Quantum Information Science. Jaewan Kim School of Computational Sciences Korea Institute for Advanced Study Seoul, Korea

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1 Quantum Information Science Jaewan Kim School of Computational Sciences Korea Institute for Advanced Study Seoul, Korea

2 Quantum Physics Information Science Quantum Information Science Quantum Parallelism Quantum computing is exponentially larger and faster than digital computing. Quantum Fourier Transform, Quantum Database Search, Quantum Many-Body Simulation (Nanotechnology) No Clonability of Quantum Information, Irrevesability of Quantum Measurement Quantum Cryptography (Absolutely secure digital communication) Quantum Correlation by Quantum Entanglement Quantum Teleportation, Quantum Superdense Coding, Quantum Cryptography, Quantum Imaging

3 Quantum Information Processing Quantum Computer Quantum Algorithms: Softwares Simulation of quantum many-body systems Factoring large integers Database search Experiments: Hardwares Ion Traps NMR Cavity QED, etc. NT IT BT

4 Quantum Information Processing Quantum Communication Quantum Cryptography Absolutely secure digital communication Generation and Distribution of Quantum Key optical fiber wireless secure satellite communication Quantum Teleportation Photons Atoms, Molecules Quantum Imaging and Quantum Metrology IT

5 Bit and Logic Gate NOT A N D OR XOR NAND C-NOT CCN ( Toffoli ) C-Exch ( Fredkin ) Universal Reversible Universal Reversible

6 Classical Computation Hilbert (1900): 23 most challenging math problems Is there a mechanical procedure by which the truth or falsity of any mathematical conjecture could be decided? T u r i n g Conjecture ~ Sequence of 0 s and 1 s Read/Write Head: Logic Gates Model of Modern Computers

7 Turing Machine Finite State Machine: Head q q q = s = d = f( q, s) g( q, s) d( q, s) d Bit {0,1} s s' Infinitely long tape: Storage

8 Quantum Information {0,1} Bit Qubit N bits 2 N states, One at a time Linearly parallel computing AT BEST N qubits Linear superposition of 2 N states at the same time Exponentially parallel computing Quantum Parallelism But when you extract result, you cannot get all of them. a 0 + b1 with a + b = 1 Deutsch 2 2

9 Quantum Algorithms 1. [Feynman] Simulation of Quantum Physical Systems with HUGE Hilber space ( 2 N -D ) e.g. Strongly Correlated Electron Systems 2. [Peter Shor] Factoring large integers, period finding t q Pol (N) t cl Exp (N 1/3 ) 3. [Grover] Searching t q N t cl ~ N/2

10 N bits N bits Digital Computer Digital Computers in parallel m{ 1 N m N N bits N bits Quantum Computer : Quantum Parallelism N bits N bits

11 ? No Male F No Female

12 ? No Male F No Young O No Female M

15 M Y F Female / Male vs. Young / Old O F No Young No Male No Female

16 Y F O M F No Young O No Male M F Y O No Female F O M

18 양자암호통신

23 ψ ψ Simplest form of quantum uncertainty principle

24 1 0 = H 0 = D Bloch sphere z 1 0 = 0 θ 0 1 = 0 ϕ y H = 2 1 x = 1 Pure states on the surface of a Bloch sphere

25 Quantum Gates Time-dependent Schrödinger Eq. i ψ = H ψ t iht / ψ( t) = e ψ(0) = U( t) ψ(0) P ( ) = i 0 I θ θ e 0 1 X = = i Z =P( π )= Y = XZ= or ixz = i H = Hadamard H 0 = = = ( ) H 1 = = = ( 0 1 ) Unitary Transform Norm Preserving Reversible

26 Hadamard Gate H1 H2... HN N = N = ( 0 ) N N N N 2 N = k( binary expression) N 2 k = 0 ( 0 1 ) ( 0 1 )... ( 0 1 ) N

27 Universal Quantum Gates General Rotation of a Single Qubit θ iφ θ cos ie sin 2 2 V( θφ, ) = + iφ θ θ ie sin cos 2 2 X c : CNOT (controlled - NOT) or XOR = 0 0 I X V X a b = a a b c

28 Scalable Qubits Initial State Quantum Network DiVincenzo, Qu-Ph/ Unitary evolution : Deterministic : Reversible Universal Gates Ψ H t X X 1 H 2C X 13 Ψ M M M Cohere, Not Decohere Quantum measurement : Probabilistic : Irreversible change

29 Quantum Key Distribution [BB84,B92] Single-Qubit Gates ~ 3 QKD[E91] Quantum Repeater Quantum Teleportation Quantum Error Correction Quantum Computer Single- & Two-Qubit Gates Single- & Two-Qubit Gates ~ 3

30 Physical systems actively considered for quantum computer implementation Liquid-state NMR NMR spin lattices I o n - t ra p Neutral-atom optical lattices Cavity QED + atoms Linear optics with single photons Nitrogen vacancies in diamond Electrons on liquid He Josephson junctions charge qubits flux qubits Spin spectroscopies, impurities in semiconductors Coupled quantum dots Qubits: spin,charge, excitons Exchange coupled, cavity coupled

31 Chuang Nature 414, (20/27 Dec 2001) OR QP/

32 Quantum-dot array proposal

33 Ion Traps Couple lowest centre-of-mass modes to internal electronic states of N ions.

34 Quantum Error Correcting Code Three Bit Code Encode Recover Decode φ 0 0 channel noise 0 0 M M U m 1m2 φ

35 Cryptography and QIP Giving disease (Q Comp), Giving medicine (QKD). Public Key Cryptosystem (Asymmetric) Computationally Secure Based on unproven mathematical conjectures Cursed by Quantum Computation One-Time Pad (Symmetric) Unconditionally Secure Impractical Saved by Quantum Cryptography

37 Quantum Key Distribution is a major paradigm shift in the development of cryptography. Conventional and quantum cryptography are a powerful combination in making a secure communications a reality. -Burt Kaliski, Chief Scientist, RSA Laboratories

38 [Symmetric] One-Time pad Alice Bob Tell me the password. Pass word + key = Eve key = pass word

39 [Symmetric] One-Time Pad Alice M = K = E = Vernam Bob E = K = M = Unconditionally Secure Impractical: Generation and Distribution (random) (random)

40 KEY Cypher T e x t Encoding M K: 0 0=0, 0 1=1, 1 0=1, 1 1= KEY

41 [Symmetric] Why is it called One-Time? E1 = M1 K E2 = M2 K E1 E2 = ( M1 K ) ( M2 K ) = M1 M2 ( K K ) = M1 M2 ( 0 ) = M1 M2

42 [Asymmetric] Public Key Cryptosystem Diffie, Hellman RSA : Rivest, Shamir, Adleman Alice1, Alice2, Message [m] Encryption Encryption Key [k] Computationally Secure Eve Cryptogram [c] Bob Message [m] Decryption Decryption Key [d=f(k)] Could be broken, especially by Quantum Computers

43 [Asymmetric] RSA Cryptosystem Message M n = p q, e d = 1 mod (p-1)(q-1) n, e : Public Key (Encryption) n, d : Secret Key (Decryption) Alice : E = M e mod n Bob : M = E d mod n M Message

44 RSA Example Bob s Keys p = 11, q = 13, n = p q = 143 d = 103, e = 7 d e mod (p-1)(q-1) = 103 x 7 mod 120 = 1 Alice s Message: M = 9 E = M e mod n = 9 7 mod 143 = 48 Bob M=E d mod n = mod 143 = 9

45 How Hard is Factoring? Almost exponentially complex with the number of digits, L RSA129 (1977) Factored 17 years later using 1,600 computers 2,000 Digit Number Impossible to factor even Vazirani With as many digital computers as the number of particles in the Universe (10 80 ) In as long time as the age of the universe (10 18 sec)

47 No Cloning Theorem An Unknown Quantum State Cannot Be Cloned. <Proof> U ( α 0 ) ( 0 ) = α α Zurek, Wootters Diks U β = β β α β 1 Let γ = ( α + β ). 2 1 Then U ( γ 0 ) = ( α α + β β ) γ γ 2

48 Controlled-NOT XOR X X = 0 0 I X AB AA B AA B = = AB AB = A B A B AB

49 Qubit Copying Circuit? x x x x 0 y x y x ψ = a 0 + b 1 0 ψ = a 0 + b 1 a 00 + b 11 ψ = a 0 + b 1

50 IF an unknow quantum state CAN be cloned Quantum States can be measured as accurately as possible??? ψ ψ, ψ, ψ, ψ measure, measure, Communication Faster Than Light? ψ = 1 A 0 B - 0 A 1 B for 0 = + A - B - - A + B for 1

51 Communication Faster Than Light when unknown quantum state can be copied. Alice wants to send Bob 1. Alice mesures her qubit in { +>, ->}. Alice s state will become +> or ->. Bob s state will become -> or +>. Let s assume it is +>. Bob makes many copies of this. He measures them in { +>, ->}, and gets 100% +>. He measures them in { 0>, 1>}, and gets 50% 0> and 50% 1>. Thus Bob can conclude that Alice measured her state in { +>, ->}.

52 Mysterious Connection Between QM & Relativity Weinberg: Can QM be nonlinear? Experiments: Not so positive result. Polchinski, Gisin: If QM is nonlinear, communication faster than light is possible.

53 Irreversability of Quantum Measurement If you describe it as that, that is not it anymore - 老子 - ψ = a 0 + b 1 = c 0' + d 1' Measure in { 0, 1 } 0 or 1 Measure in { 0', 1' } 0' or 1'

54 Quantum Cryptography Quantum Key Distribution BB84, B92, E91 No Cloning & Irreversability of Quantum Measurement Alice EVE (eavesdropper) Single Photon, Ent., Coh. Bob 1. Interconvertibility between stationary and flying qubits. 2. Faithful transmission of flying qubits. DiVincenzo, Qu-Ph/

55 BB84: 4 Polarizations 0 o,90 o 0 o,45 o 0 o,45 o D1 PC1 PC2 Laser PC3 P B S A1 A2 B Alice A1 = A2 = P = Bob B = D = D0

56 BB84 full implementation Authenticated Public Discussion Mesurement in random basis Y e s Same basis? No Null result or different basis Discard Transmission of raw data: Random coding on single photons Initial Authentication NEEDED Check random sample Error < Q% Y e s Reconciliation of bit strings Parity check n bits Secret key s bits No Stop Estimate information leakage k bits Privacy Amplification Keep the parities

57 World s first QKD (1989): Bennett(IBM) et al. 32 cm in free space with 4 polarizations

58 KIAS-ETRI, December km Quantum Cryptography

62 Deutsch Algorithm Global property of f

63 Quantum Fourier Transform ~n 2 =(log 2 N) 2 vs N log 2 N ***can be used only as a subroutine, used for factoring

64 Molding a Quantum State Ψ H t X X 1 H 2C X 13 Ψ M M M Molding

65 Sculpturing a Quantum State - Cluster State Quantum Computing Initialize each qubit in + state. 2. Contolled-Phase between the neighboring qubits. 3. Single qubit manipulations and single qubit measurements only [Sculpturing]. No two qubit operations!

66 Entanglement EPR & Nonlocality Alice Bob 1 2 ( ) ψ A B A B A B φ Local Hidden Variable Bell s Inequality Aspect s Experiment Quantum Mechanics is nonlocal! 1 GHZ State: ( + ) A B C A B C

67 Bell s inequality, CHSH inequality (Classical) Violation of inequality (Quantum)

68 Alice x Entanglement Quantum correlation Bob y Classical physics: x and y are decided when picked up. 0 1 or 1 0 A B A B Quantum physics: x and y are decided when measured. {0,1} basis 0 or 1 1 {+,-} basis + or - Ψ = ( A B A B) 2 1 = ( + + A B A B) = 0 + 1, = ( ) ( )

69 Entanglement and Two-Qubit Gate a 0 + b 1 0 a 0 + b a b 1 1 a b 1 1 1

70 Classical Teleportation? Alice X a m m 1 bit Bob b m 1 X 1 bit Correlated P a i r X = 0 or 1 1 bit a 1 bit {0,1} b 1 bit x a m b = x 1 1 = x 1

71 Quantum Teleportation Transportation Continuous movement through space P P Quantum Teleportation B 1, B 2 X A BX

72 Quantum Teleportation Transmit an unknown qubit Alice Bell State Measurement Φ ±, Ψ ± without sending the qubit 2 Bits Classical Information Entangled Pair unknown Ω = α 0 + β 1 Bob Unitary Transform I, Z, X, Y Bennett et al. Ω = α 0 + β 1 E P R - S o u r c e

76 θ = ο X A B

77 θ = ο X A B

78 θ = ο X A B

79 θ = ο Bell Measurement 0 Leave it as it is. X A B

80 θ = ο Bell Measurement 1 Rotate it by -90 o. X A B

81 θ = ο X A B

82 Single Particle Entanglement 1 Single Photon Single Electron ( + + H = t c1 c2 + c2c1 ) 2 Beam Splitter 2 1 ψ = ( ) 2 1 or ψ = f e + e f 2 ( ) φ = + 2 ( )

83 Quantum Teleportation using Single Particle Entanglement Lee, JK, Qu-ph/ ; Phys. Rev. A 63, (2001)

84 PRL88, (2002) or QP/ "Active" Teleportation of a Quantum Bit S. Giacomini, F. Sciarrino, E. Lombardi and F. De Martini

85 Quantum Teleportation of Single Particle Entanglement Lee, Qu-ph/ ; Phys. Rev. A 64, (2001)

86 Quantum Superdense Coding Transmit two bits by sending one qubit Alice I, X, Y, Z 1 qubit Entanglement Bob X H AB A 2 bit A B EPR

87 Quantum Superdense Coding I 0 0 A: = (0 + 1) 0 X : A = (1 + 0) : : A B A B XAB HA YA : = (1 0)1 11 Z A: = (0 1) Alice s manipulation Alice sends the qubit to Bob 1 qubit Bob s manipulation 4 states differentiated 2 bit

88 Tripartite Qubit Entanglement Z-basis: 0 with 1, 1 with + 1 X-basis: a with 1, b 0 1 with a + b, 1 a b Y-basis: c 0 + i 1 with 1, d 0 i 1 with + 1 ( ) 0 c + d, 1 i c b Not expectation values A GHZ 1 = { } : Z-basis 2 ( a + b)( a + b)( a + b) + ( a b)( a b)( a b) aaa + abb + bab + bba : xxx = 1 B C xxx xyy yxy yyx = ( a + b)( c + d)( c + d) ( a b)( c d)( c d) acd + adc + bcc + bdd : xyy =+ 1 = 1 =+ 1 = ( ) ( ) ( ) = A B C x y x y x y 1?

Bipartite Qubit Entanglement Not expectation values counterfactual argument 2 2 2 Ψ = a 00 + b 01 + c 10 + d 11 with a + b + c = 1 2 2 = 0 00 Ψ = 0 = a 0 0 + b 0 1 00 Ψ = 0= a 0 0 + c 0 1 2 2 p = 00

89 Bipartite Qubit Entanglement Not expectation values counterfactual argument Ψ = a 00 + b 01 + c 10 + d 11 with a + b + c = = 0 00 Ψ = 0 = a b Ψ = 0= a c p = 00 Ψ = a A 0 0 B 0 0 = A B Maximize p! p p A B B A A B A B A = pa = pb p = = p + p p = p + p p = p A A B 1 2 A A B 0 B papb(1 pa)(1 pb) 1 p p 1± pa = pb = 1, pa = pb = = g: golden mean p = g 9 % A B

92 Zurek, QP/

93 j 1 = 3 + R 4 2 ( ) ìé ï 1 1 ù 3 ïê 2 2 rlr = j j = í ï ê 4 ïê ï 1 1 ïë ïî ê 2 2 ú û R R 4 4 R

94 Parametric Down Conversion : Splitting a photon into two w w 2 w w 2

95 j 1 = r = j j LR ( ) L R L R w 3 L R 4 L 4 R C K Hong and T Noh (1998) Y-H Kim and Y Shi (2000)

96 Delayed Choice Quantum Erasure REVISITED Kim et al., PRL84, 1(2000); C.K.Hong and T.G.Noh, JOSA B15, 1192(1998)

97 j 1 = ( ) L R L R 1 ì æ ö æ ö üïì + ï ü U U ï L L L i 3 L L L r i 4 ïïhc.. ï LR = L j j L = í ýí ý 2 ç 2 2 è rlr = 2 ø R j j çè R ø ïï ïîï ïî ïþ ïþ D 3 2 BSA D R L D 0 D 2 BS 4 4 L 4 R D 4 U L BSB 1 é 1 i 1 i ù i 1 i = 0 i i êë 2 2úû

98 + r = 1 U j j U 1 RD1 L L L L 1 = { 3 i 4 }{ 3 i 4 } ìé ï 1 i ù 3 ïê 8 8 = ï í ê 4 ï ïê - i 1 ïë ïî ê 8 8 ú û R R R R 3 4 D 3 2 BSA D R L D 0 D 2 BS 4 4 L 4 R D 4 BSB 1 + r = 2 U j j U 2 RD2 L L L L 1 = { i 3 4 }{ i 3 4 } ìé ï 1 i ù 3 ïê = ï í ê 4 ïê ï i 1 ïë ïî ê 8 8 û ú R R R R 3 4

99 D 3 2 BSA D R L D 0 D 2 BS 4 4 L 4 R D 4 BSB 1 é 1 i ù é1 i ù é1 0 ù = + = ( 1 2) i 1 i ê- ë 8 8úû êë 8 8 úû êë 4úû r R D +D

100 1 é1 0ù r = 3 U U + 3 = 3 3 = 4 3 L j j RD L L L R R 4 êë 0 0úû D 3 2 BSA D R L D 0 D 2 BS 4 4 L 4 R D 4 BSB 1 1 é0 0 ù r = 4 U U + 4 = 4 4 = 4 L j j RD L L L R R êë 4úû

101 Environment Decoherence D 3 BSA 2 3 D 2 D 1 D 0 3 L j 1 = r = j j LR 3 R ( ) L R L R BS 4 L 4 4 R D 4 BSB é1 0ù r { } R = trl rlr = + = R R R R 2 2 ê ë0 1 ú û é1 0 ù é1 0 0 ù 4 é ù é ù 4 2 = + + ( ) = ê 0 0ú ê 0 1 ê 4ú ë û ë 4ú ë û û êë 2úû r R D +D +D +D

102 Entanglement and Decoherence A Zurek B or Environment When system A is entangled with environment, state of A cannot be described by a state vector, but by a density matrix. Ψ AB = a 0 A 0 B + b 1 A 1 B = ψ A φ B ρ A = T r B ρ AB = T r B Ψ AB AB Ψ = a 2 0 = ψ A A ψ 0 b 2 ba* ab*

103 Quantum Repeater 1. Quantum Repeater ~100 km Indefinite distance Briegel, Durr, Cirac, and Zoller, quant-ph/ Transmission: Photon Storage, Processing: Atomic Physics, etc. 2. Multiuser Quantum Network 1:1 multiusers & Entanglement Purification Phoenix, Barnett, Townsend, and Blow, JMO 42, 1155 (1995) Biham, Huttner, and Mor, Phys. Rev. A 54, 2651 (1996)

104 Optical Imaging by Two-Photon Entanglement Shih et al., PRA52, R3429 (1995).

105 yes no no yes

106

107 Quantum Lithography

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111

112 Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit Agedi N. Boto,1 Pieter Kok,2 Daniel S. Abrams,1 Samuel L. Braunstein,2 Colin P. Williams,1 and Jonathan P. Dowling1,* PRL85, 2733 (2000) Two-Photon Diffraction and Quantum Lithography Milena D Angelo, Maria V. Chekhova,* and Yanhua Shih PRL87, (2001)

113 Quantum Metrology We are applying our understanding of the quantum nature of entangled-photon pairs for the characterization of optical-detector quantum efficiency, source radiation density, and photonic-material characteristics, all without the necessity of employing a reference. The use of this unique light source provides particular advantages over the classical counterparts that are traditionally used. A n o t h e r e x a m p l e i s p r o v i d e d b y a t e c h n i q u e w e c a l l quantum ellipsometry. The high accuracy required in traditional ellipsometric measurements necessitates the absolute calibration of both the source and the detector. These requirements can be circumvented by using spontaneous parametric down-conversion in conjunction with a novel polarization interferometer and coincidencecounting detection scheme. Entangled-photon ellipsometry A. F. Abouraddy, K. C. Toussaint, Jr., A. V. Sergienko, B. E. A. Saleh, and M. C. Teich J. O p t. S o c. A m. B. 19, (2002). [PDF] Ellipsometric measurements by use of photon pairs generated by spontaneous parametric downconversion Ayman F. Abouraddy, Kimani C. Toussaint, Jr., Alexander V. Sergienko, Bahaa E. A. Saleh, and Malvin C. Teich Opt. Lett. 26, (2001). [PDF]

114 Quantum Physics Information Science Quantum Information Science Quantum Parallelism Quantum computing is exponentially larger and faster than digital computing. Quantum Fourier Transform, Quantum Database Search, Quantum Many-Body Simulation (Nanotechnology) No Clonability of Quantum Information, Irrevesability of Quantum Measurement Quantum Cryptography (Absolutely secure digital communication) Quantum Correlation by Quantum Entanglement Quantum Teleportation, Quantum Superdense Coding, Quantum Cryptography, Quantum Imaging

Quantum Computer. Jaewan Kim School of Computational Sciences Korea Institute for Advanced Study

Quantum Computer. Jaewan Kim School of Computational Sciences Korea Institute for Advanced Study Quantum Computer Jaewan Kim jaewan@kias.re.kr School of Computational Sciences Korea Institute for Advanced Study KIAS (Korea Institute for Advanced Study) Established in 1996 Located in Seoul, Korea Pure