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Anil Kumar Indian Institute of Science, Bangalore

Former QC Students . Dr. Arvind Dr. Kavita Dorai Dr. T.S. Mahesh Dr. Neeraj Sinha Dr. K.V.R.M.Murali Dr. Ranabir Das Dr. Rangeet Bhattacharyya . - IISER Mohali IISER Mohali IISER Pune - CBMR Lucknow IBM, Bangalore NCIF/NIH USA SUNY Stony Brook.

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Anil Kumar Indian Institute of Science, Bangalore

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  1. Former QC Students Dr. Arvind Dr. Kavita Dorai Dr. T.S. Mahesh Dr. Neeraj Sinha Dr. K.V.R.M.Murali Dr. Ranabir Das Dr. Rangeet Bhattacharyya • - IISER Mohali • IISER Mohali • IISER Pune • - CBMR Lucknow • IBM, Bangalore • NCIF/NIH USA • SUNY Stony Brook Dr.Arindam Ghosh - SUNY Buffalo Dr. Avik Mitra - Varian Pune Dr. T. Gopinath - Univ. Minnesota Quantum Computations By NMR:Status and Challenges. Anil KumarIndian Institute of Science, Bangalore Current QC Students Ms. Jharana Rani Samal* (* Nov., 12, 2009) Mr. Ram K. Rao Mr. V.S. Manu Collaborators @ IISc: Prof. K.V. Ramanathan Prof. N. Suryaprakash Prof. Apoorva Patel And Prof. Malcolm H. Levitt - UK

  2. DSX 300 AV 700 NMR Research Centre, IISc DRX 500 AV 500 AMX 400

  3. Introduction to QC

  4. All present day computers (classical computers) use binary (0,1) logic, and all computations follow this Yes/No answer. Feynman (1982) suggested that it might be possible to simulate the evolution of quantum systems efficiently, using a quantum simulator What is Special about Quantum Systems? Entanglement Coherent Superposition EPR States:2(-1/2)(00+ 11) NOT resolvable into tensor product of individual particles(01+ 11) (02+ 12) Classical bit Qubit |1  |0  c1|0  + c2|1  0 1 Teleportation Quantum Computation New Possibilities

  5. Quantum Algorithms 1. PRIME FACTORIZATION Classically : exp [2(ln c)1/3(ln ln c)2/3] 400 digit 1010years(Age of the Universe) Shor’s algorithm : (1994) (ln c)3 3 years 2. SEARCHING ‘UNSORTED’ DATA-BASE Classically : N/2 operations Grover’s Search Algorithm : (1997) Noperations 3. DISTINGUISH CONSTANT AND BALANCED FUNCTIONS: Classically : ( 2N-1 + 1) steps Deutsch-Jozsa(DJ) Algorithm : (1992) . 1 step 4. Quantum Algorithm for Linear System of Equation: A.W. Harrow, A. Hassidim and Seth Lloyd; PRL, 103, 9 Oct . (2009). Exponential speed-up

  6. Experimental Techniques for Quantum Computation: 2. Cavity Quantum Electrodynamics (QED) 1. Trapped Ions 3. Quantum Dots 4. Nuclear Magnetic Resonance 5. Josephson junction qubits 6. Fullerence based ESR quantum computer

  7. 0 NUCLEAR SPINS 1.Nuclear spins have small magnetic moments (I) and behave as tiny magnets. B1 2.When placed in a large magnetic field B0 , spins are oriented either along the field (|0 state) or opposite to the field (|1 state) . 3.A transverse radiofrequency field (B1) tuned at the Larmor frequency of spins can cause transition from |0 to |1 (NOT operation). 4.Spins are coupled to other spins by indirect spin-spin (J) coupling, and controlled (C-NOT) operations can be performed using J-coupling. SPIN IS QUBIT ANZMAG-2008

  8. Why NMR? > A major requirement of a quantum computer is that the coherence should last long. > Nuclear spins in liquids retain coherence ~ 100’s millisec and their longitudinal state for several seconds. > A system of N coupled spins (each spin 1/2) form an N qubit Quantum Computer. > Unitary Transform can be applied using R.F. Pulses and J-evolution and various logical operations and quantum algorithms can be implemented.

  9. Achievements of NMR - QIP 10. Quantum State Tomography 11. Geometric QC 12. Adiabatic QC 13. Quantum State discriminator 14. Error correction 15. Teleportation 16. Quantum Simulation 17. Quantum Cloning 18. Shor’s Algorithm 1. Preparation of Pseudo-Pure States 2. Quantum Logic Gates 3. Deutsch-Jozsa Algorithm 4. Grover’s Algorithm 5. Hogg’s algorithm 6. Berstein-Vazirani parity algorithm 7. Quantum Games 8. Creation of EPR and GHZ states 9. Entanglement transfer              Maximum number of qubits achieved in our lab: 8

  10. NMR sample has ~ 1018 spins. Do we have 1018 qubits? No - because, all the spins can’t be individually addressed. Progress so far Spins having different Larmor frequencies can be individually addressed as many “qubits” One needs resolved couplings between the spins in order to encode information as qubits.

  11. NMR Hamiltonian H = HZeeman + HJ-coupling =  wi Izi + Jij Ii Ij Two Spin System (AM) bb = 11 i i < j Weak coupling Approximation wi - wj>> Jij A2= 1M M2= 1A ab = 01 ba = 10 H=  wi Izi+ Jij Izi Izj M1= 0A A1= 0M i i < j aa = 00 Spin States are eigenstates A1 M1 A2 M2 Under this approximation all spins having same Larmor Frequency can be treated as one Qubit wM wA

  12. 13CHFBr2 An example of a three qubit system. A molecule having three different nuclear spins having different Larmor frequencies all coupled to each other forming a 3-qubit system Homo-nuclear spins having different Chemical shifts (Larmor frequencies) also form multi-qubit systems

  13. 3 Qubits 111 011 110 101 010 001 100 000 2 Qubits 1 Qubit CHCl3 11 1 10 01 0 00

  14. The two methods Coupling (J) Evolution Transition-selective Pulses Examples XOR 11 I1z+I2z p y I1 I1z+I2x 01 10 x y (1/2J) I1z+2I1zI2y 00 I2 x 1/4J 1/4J I1z+2I1zI2z 11 p I1 01 NOT1 10 I2 p 00

  15. y x -y 11 I1 y x -y I2 y y -x -x I1 I2 I3 01 10 SWAP p1 p3 p2 00 111 p 011 110 101 Toffoli 010 001 100 000 y y -x x non-selective p pulse + a p on 000  001 111 I1 011 110 101 I2 OR/NOR 010 001 100 p I3 000

  16. Pure States: Tr(ρ ) = Tr ( ρ2 ) = 1 For a diagonal density matrix, this condition requires that all energy levels except one have zero populations. Such a state is difficult to create in NMR Pseudo-Pure States Under High Temperature Approximation ρ = 1/N ( α1 + Δρ ) Here α = 106 and U 1 U-1 = 1 We create a state in which all levels except one have EQUAL populations. Such a state mimics a pure state.

  17. In a two-qubit system: Equilibrium: ρ = 106+ Δρ = {2, 1, 1, 0} (ii) Pseudo-Pure Δρ = {4, 0, 0, 0} 1 4 0 0 2 0 1 0  01  10  10  11  00  01  00  11 Pseudo-Pure State How to Create ? i) Spatial averaging ii) Temporal averaging iii) Logical labeling iv) Spatially averaged- Logical labeling

  18. Preparation of Pseudo-pure states • Spatial Averaging • Logical Labeling • Temporal Averaging • Pairs of Pure States (POPS) • Spatially Averaged Logical Labeling Technique (SALLT) Cory, Price, Havel, PNAS, 94, 1634 (1997) N. Gershenfeld et al, Science, 275, 350 (1997) Kavita, Arvind, Anil Kumar, PRA 61, 042306 (2000) E. Knill et al., PRA, 57, 3348 (1998) B.M. Fung, Phys. Rev. A 63, 022304 (2001) T. S. Mahesh and Anil Kumar, PRA 64, 012307 (2001)

  19. Spatial Averaging:Cory, Price, Havel, PNAS, 94, 1634 (1997) 1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 1 1 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 -1 3 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1 1 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 -1 (p/3)X(2) (p/4)X(1) (p/4)Y(1) p 6 1 2 3 4 5 1/2J Gx I1z = 1/2 I2z = 1/2 2I1z I2z = 1/2 Pseudo-pure state I1z + I2z + 2I1zI2z = 1/2 I1z + I2z + 2I1zI2z Eq.= I1z+I2z

  20. 2. Logical Labeling • N. Gershenfeld et al, • Science, • 1997, 275, 350 • Kavita, Arvind, • and Anil Kumar • Phys. Rev. A, • 2000, 61, 042306 DRX-500 SIF

  21. 0 2 1 1 1 0 1 0 2 1 2 2 2 1  00  10  10  00  01  11  01  01  11  11  10  01  11  10 + + 6 2  00  00 = 3. Temporal Averaging p p Pseudo-pure state E. Knill et al., PRA, 57, 3348 (1998)

  22. 4. Pseudo Pure State by SALLT: (Spatially Averaged Logical Labeling Technique) This method does not scale with number of qubits Subsystem Pseudo-pure states of 2 qubits T. S. Mahesh and Anil Kumar, PRA 64, 012307 (2001)

  23. Relaxation of Pseudopure states 1 -1 -1 -1 -3 1 3 1 1 -3 1 1 1 -3 1 1 Cross Correlations retard the relaxation of some PPS Open circles 00; Filled circles 11 PPS Arindam Ghosh and Anil Kumar, J. Magn. Reson., 173, 125 (2005).

  24. Logic Gates By 1D NMR

  25. Logic Gates 1 1 2 2 1 INPUT OUTPUT INPUT OUTPUT 2 p p 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 1 0 1 1 1 0 0 0 p p 11 11 1 1 1 1 1 0 1 1 0 1 1 1 01 10 01 10 p 2 2 p 00 00 XOR2 (Exclusive OR or C-NOT) NOT1 UXOR2 UNOT1 e1 , e2 e1 , e1  e2 e1 , e2 e1 , e2 1 N 2

  26. Logic Gates Using 1D NMR  NOT(I1)  XOR2  XOR1 Kavita Dorai, PhD Thesis, IISc, 2000.

  27. e1 , e2 e2 , e1 1 0 2  11 1 1  01  10 2  00 0 0 0 0 INPUT OUTPUT 1 0 0 1 1 0 0 1 1 1 1 1 p Logical SWAP XOR+SWAP p p1 p2 p3 Kavita, Arvind, and Anil Kumar Phys. Rev. A, 2000, 61, 042306

  28. 1 1 2 2 3 3 Toffoli Gate = C2-NOT e1 , e2 , e3 e1 , e2 , e3  (e1e2) ^ Input Output 000 000 001 001 010 010 011 111 100 100 101 101 110 110 111 011 AND Eqlbm. p NAND Toffoli Kavita Dorai, PhD Thesis, IISc, 2000.

  29. Logic Gates By 2D NMR

  30. 2D NMR Quantum Computing Scheme Detection Preparation Evolution Mixing -y y y I 0 t t 2 1 I , I etc. Computation 1 2 G z Creation of Labeling Reading Initial States of the Computation output initial states states Ernst & co-workers, J. Chem. Phys., 109, 10603 (1998).

  31. Two-Dimensional Gates A complete set of 24 Reversible, One-to-one, 2-qubit Gates T. S. Mahesh, Kavita Dorai, Arvind and Anil Kumar, JMR, 148, 95, (2001)

  32. 111 111 110 110 101 101 100 100 011 011 010 010 001 001 000 000 110 101 000 100 010 001 111 110 101 100 011 010 001 000 111 011 NOP Three-qubit 2D-Gates: NOT1 I0 I1 INPUT I3 I2 OR/NOR TOFFOLI w1 T. S. Mahesh,et al, JMR,148, 95, 2001 w2 OUTPUT

  33. I F (Io) C = C F (I1) F (I2) 2D Gates by Hadamard Spectroscopy Total time: Less than 2 min. 2D Method takes 2 Hours T. Gopinath and Anil Kumar, J. Magn. Reson., 183, 259 (2006).

  34. Quantum Algorithms • DJ • (b) Grover’s Search

  35. DJ algorithm on ONE qubit with one work bit: Constant Balanced x f (x) f (x) f (x) f (x) 1 2 3 4 0 0 1 0 1 1 0 1 1 0 U f ñ ñ ® ñ Å ñ |x |y |x |y f(x) is input qubit and |y ñ ñ |x is work qubit I/P O/P Constant Balanced Å Å Å Å x y f (x) f (x) f (x) f (x) y f (x) y f (x) y f (x) y f (x) 1 2 3 4 1 2 3 4 0 0 0 0 1 1 0 0 1 1 0 1 0 1 1 0 0 1 1 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 Operator U f 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 p p pulse on pulse on one of the Pulses x x p No pulse x work qubit transitions of the work qubit Cleve Version

  36. Deutsch-Jozsa Algorithm One qubit DJ Two qubit DJ Experiment Eq Kavita, Arvind, Anil Kumar, Phys. Rev. A 61, 042306 (2000)

  37. Grover’s Algorithm Steps: |00..0 Pure State Superposition Avg Grover iteration Selective Phase Inversion Inversion about Average N times Measure

  38. Grover search algorithm using 2D NMR Ranabir Das and Anil Kumar, J. Chem. Phys. 121, 7603(2004)

  39. Typical systems used for NMR-QIP using J-Couplings 2 qubits 4 qubits 7 qubits 7 qubits Chuang et al, quant-ph/0007017 6 3 qubits 7 5 qubits Marx et al, Phys. Rev. A 62, 012310 (2000) Knill et al, Nature, 404, 368 (2000)

  40. How to increase the number of qubits? • Use Molecules Partially Oriented (~ 10-3) • in Liquid Crystal Matrices • Quadrupolar Nuclei (spin >1/2). Reduced Quadrupolar Couplings • Spin =1/2 Nuclei. Reduced Intramolecular Dipolar Couplings

  41. Quadrupolar Systems

  42. Using spin-3/2 (7Li) oriented system as 2-qubit system Pseudo-pure states Neeraj Sinha,T. S. Mahesh, K. V. Ramanathan, and Anil Kumar, JCP, 114, 4415 (2001).

  43. 2-qubit Gates using 7Li oriented system Neeraj Sinha, T. S. Mahesh, K. V. Ramanathan, and Anil Kumar, JCP, 114, 4415 (2001).

  44. -7/2 -5/2 -3/2 -1/2 1/2 3/2 5/2 7/2 -7/2 -5/2 -3/2 -1/2 1/2 3/2 5/2 7/2 000 010 011 001 101 110 111 100 111 110 101 100 011 010 001 000 7 6 5 4 3 2 1 7 6 5 4 3 2 1 133Cs system – spin 7/2 system [Cs pentadeca-fluoro- octonate + D2O] Equilibrium Half-Adder Subtractor Optimal Labeling Conventional Labeling Half-Adder p(1,3,2) 3-pulses Subtractor p(6,4,2) 3-pulses Half-Adder p(5,6,5,7,6) 5-pulses Subtractor p(3,2,3,5,4,3,2,5,7) 9-pulses Murali et.al. Phys. Rev. A. 022313 (2003) R. Das et al, PRA 012314 (2004)

  45. Collins version of 3-qubit DJ implemented on the 7/2 spin of Cs-133. B B C There are 2 constant and 70 Balanced functions. Half differ in phase of the Unitary transform. 12 are shown here. 1-Constant and 11- Balanced B B B B B B B B B Gopinath and Anil Kumar, JMR, 193, 163 (2008)

  46. Dipolar Systems

  47. Advantages of Oriented Molecules • Large Dipolar coupling - ease of selectivity - smaller Gate time • Long-range coupling - more qubits • Disadvantage • For Homo-nuclear spin system • Spins become Strongly coupled • A spin can not be identified as a qubit • 2N energy levels are collectively treated as an N-qubit system Weak coupling Approximation wi - wj>> Dij Solution

  48. 3-Qubit Strongly Dipolar Coupled Spin System Bromo-di-chloro-benzene C2-NOT gate (|110  |111) POPS (|000 000| - |001 001| ) GHZ state (|000+ |111) populations coherences Z-COSY (90-t1-10-τm-10-t2) was used to label the various transitions T.S. Mahesh et.al., Current Science 85, 932 (2003).

  49. 5-qubit system HET-Z-COSY spectrum for labeling (in liquid crystal) Eqlbm E-level diagram Proton transitions Fluorine transitions

  50. Entanglement transfer |0 |0 |0 |0 Starting from 4-qubit PPS prepared by SAALT: H Transfer  Entanglement between 1st and 4th qubit Entanglement between 2nd and 3rd qubit |0000+ |0110 |0000+ |1001 (p)27(p)40(p)27 w2+ w27 w2+w40 Ranabir Das , Rangeet Bhattacharya and Anil Kumar, JMR. 170, 310-321 (2004).

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