All Optical Quantum Gates

Transcription

1 All Optical Quantum Gates T.C.Ralph Centre for Quantum Computer Technology Department of Physics University of Queensland

2 LOQC People Staff T.C.Ralph A.G.White G.J.Milburn Postdocs J.L.O Brien G.J.Pryde A.Gilchrist H.Jeong K.Pregnell Collaborators D.F.V.James ~ Los Alamos S.Bartlett ~ University of Queensland C.Myers ~ University of Waterloo M.Nielsen ~ University of Queensland D.Branning ~ University of Illinois Students N.K.Langford T.Weinhold A.Hayes A.Lund P.Rohde R.Dalton

3 LOQC People Staff T.C.Ralph A.G.White G.J.Milburn Postdocs J.L.O Brien G.J.Pryde A.Gilchrist H.Jeong K.Pregnell Collaborators D.F.V.James ~ Los Alamos S.Bartlett ~ University of Queensland C.Myers ~ University of Waterloo M.Nielsen ~ University of Queensland Students N.K.Langford T.Weinhold A.Hayes A.Lund P.Rohde R.Dalton

4 Overview Introduction Optical CNOT gate How it works in theory How it works in practice

5 Overview Introduction Optical CNOT gate How it works in theory How it works in practice Process Tomography Error correction Future - scale-up

6 Single Photon States spatial distribution temporal distribution single spatio-temporal mode transform limited

7 Single Photon States spatial distribution temporal distribution single spatio-temporal mode transform limited If there is exactly 1 quanta of energy in the mode it is a single photon state

8 Single Photon States spatial distribution temporal distribution single spatio-temporal mode transform limited If there is exactly 1 quanta of energy in the mode it is a single photon state 1>

9 Single Photon States spatial distribution temporal distribution single spatio-temporal mode transform limited If there is exactly 1 quanta of energy in the mode it is a single photon state 11>

10 Experimental Reality Check Presently there are no sources of single photon states as just described Best that can be done: ρ = P P P 1 60% A.I.Lvovsky, H.Hansen, T.Aichele, O.Benson, J.Mlynek, and S.Schiller Phys. Rev. Lett. 87, (2001)

11 Experimental Reality Check Presently there are no sources of single photon states as just described Best that can be done: ρ = P P P 1 60% Also need detectors that can count photons with high efficiency - currently ~ 90% efficiency

12 Post-selection α> = 0> + α 1> α 2 2> +.

13 Post-selection α << 1 α> = 0> + α 1> α 2 2> +.

14 Post-selection α << 1 α> = 0> + α 1> α 2 2> +. G.I.Taylor, Proc.Cambridge Phil.Soc.15, 114 (1909).

15 Post-selection χ << 1 Down-conversion splits 2ω ω φ > = 00> + χ 11> + Ghosh and Mandel, PRL, 59, 1903 (1987)

16 Post-selection χ << 1 φ > = 00> + χ 11> + Down-conversion splits 2ω ω Measure in coincidence Ghosh and Mandel, PRL, 59, 1903 (1987)

17 Post-selection χ << 1 Down-conversion splits 2ω ω φ > = 00> + χ 11> + 2 photon experiment Measure in coincidence Ghosh and Mandel, PRL, 59, 1903 (1987)

18 Post-selection χ << 1 Down-conversion splits 2ω ω φ > = 00> + χ 11> + 2 photon experiment Measure in coincidence Ghosh and Mandel, PRL, 59, 1903 (1987) See also: Santori, Fattal, Vuckovic, Solomon and Yamamoto, Nature, 419, 594 (2002)

19 Photon source: bright beam-like source tune crystal to obtain good modes S. Takeuchi, Optics Letters 26, 843 (2001); C. Kurstsiefer et al., J. Mod. Opt. 48, 1997 (2001)

20 Photon source: bright beam-like source QuickTime and a TIFF (Uncompressed) decompressor are needed to see this picture. tune crystal to obtain good modes S. Takeuchi, Optics Letters 26, 843 (2001); C. Kurstsiefer et al., J. Mod. Opt. 48, 1997 (2001)

21 Photon source: bright beam-like source QuickTime and a TIFF (Uncompressed) decompressor are needed to see this picture. tune crystal to obtain good modes spatially filter with fibres frequency filter (~0.4 nm) S. Takeuchi, Optics Letters 26, 843 (2001); C. Kurstsiefer et al., J. Mod. Opt. 48, 1997 (2001)

22 Different CNOT Experiments Pittman, Fitch, Jacobs, and Franson, PRA 68, (2003). 3 photon gate, operates in coincidence. O Brien, Pryde, White, Ralph and Branning, Nature 426, 264 (2003). 2 photon gate, operates in coincidence Gasparoni, Pan, Walther, Rudolph, and Zeilinger, Phys. Rev. Lett. 93, (2004) 4 photon gate, operates in coincidence (though in principle could be heralded)

23 Photons as qubits We can encode qubits as the polarization states of single photons 0 = H> 1 = V> α H> + β V> Arbitrary one qubit operations can be realized with half and quarter wave-plates PBS α 01> + β 10> Dual-rail encoding

24 CNOT Gate

25 Optical CNOT Gate Control In PBS 1/3 PBS Control Out 1/3 Target In HWP 22.5 PBS 1/3 PBS HWP 22.5 Target Out T.C.Ralph, N.K.Langford, T.B.Bell and A.G.White, PRA 65, (2002)

26 single photons of arbitrary polarization simultaneously injected Optical CNOT Gate Control In PBS 1/3 PBS Control Out 1/3 Target In HWP 22.5 PBS 1/3 PBS HWP 22.5 Target Out T.C.Ralph, N.K.Langford, T.B.Bell and A.G.White, PRA 65, (2002)

27 single photons of arbitrary polarization simultaneously injected Optical CNOT Gate Control In PBS 1/3 PBS Control Out non-classical interference 1/3 Target In HWP 22.5 PBS 1/3 PBS HWP 22.5 Target Out T.C.Ralph, N.K.Langford, T.B.Bell and A.G.White, PRA 65, (2002)

28 single photons of arbitrary polarization simultaneously injected Optical CNOT Gate successful events post-selected as simultaneous clicks on photon counters Control In PBS 1/3 PBS Control Out non-classical interference 1/3 Target In HWP 22.5 PBS 1/3 PBS HWP 22.5 Target Out T.C.Ralph, N.K.Langford, T.B.Bell and A.G.White, PRA 65, (2002)

29 Optical CNOT Gate Photon pairs from downconverter Polarization beam displacers Photon counting

30 Optical CNOT Gate J.L.O Brien, G.J.Pryde, A.G.White, T.C.Ralph, D.Branning, Nature 426, 264 (2003).

31 Classical CNOT Operation Truth Table 0,0 0,0 0,1 0,1 1,0 1,1 Ideal operation 1,1 1,0 Experiment

32 Quantum CNOT Operation Entanglement Production ( 0 1 ) 1 C T 0 C 1 T 1 C 0 T

33 CNOT Truth Table (average Fidelity = 92%) Singlet state from CNOT H> V>- V> H> (Fidelity = 94%) Ideal experimental: real imaginary

34 Process Tomography

35 Process Tomography environment ρ ε(ρ)

36 Process Tomography environment ρ? ε(ρ)

37 Process Tomography environment ρ? ε(ρ) ε(ρ) = Σ χ Α ρ Α

38 Process Tomography environment ρ? ε(ρ) ε(ρ) = Σ χ Α ρ Α Process matrix

39 Process Tomography environment ρ? ε(ρ) Quantum Computation and Quantum Information, Nielsen and Chuang ε(ρ) = Σ χ Α ρ Α Process matrix - determined by 256 measurement combinations

40 Find physical matrix closest fit Constraint 1: Positive map Process Tomography environment Constraint 2: Trace preserving ρ? ε(ρ) O Brien et al, to appear Phys.Rev.Lett. (04) quant-ph/ ε(ρ) = Σ χ Α ρ Α Process matrix - determined by 256 measurement combinations

41 Process Tomography ρ CNOT ε(ρ) ε(ρ) =U cnot ρ cnot U cnot = 0.5(II + IX + ZI - ZX)

42 Process Tomography II IX IY IZ XI XX XYXZ YI YX YY YZ ZI ZX ZY ZZ II IX IY IZ XI XX XY XZ YI YX YY Ideal Measured (Re) YZ ZI ZX ZY ZZ ε(ρ) =U cnot ρ cnot U cnot = 0.5(II + IX + ZI - ZX)

43 Process Tomography II IX IY IZ XI XX XYXZ YI YX YY YZ ZI ZX ZY ZZ II IX IY IZ XI XX XY XZ YI YX YY Ideal Measured (Re) YZ ZI ZX ZY ZZ ε(ρ) =U cnot ρ cnot U cnot = 0.5(II + IX + ZI - ZX)

44 Process Tomography = 87% Measured (Re) Measured (Im) average fidelity = (d F p +1)/(d+1) = 90% purity = 0.83 maximum increase in tangle = 0.73 O Brien, Pryde, Gilchrist, James, Langford, Ralph, White, PRL, 93, (2004)

45 Process Tomography II Control In PBS τ 1 1/3 PBS Control Out 1/3 τ 3 τ 4 Target In τ 2 HWP 22.5 PBS 1/3 τ 5 PBS HWP 22.5 Target Out *Model gate including spatio-temporal structure *Perform tomography on the model - equivalent result for process with less data P.P.Rohde, J.L.O Brien, G.J.Pryde,T.C.Ralph, quant-ph/

46 Error Correction

47 Z-measurement Error Correction 0 1 α ( HH> + VV>) + β ( VH> + HV>) Knill, LaFlamme and Milburn, Nature 409, 46 (2001)

48 Z-measurement Error Correction α ( HH> + VV>) + β ( VH> + HV>) H α H> + β V> Knill, LaFlamme and Milburn, Nature 409, 46 (2001)

49 Z-measurement Error Correction α ( HH> + VV>) + β ( VH> + HV>) V α V> + β H> Knill, LaFlamme and Milburn, Nature 409, 46 (2001)

50 Z-measurement Error Correction α ( HH> + VV>) + β ( VH> + HV>) V α V> + β H> Teleported gates fail by making a Z-measurement Knill, LaFlamme and Milburn, Nature 409, 46 (2001) Pittman, Jacobs and Franson, PRA, 64, (2001)

51 Z-measurement Error Correction α ( HH> + VV>) + β ( VH> + HV>) V α V> + β H> Teleported gates fail by making a Z-measurement Knill, LaFlamme and Milburn, Nature 409, 46 (2001) Pittman, Jacobs and Franson, PRA, 64, (2001) LOQC cluster states Nielsen, PRL, 93, (04)

52 Z-measurement Error Correction α ( HH> + VV>) + β ( VH> + HV>)? α H> + β V> or α V> + β H> Photon Loss error correction Knill, LaFlamme and Milburn, Nature 409, 46 (2001) Φ> LL = α 0> L 0> L 0> L + β 1> L 1> L 1> L

53 Experimental Z-measurement Error Correction (i) Encoding α H> + β V> H> + V> α ( HH> + VV>) + β ( VH> + HV>) Non-deterministic CNOT

54 Experimental Z-measurement Error Correction (i) Encoding Input: 0> 0> + 1> 0> + i 1> Average Fidelity 88 ± 3%

55 Experimental Z-measurement Error Correction (ii) Decoding H or V? H> H> + V> real superpositions imaginary superpositions V> H> - V>

56 Experimental Z-measurement Error Correction (ii) Decoding? H or V H> H> + V> real superpositions imaginary superpositions V> H> - V>

57 Experimental Z-measurement Error Correction (ii) Decoding Syndrome measured but not corrected Average Fidelity 96 ± 3 % O Brien, Pryde White and Ralph, quant-ph/

58 Linear Optics QC~3 main ideas of KLM Q1> Q2> 1> 1 LINEAR OPTICAL NETWORK *Non-deterministic gates. *Don t always work, but heralded when they do. 1> 1 Knill, LaFlamme and Milburn, Nature 409, 46 (2001)

59 Linear Optics QC~3 main ideas of KLM Q1> Q2> 1> 1 LINEAR OPTICAL NETWORK *Non-deterministic gates. *Don t always work, but heralded when they do. 1> 1 *Non-deterministic teleported gates. When they don t work they measure the qubit. Q> a> E> F> { B X Z a> Knill, LaFlamme and Milburn, Nature 409, 46 (2001)

60 Linear Optics QC~3 main ideas of KLM Q1> Q2> 1> 1 LINEAR OPTICAL NETWORK *Non-deterministic gates. *Don t always work, but heralded when they do. 1> 1 *Non-deterministic teleported gates. When they don t work they measure the qubit. Q> a> E> F> { B X Z a> Knill, LaFlamme and Milburn, Nature 409, 46 (2001)

61 Linear Optics QC~3 main ideas of KLM Q1> Q2> 1> 1 LINEAR OPTICAL NETWORK *Non-deterministic gates. *Don t always work, but heralded when they do. 1> 1 *Non-deterministic teleported gates. When they don t work they measure the qubit. Q> a> E> F> { B X Z a> u ( 00>+ 11>) + v ( 01>+ 10>) Knill, LaFlamme and Milburn, Nature 409, 46 (2001) } Q> *Error encoding against qubit measurment

62 Linear Optics QC~3 main ideas of KLM Q1> Q2> 1> 1 LINEAR OPTICAL NETWORK *Non-deterministic gates. *Don t always work, but heralded when they do. 1> 1 *Non-deterministic teleported gates. When they don t work they measure the qubit. Q> a> E> F> { B X Z a> u ( 00>+ 11>) + v ( 01>+ 10>) Knill, LaFlamme and Milburn, Nature 409, 46 (2001) } Q> *Error encoding against qubit measurment

63 thanks