 n 0 n 59

60 Quantum Algorithms Deutsch-Jozsa Algorithm Circuit Case 2: If f is balanced then the positive and negative contributions to the amplitude for 0 n cancel, leaving an amplitude of 0 - A measurement must yield a result other than 0 on at least one qubit Summary: - If Alice measures all zeros then the function is constant - Otherwise the function is balanced. - Deutsch's problem on a quantum computer can be solved in one evaluation. 60

61 Quantum Algorithms Other Quantum Algorithms - Generally there are three classes * Discrete Fourier Transform Algorithms ~Deutsch-Jozsa Algorithm ~Shor's Algorithm for Factoring ~Shor's Discrete Logarithm Algorithm * Quantum Search Algorithms * Quantum Simulation Algorithms ~Quantum Computer is used to simulate quantum systems 61

62 Experimental Quantum Information Processing The Stern-Gerlach Experiment Optical Techniques Nuclear Magnetic Resonance Quantum Dots Traps: Ion Traps & Neutral Atom Traps 62

63 NMR Quantum Computing Lecture 2 63

64 Nuclear Magnetic Resonance Quantum Computers Qubit representation: spin of an atomic nucleus Unitary evolution: using magnetic field pulses applied to spins in a strong magnetic field. Chemical bonds between atoms couple the spins State preparation: using a strong magnetic field to polarize the spins Readout: using magnetic-moment induced free induction decay signals 64

65 Nuclear Magnetic Resonance Q.C. Physical Apparatus pre - amplifier RF -source Computer Liquid sample 12 C, RF - coil 19 F, 15 N, 31 P B =11.8 Tesla (uniform to 1 part in 10 9 ) Regard as an ensemble of n-bit quantum computers amplifier Typical Experiment Wait a few minutes for the sample to come to thermal equilibrium 2. Send RF pulses to manipulate nuclear spins into desired state. 3. Switch off the amps and switch on the preamplifier to measure the free-induction decay 65

66 Nuclear Magnetic Resonance Q.C. Spectrometer Physical Apparatus Nuclear Spins as qubits ADC for data acquisition RF synthesizer and amplifier Gradient control 0 1 B wave guides sample test tube 9.6 T I J IS S RF Wave RF wave High field magnet 2-3 Dibromothiophene 66

67 Internal Hamiltonian The evolution of a spin system is generated by Hamiltonians Internal Hamiltonian: H int =ω I I z +ω S S z +2π J IS I z S z interaction with B field 9.6 T I J IS S spin-spin coupling 2-3 Dibromothiophene 67

68 External Hamiltonian Experimentally Controlled Hamiltonian: H ext (t) =ω RFx (t) (I (I x +S x )+ spins couple to RF field )+ω RFy (t) (t) (I(I y +S y ) Total Hamiltonian: H total (t) = H int + H ext (t) H total (t (t) controlled via H ext (t) 9.6 T I J IS S RF wave 2-3 Dibromothiophene 68

69 Tomography Not all elements of the density matrix are observable on an NMR spectra. 2 σ x σ 2 3 x σ z To observe the other elements of the density matrix requires repeating the experiment 7 times with readout pulses appended to the pulse program. This is done without changing any other parameters of the pulse program. 69

70 One Example of NMR QC: Quantum Games: theoretical and experimental results 70

71 Outline Introduction of quantum games Classical game: Prisoner s s Dilemma Maximal entangled quantum game Some of our results Theoretical extensions with non-maximal entanglement, more players, larger strategy space, and so on. Experimental realization of quantum game Future Plan and discussion 71

72 Game theory Prisoner's dilemma --an important branch of applied mathematics. It is the theory of decision-making and conflict between different agents. Since the seminal book of Von Neumann and Morgenstern, modern game theory has found applications ranging from economics through to biology. It concludes: Players, Strategy space, Payoff function Classifications: Time (static & Dynamic). Information (complete &incomplete) Prisoner s s Dilemma --a a famous game in game theory. 72

73 Table: Payoff matrix for the Prisoner's Dilemma. The first entry in the parenthesis denotes the payoff of Alice and the second to Bob's. Alice C D C (3,3) (5,0) Bob D (0,5) (1,1) Nash Equilibrium: mutual defect (D,D) Nash Equilibrium implies that no player can increase his payoff by unilaterally changing his strategy. Pareto optimal: mutual cooperation (C,C) A pair of strategies is called pareto optimal if it is not possible to increase one player s s payoff without lessening the payoff of the other player. Prisoner s s Dilemma: Nash Equilibrium strategy profile is not equivalent to Pareto optimal 73

74 Maximal entangled quantum game Quantum game theory Recently, new effect involving quantum information on has been discovered theoritically in the area of game theory by some pioneers. 1. L.Goldenberg, L.Vaidman, S.Wiesner, Phys.Rev.Lett.. 82, 3356 (1999). 2. D.A.Meyer, Phys.Rev.Lett.. 82, 1052 (1999). 3. J.Eisert, M.Wilkens, M.Lewenstein, Phys.Rev.Lett.. 83, 3077 (1999). Maximal entangled quantum game Eisert et al. showed that the classical problem of Prisoner's Dilemma is a subset of the quantum game by using a physical model of the quantum game,, and there is no longer a dilemma when employ a maximally entangled game. 74

75 Interesting results For a separable game with γ =0, there exists a pair of quantum strategies (D,D) is a Nash Equilibrium and yields payoff (1,1) which is not Pareto optimal. Indeed, this quantum game behaves classically. For a maximally entangled quantum game π γ = 2 with, (Q,Q) is the Nash equilibrium of the game and has the property to be Pareto optimal. So Prisoner s s Dilemma is removed if quantum strategies are allowed for. 75

76 Correlation between quantum game and quantum entanglement Yet it is legitimate for us to ask: --Whether a quantum game will still outperform its classical version if it is not maximally entangled? and how a quantum game depends on the entanglement of the game's state? 76

77 A physical model of quantum game J. Eisert have proposed a physical model of this game and the elegant quantum network is illustrated as: U A and U B are the strategy moves available to the players: iφ ( ) = e cos θ sin θ U θ, φ 2 2 Unitary operator J { i D 2} sin θ 2 e iφ cos θ = exp γ D/ D= Uπ (,0) = ^ 77

78 Two player s s initial state is ( ) ( )11 ψi = J 00 = cos γ isin γ 2 The entanglement of the game's initial state can be denoted as γ γ γ γ sin therefore, can be denoted as a measure for the entanglement. The final state is ψ = J + U U J f 2 ln sin cos 2 ( ) 00 Then the expected payoff for Alice and Bob are A B 2 ln cos γ ξ Α = 3P00 + 5P10 + P11 2 ξ B = 3P00 + 5P01 + P Pij = ij ψ f

79 Nash Equilibrium: Uˆ A Uˆ B Theoretical Results Dˆ Dˆ, 0 γ γ th1 Dˆ Qˆ, γ th1 γ γ th2 =, Q Dˆ, γ th1 γ γ th2 Qˆ Qˆ, γ th2 γ π 2 There exist two threshold: γ sin ( 1 5 ) γ sin ( 2 5 ) th 1 = Arc Dˆ 0 = 1 th = 2 Arc 1, 0 i Qˆ = 0 Expected payoff as game s s entanglement varies 0 i 79

80 Other Theoretical Results Differnet sets of strategies. J. Du et al., Physics Letter A, 289 (2001) 9 Multi players more than 2-player. J. Du et al., Physics Letter A, 302 (2002) 229 Phase-transition-like behavior of quantum games J. Du et al., Journal of Physics A: Mathematical and General 36, (2003). One Review J. Du et al., Fluctuation and Noise Letters Vol 2, Iss 4, R189-R203. Quantum games in econophysics H. Li, J. Du and S. Massar, Physics Letter A, 306 (2002) 73 J. Du et al., Physics Review E 68, (2003) 80

81 Experimental realization Physics Review Letter 88,, ( ) Technologies for quantum information processing(qip) -There are a number of proposed device technologies for QIP. --Of them, NMR have given the many successful results experimentally for QIP, such as quantum teleportation, quantum error correction, quantum simulation, quantum algorithm etc. We add game theory to the list: Quantum games was experimental realized on nuclear magnetic resonance quantum computer. 81

82 Qubits Two-qubit qubit: : Nuclear Coupled Spins Partially deuterated cytosine molecule contains two protons, in a magnetic field, each spin state of proton could be used as a qubit. Distinguish each qubit Different Larmor frequencies (the chemical shift) enable us to address each qubit individually. Quantum logic gates Radio Frequency (RF) fields and spin-- --spin couplings between the nuclei are used to implement quantum logic gates. 82

83 Quantum network and gates Quantum network Entangled gate: ˆ nπ J = exp { iγd D / 2}, γ =, n = {0,1,... 18} The strategy moves U A and U B are Uˆ A Uˆ B Dˆ Dˆ, Dˆ Qˆ, = Q Dˆ, Qˆ Qˆ, 0 n 5 6 n 7, 6 n 7 7 n 18 i Dˆ =, Qˆ = i Each gate can be realized by NMR technique

84 Experiments for quantum game Experimentally, we performed nineteen separate sets of experiments which was distinguished by: ˆ nπ J = exp iγd D / 2, γ =, n = {0,1, { } } In each set, the full process of the game was executed. 1. Create an effective pure state 2. Prepare the initial entangled state by applying gate J 3. Players Alice and Bob executed their strategic moves U A and U B 4. Apply the unentangled gate J + 5. Measure the final state and calculate the expected payoff

85 NMR Spectrometer 85

86 Experimental results The player Alice's payoffs as a function of the parameter γ. It is easy to see that γ = 0 (n=0) corresponds to Eisert et al.'s separable game and γ = π 2 (n=18) corresponds to their maximally entangled quantum game. 86

87 Good agreement between theory and experiment. Experimental Error: --an estimated error is less than 0.08, the errors are primarily due to inhomogeneity of magnetic field, imperfect RF selective pulses, and the variability over time of the mesurement process. Decoherence: --each experiment took less than 300 milliseconds, which was well within the the decoherence time (3 seconds). This experiment was referred by : Physics News update (APS), Physics web (IOP), New Scientist, Science Update (Nature).( Physics world Experimental results 87

88 Nature Science Update 88

89 2001.9: APS- Physics News Update 89

90 New Scientists 90

91 Thanks 91