Quantum Computing New Frontiers

Transcription

1 Quantum Computing New Frontiers M. Suhail Zubairy Department of Physics and Institute for Quantum Studies Texas A&M University

2 Quantum Computation Computational power could exceed that of Turing machines History: R. P. Feynman (98) Potential of quantum computer Simulation of quantum-mechanical objects by other quantum systems D. Deutsch (985) Description of universal quantum computer

3 Hunt for something interesting for quantum computers to do P. Shor (994) Quantum algorithm to perform efficient factorization L. K. Grover (997) Quantum search algorithm 994-present Quantum logic gates Quantum lithography Quantum dense coding Quantum teleportation

4 Present Status Entanglement: 5-bit entanglement NMR Quantum logic gates: ion traps, cavity QED, NMR Shor s algorithm: NMR (factoring of 5) Grover s algorithm: NMR ( qubit), multilevel atomic system without entanglement

5 Quantum Computation Classical bits: 0, Quantum bits (qubit): 0, quantum states Qubits in the laboratory: Photon number state: Photon linear polarization: Electron, nuclear spin: Atomic energy levels: 0 V H 0

6 Why Quantum Computer? () Qubits can form coherent superposition Ψ = C + C + C 0 0 C 0 = () Qubits can form quantum entanglement ( 0, +, ) or ( 0,0 +, ) 0 ψ (,) ψ () ψ () N qubits can store N numbers simultaneously For qubits we can have C 0 0,0 + C 0, + C,0 + C3,

7 Two particles: Four particles: 4 + ( C C 4 0 0,0,0,0 0,,0,0 + C N = 000 particles: ( C 0,0 + C 0, + C,0 +, ) 0 C3 0,0,0, C 5 0,,0, + C 0,0, C > ,0,,,,, 0 56 ~ number of atoms in the universe + C 3 5 )

8 Problems For coherent superposition ψ = C C experimental outcome is probabilistic Probability of finding 0 is Probability of finding is C 0 C Decoherence any interaction with surroundings will destroy coherent superposition and entanglement

9 Conditions for a Quantum Computing System Be a scalable physical system with well defined qubits Be initializable to a simple state such as Have much longer decoherence times (i.e., one can do many operations before losing quantum coherence) Have a universal set of quantum gates Permit high quantum efficiency, qubit specific measurements

10 Basic building blocks: Quantum logic gates (QLG) Due to coherent nature of quantum mechanics, quantum logic gates should be reversible Classical gates: AND a b a b Not reversible!! XOR a b a b

11 Basic Building Blocks Quantum logic gates (QLG) Due to coherent nature of quantum mechanics, quantum logic gates should be reversible One-bit gate: A U A θ U θ a b Quantum phase gate: Q φ U U 0 = cos( θ / ) 0 sin( θ / ) θ + θ = cos( θ / ) sin( θ / ) 0 Q 0,0,0 iφδ a b, a, b e φ = 0,0,0,,, δ 0,, a, b iφ e 0,,

12 Quantum Phase Gate a First qubit: Atomic state c = 0, b = Second qubit: Cavity photon number state 0, Cavity field is nearly resonant with a b transition (Dispersive coupling) a b c Δ

13 Quantum Phase Gate a Effective Hamiltonian: ηg = Δ c, 0 c,0 + H eff b, 0 b,0 c, c, iφ b, e b, ( + aa a a a a b b ) ( 0,0 0,0 ) (,0,0 ) ( 0, 0, ) iφ (, e, ) φ = ( g / Δ)t Quantum phase gate!! A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J.M. Raimond and S. Haroche, Phys. Rev. Lett. 83, 566 (999) a b c Δ

14 Cavity QED Based Quantum Phase Gate a b c ω ab = ν Δ ω bc 0 = ν ν ν + Δ Qubits:, in two modes of the cavity Three-level atom in ground state c passes through the cavity 0 ν ν,0 0, 0, 0,,0, 0 0 a b c Δ,, if ν Δ >> ν g g π-pulse phase shift M. S. Zubairy, M. Kim, and M. O. Scully, Phys. Rev. A 68, (003)

15 Quantum dense coding One photon can carry one bit of information: 0 or Can one photon carry two bits of information? Initially Alice and Bob have entangled pair of photons [ 0, +, ] A B A 0 B C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. 69, 88 (99)

16 Quantum dense coding Alice can now send two bits of information by sending her photon to Bob after making one of four operations: 0,0 no change,0 A 0 A A, A 0 Bob measures Bob measures [ 0 A, B + A, 0 B ] A, B + 0 A, 0B 0 A 0 A 0 A A,, 0 A A, A 0 A Bob measures Bob measures [ 0 ] A, B 0 A, 0 A, B A, 0B B 0,, [ ] [ ] All operations are unitary Bob needs to make Bell basis measurement!! C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. 69, 88 (99)

17 Shor s Algorithm for Factoring an Integer Problem: Factoring a number N into its prime factors 8463 = Input length log N Conventional computers: Factoring algorithm runs in steps. O exp ( ln N ) 3 ( ln ln N ) 3

18 Factoring an integer 9 + (54 digit number) requires 700 workstations 00 digit number requires about 6 years 50 digit number requires about 80 years (RSA) 000 digit number requires about 0 years (Age of the universe ~0 0 years) Quantum Algorithm: (Shor, 994) Requires about O((log N) ) steps 000 digit number requires only few million steps

19 Why is this problem important? Present day secure information transfer is based on public key system (RSA) Encryption Key (e,n) Decryption Key (d,n) N = p q is a large number primes (56 digits) Code is broken if we can factorize N

20 Searching of an unsorted data base Grover s Algorithm L. K. Grover, Phys. Rev. Lett. 79, 35 (997). Search Problem There is an unsorted data base containing N items out of which just one item satisfies a given condition. That one item has to be retrieved. On a classical computer, it would require an average of N/ searches. On a quantum computer, O( N ) steps are required. Also require log N bits.

21 Quantum search without entanglement M. O. Scully and M. S. Zubairy, Phys. Rev. A 64, 0304 (00) Classical search pe τ = π η 3 Rabi freq N pulses requiredone for each atom Energy required ε0e Aτc N = ε Quantum search p η N E N τ = π N pulses required Energy required ε ε0e Aτc N = N

22 Grover s Algorithm (N=4) Two qubits: () Prepare a quantum superposition of all states 0,0 0,,0, , 0 0,, 0, (3) Inversion about mean 0,0, 0,, () Invert the desired state (say 0, ),0,, 0,0 0,,0, + + mean x 0 = = 4 0,

23 Start with Unitary Operations Ψ0 = 0,...0 { k () Walsh-Hadamard transformation W on all qubit = 0 = ( 0 + ) ψ s = 0,0 + 0, +,0, 0 ( + ), () Invert the target state i For t = 0, U f ψ (3) Inversion about mean U t = π k N = U f e = t [ 0,0 0, +,0, ] = + i s s s = e π = s s t U s U f [ ] + t t Ψ = 0,

24 Quantum interference Quantum eraser Quantum lithography with classical light Sub-wavelength resolution in microscopy

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26 M. O. Scully and K. Druhl, Phys. Rev. A 5, 08 (98) Quantum interference Quantum eraser

27 Science, Feb., 005

28 Delayed choice Quantum eraser Y.-H. Kim, R. Yu, S. P. Kulik, Y. Shih, M. O. Scully, Phys. Rev. Lett. 84, (000)

29 Brian Greene in The Fabric of the Cosmos (004) These experiments are a magnificent affront to our conventional notions of space and time for a few days after I learned of these experiments, I remember feeling elated. I felt I'd been given a glimpse into a veiled side of reality.

30 Microscopy -- The diffraction limit Young s interference: st order correlation I ( r + E ' ', r ) = E() ( ) = 4 E cos [ k( r r ) / ] ' = 4 E cos [ k( d sinθ ) / ] The first minimum occurs at k( d sinθ ) / = π / λ λ d = sinθ The diffraction limit.

31 Sub-wavelength microscopy Two-photon correlation: Two Cascade Atoms k( d sinθ ) = π / k( d sinθ ) / = π / The first minimum occurs at instead of λ λ d = 4sinθ 4 Resolution improves by two fold!. U. W. Rathe and M. O. Scully, Lett. Math. Phys. 34, 97 (995).

32 Optical lithography I x e e kx ikx ikx ( ) + = (+ cos( )) Rayleigh criterion: Minimal resolvable feature size occurs at a spacing corresponding to the distance between an intensity maximum and an adjacent intensity minimum For grazing incidence: kxmin x = π min = λ /

33 Classical two-photon exposure Two-photon absorption probability: ikx ikx 4 P I e + e = 3+ 4cos( kx) + cos(4 kx) IF cos(kx) term can be neglected, then P 3+ cos(4 kx) x = λ /4 min Factor of improvement!! E. Yablonovitch and R. B. Vrijen, Opt. Eng. 38, 334 (999)

34 Two-Photon Quantum Lithography Entangled state Ψ= 0A B + A 0B Deposition rate on a two-photon absorbing substrate P ( c d ) ( c d) Ψ + + Ψ =+ cos(4 kx) ikx c= ( a ib) e / ; d= ( ia+ b)/ A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 733 (000).

35 Difficulties: Entangled state generation for ( N,0>+ 0,N>) is tedious. Two- or N-photon absorption probability is very low for low intensity beams.

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37 Quantum Lithography with Classical Light P. R. Hemmer, A. Muthukrishnan, M. O. Scully, and M. S. Zubairy, Phys. Rev. Lett. 96, (006)

38 5 May 006 Vol. 3, 6 (006)

39 One-dimensional patterns: Q. Sun, P. R. Hemmer and M. S. Zubairy, (007).

40 Two-dimensional patterns: Q. Sun, P. R. Hemmer and M. S. Zubairy, (007).

41 Measurement of the separation between atoms beyond diffraction Applications: limit Microscopy Nano-science Molecular level biology technology Lithography Can the optical imaging resolution exceed Rayleigh diffraction limit? J.-T. Chang, J. Evers, M. O. Scully, M. S. Zubairy, Phys. Rev. A 73, (R) (006) J.-T. Chang, J. Evers, and M. S. Zubairy, Phys. Rev. A 74, (006).

42 S. Qamar, S.-Y. Zhu, and M. S. Zubairy, Phys. Rev. A 6, (000). Atom localization via resonance fluorescence Atom with a wavefunction f(x) initially in level a> Rabi passes frequency through Ω the = Ωstanding 0 sin (kx) is position wave. dependent The spontaneously emitted photon carries information about An During atom the is localized passage to through standing wave it sub-wavelength position region!! undergoes Rabi oscillations and eventually decays to level b> a> a> Ω b> b> Ω Ω= E / h kx

43 Atom microscopy J.-T. Chang, J. Evers, M. O. Scully, M. S. Zubairy, Phys. Rev. A 73, (R), (006) Two Identical atoms: and with distance r ρ Establish a standing wave driving field along the direction of Ω=Ω sin( kx. ) Ω =Ω sin( kx ) x sin - ( / ) = Ω Ω x - = sin ( Ω / ) Ω k k distance= x x Ω Ω

44 Atomic energy part: Dipole-dipole interaction: Atom-field interaction:

45 Fluorescence spectrum properties (i) Large distance: 0 λ λ r ~ 0 Ω Exactly as the same as that in the simple scheme Ω=Ω sin( kx. ) Ω =Ω sin( kx ) x sin - ( / ) = Ω Ω x - = sin ( Ω / ) Ω k k distance= x x Ω Ω

46 Fluorescence Spectrum Properties (ii) Very small distance: Ω >> Ω,Ω r λ 0 Ω r We should have Ω = ω0 For γ ~ 0 7 Hz, Ω ~0 3 Hz, we estimate r ~ λ/550

47 Fluorescence Spectrum Properties (iii) Intermediate distance r 30 0 λ λ Ω Ω,Ω The spectrum turns out to be very complicated. Ω However, increase the intensity of the driving field Ω, Ω >> Ω Again, r

48 Quantum thermodynamics

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50 Carnot Cycle: T T h Q in = T h ΔS Isothermal expansion at T=T h, absorbing heat Q in Adiabatic expansion until T=T c T c 4 Q out = T c ΔS 3 Isothermal compression at T=T c, rejecting Q out S Adiabatic compression until T=T h Efficiency: η c = T T c h

51 Photo-Carnot Engine Radiation pressure drives piston Single-mode radiation field at T rad by flowing atoms Atoms reheated by hohlraum at T h Steam engine with photon gas

52 Photo-Carnot Engine with Coherence without coherence with coherence a a Ω T h Ω T h b c b Q in μ-wave ρbc T c Q out W T c Q out W η c = T T c h η φ = η c T T c h 3nφ ρbc cosφ

53 Photo-Carnot Engine with Coherence a Ω c b 3-level atoms. Cavity frequency tuned to midway between ground-state doublet μ-wave ρbc Atoms leave hohlraum at T=T h Atoms acquire coherence ρ bc = ρ bc e iφ T c Q out W Coherent interaction with cavity radiation (like EIT) Temperature characterizing cavity radiation no longer T h

54 Two level n ( n + ) ( T n & = αρaa ( Th ) αρbb h) where α = rg τ At steady state and high T: n kth = η Ω η c = T T c h Ω a b Three level with ρ bc = ρ bc e iφ n& = φ At steady state, high T ( n ) ( ) φ + α ρbb + ρcc ρbc cosφ n φ αρ + aa ( aa = ) ( ) ρ n 3 ρ cosφ 3 φ = kt h n ηω φ bc a Ω c b ( ) Cavity radiation temperature T = T h 3n ρ cosφ leads to efficiency φ η φ φ bc T = ηc T c h 3nφ ρbc cosφ

55 The Bottom Line: (Photo) Carnot: η c = T T c h Photo-Carnot with coherence: η = η φ c T T c h 3nφ ρbc cosφ Hence for φ = π η φ >η c T = c T even if!! h W net = η T ΔS nα π c μ ΔΩ Ω ηω But: W prep > W net

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57 Implementation of Grover s algorithm (M. O. Scully and M. S. Zubairy, Proc. Nat. Acad. Sc. (USA), 98, 9490 (00)) w C αβ N U π π, 4 U π π, 4 w Quantum Phase Gate Q π C αβ U π,0 U π,0 Walsh-Hadamard transformation Oracle Inversion about mean ( i) w 0,0 = 0,0 0,,0, ( ii) Cαβ w 0, = 0, 0 0,, 0, + + ( iii) N Cαβ w 0, = 0, U π π, 4 U π π, 4 Quantum Phase Gate Q π N U π π, 4 U π π, 4 Two qubits are represented by the modes of a cavity Single and two-bit gates are implemented by passing appropriate α atoms through the cavity β

58 Discrete Quantum Fourier Transform a q q q n n n l = 0 3 e qp πi n... q q q q q i : qubits 0 p q q q 0 A B A B B 0 0 A0 ν ν ν 0 0 A = A = A = = U 0... π B lm = Q 4 π m l δ ν a ν 0 a 0 U π π U π,,0 4 a δ ν ν 0 U π π U π,,0 4 δ = δ ν ν U π π U π,,0 4 δ δ ν ν 0 m=+ m=0 F= m=- ν ν ν ν M. O. Scully and M. S. Zubairy, Phys. Rev. A 65, 0534 (00) b F=0

59 Implementation of Grover s Algorithm a a M. O. Scully and M. S. Zubairy, Proc. Nat. Acad. Sc. (USA). 98, 9490 (00)

60 Grover s Algorithm for N = n In terms of one-bit and two-bit gates: K m mth order quantum phase gate K α, α,... α = α m iπδ δ... δ m e α α α i = m α, α,... α 0 or ( i =,, Λ n) m Diao, Zubairy, and Chen, Zeit. F. Naturfor. 75a, 70 (00)

61 Multiphoton quantum phase gate a For photons, effective Hamiltonian H = ηδ a a a a... a a Q m m Consider a Λ system ( m) i( δ mτ ) αα... αm η α, α,... α m = e α, α,... (Schmidt and A. Imamoglu, Opt. Lett., 936 (996)) m Effective Hamiltonian + + H Compare with nonresonant case = ηδ a aa a with g g δ = δ ~ nr 3 Δ Ω Δ Homogeneously broadened medium: m α m δ 3τ 3 Zubairy, Matsko, and Scully, Phys. Rev. A 65, (00) Enhancement δ δ nr rad 9 0

62 References Entanglement Reconstruction of entangled state via Autler-Townes spectroscopy, M. Ikram and M. S. Zubairy, Phys. Rev. A 65, (00). Quantum disentanglement eraser, M. S. Zubairy, G. S. Agarwal, and M. O. Scully, (003). Quantum logic gates Resonant enhancement of high order optical nonlinearities based on atomic coherence, M. S. Zubairy, A. B. Matsko, and M. O. Scully, Phys. Rev. A 65, (00). A cavity QED based quantum phase gate, M. S. Zubairy, M. Kim, and M. O. Scully, Phys. Rev. A 68, 0438XXX (003).

63 References Quantum search A quantum search protocol for an atomic array, M. O. Scully and M. S. Zubairy, Phys. Rev. A 64, 0304 (00). Quantum optical implementation of Grover s algorithm, M. O. Scully and M. S. Zubairy, Proc. Natl. Acad. Sci. (USA) 98, 9490 (00). A quantum circuit design for Grover s algorithm, Z. Diao, M. S. Zubairy, and G. Chen, Zeit. f. Naturforsc. 75, 70 (00). Quantum Fourier transform Cavity QED implementation of the discrete quantum Fourier transform, M. O. Scully and M. S. Zubairy, Phys. Rev. A 65, 0534 (00). Quantum teleportation The state evolution formulation of teleportation for continuous variables, J. Zhang, C. Xie, F. Li, S.-Y. Zhu, and M. S. Zubairy, Europhys. Lett. 56, 478 (00). Teleportation of an atomic momentum state, S. Qamar, S.-Y. Zhu,and M. S. Zubairy, Phys. Rev. A 67, 0438 (003).

64 Research supported by: DARPA (QuIST) Air Force Research Laboratories (Rome, New York) TAMU Telecommunication and Informatics Task Force (TITF) initiative Office of Naval Research

65 One-bit unitary gate Possible initial states: 0 b OR b First Step 0 b 0 b b i 0 a Second Step (Interaction with classical field; θ = Ωt ) i a cosθ a + ie φ sinθ b iφ b ie sin θ a + cosθ b Third Step 0 b 0 b 0 a i END RESULT: iφ ( e sinθ + cos 0 ) b b iφ 0 b θ b ( cosθ + e sinθ 0 )b M. S. Zubairy, M. Kim, and M. O. Scully, Phys. Rev. A 68, (003)

66 Universality of quantum phase gate (QPG) Consider Q π 0 = Recall Uϕ ϕ cos = ϕ sin ϕ sin ϕ cos U π = 0,0 ( 0,0 0, ) ( 0,0 0, ) 0,0 0, ( 0,0 + 0, ) ( 0,0 + 0, ) 0,,0 (,0, ) (,0 +, ),, +, (,0, ) (,0 ),0

67 Bell State Discriminator Bell basis (two qubits are the photons in two modes of the cavity): ψ ψ The operator ( 0,0, ) 3 = ( 0,, ) = 0 U Q U ( 0,0 +, ) 4 = corresponds to the passage of appropriate atoms through the cavity U ψ ψ 3π / 4 π π / 4 π / 4 ( 0, +, ) = 0 Leading to ψ 0,0 ψ 0, ψ ψ, 3,0 4 M. S. Zubairy, M. Kim, and M. O. Scully, Phys. Rev. A 68, 0438XX (003)

68 ψ ( A, B) = 8 7 a= 0 Quantum Fourier transform a, f ( a) (Entanglement) f (7) f (6) a ψ 8 7 a= 0 e ac πi 8 8 c ( A, B) = 7 a= 0 c= 0 For f ( a) = f ( a + r) measurement of state of register A yields: ψ 7 e ac πi 8 c, f ( a) c0 = 0, 8/ r, 8/ r Λ ( r )8/ ( A, B) = [ 0, f (0) + 0, f () + 4, f (0) 4, f () ] (QFT) Let r = f(0) = f() = f(4) = f(6) f() = f(3) = f(5) = f(7) Outcome of measurement of register A is 0 or 4 with equal probability. r f (5) f (4) f (3) f () f () f (0)

69 How to Factorize N? Select an integer x that is co-prime with N a Find sequence formed by function f ( a) = x (mod N) Sequence has the form Sequence has periodic structure with period r Assume period r to be even or a x a f ( a) : : : 0,,, x,, x, = ( x Λ Λ Λ 4 43 Λ Λ Λ r / r / ( x ) or ( x + ) must have common factors with N a PROBLEM: Find r (period of function f ( a) = x (mod N) ), x r terms, Λ Λ r, x, = x r (mod N) r / )( x r / + ) r, x, (mod N) r, x r r +,, x r+ Λ, Λ

70 Example so a 3 a f ( a) : : : 0,,, 3,, 9, 3, 7, N = 9 x = 3, 3, 9, 7, 8, 6, r=6 = 36 (mod 9) 0 = (3 3 - )(3 3 + ) (mod 9) 0 = 6 8 (mod 9) 0 = ( 3) ( 7) (mod 9) 3, 7 are factors of 9 4, 8, 5, 6 43,,7Λ Λ, 79, 3, 87, Λ Λ Λ Λ Λ

71 ) Perform discrete quantum Fourier transform on first register (reversible via inverse transform) = 0 q a q ac i c e q a π = = = ) (, ) (, q a q c q ac i q a a f c e q a f a q π 0, ( ) q a a f a q = ) Prepare the following state in two registers ) (mod ) ( N x a f a = Highly entangled state!!!

72 3) q q ac π i q e c, f( a) q a= 0 c= 0 Retrieve the output from the quantum computer by measuring the state of all elements in the first register If f(a)=f(a+r), the sum over a will yield constructive interference from πiac coefficients q e c / Only when is a multiple of the reciprocal / r period, all other values of c / q will produce destructive interference 0 r r q 3 r c q