Quantum Information Processing using Nuclear Magnetic Resonance (NMR): What can be done and How to do it?

Transcription

1 Quantum nformation Processing using Nuclear Magnetic Resonance (NMR): What can be done and How to do it? Anil Kumar Centre for Quantum nformation and Quantum Computing (CQQC) ndian nstitute of Science, Bangalore Quantum nformation Processing: School HR-Allahabad- Feb 5, 0

2 Centre for Quantum nformation and Quantum Computing (CQQC) (At ndian nstitute of Science, Bangalore) nvestigators: Prof. Anil Kumar Prof. Vasant Natarajan Dr. P.S. Anil Kumar Dr. Arindam Ghosh Prof. K.V. Ramanathan Prof. Apoorva Patel Prof. Rahul Pandit Prof. H.R. Krishnamurthy Prof. N.D. Hari Dass Prof. C.E. Veni Madhavan Dr. Subroto Mukerjee Prof. Diptiman Sen Funded by Department of Science and Technology, New Delhi June-00- June-05

3 Activities of the Centre. Bring the experimentalists and the theorists working in the area of Quantum nformation, Quantum Computation and Foundations of Quantum Mechanics together. We have already organized about a dozen lectures (nearly one per month).. Exchange of information, collaboration and cross-fertilization of ideas leading to consolidation of efforts in this important area. 3. nvite National and nternational experts and have colloquia, seminars and lectures, with the aim of exchanging information, ideas and collaboration. We have provision for Visiting Scientists, Post-Docs and Project Assistants positions. 4. Organize schools and workshops to train young researchers in the field. 5. Channelize/assist/expedite/inspire and focus research in different experimental areas of Quantum nformation and Computation, such as (a) quantum dots, (b) NMR, (c) superconducting devices, (d) lasers and (e) trapped ions, with added experimental facilities.

4 The Centre for QQC has the unique feature of combining experimental and theoretical work under one umbrella. The various areas in which the Centre is consolidating its efforts are:

5 Experimental nvestigations. Nuclear Magnetic Resonance: Prof. Anil Kumar and Prof. K.V. Ramanathan. on Trap: Prof. Vasant Natarajan 3. Superconducting Qubits: Dr. P.S. Anil Kumar 4. Quantum Dots: Dr. Arindam Ghosh

6 Theoretical nvestigations. Quantum Algorithms: Prof. Apoorva Patel. Study of entanglement in quantum many body systems: Prof. Rahul Pandit and Dr. Subroto Mukerjee 3. Theoretical support to experimental efforts: Prof. H.R. Krishnamurthy 4. Foundational aspects of quantum theory: Prof. Hari Dass 5. Computer science effort: Prof. Veni Madhavan

7 Liquid-State Room-Temperature NMR: Using spins in molecules as qubits: -- Pseudo-Pure States (PPS) -- One qubit Gates -- Multiqubit Gates -- mplementation of DJ and Grover s Algorithms -- How to increase the number of Qubits -- Quadrupolar Nuclei as multiqubits -- Spin as qudit -- Dipolar Coupled spin ½ Nuclei- up to 8 qubits -- Geometric Phase and its use in Quantum Algorithms -- Quantum Games -- Adiabatic Algorithms

8 Acknowledgements Former QC- Sc-Associates/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 Dr.Arindam Ghosh Dr. Avik Mitra Dr. T. Gopinath Dr. Pranaw Rungta Dr. Tathagat Tulsi - SER Mohali - SER Mohali - SER Pune - CBMR Lucknow - BM, Bangalore - NCF/NH USA - SER Kolkata - NSER-Bhubaneswar - Varian Pune - Univ. Minnesota - SER Mohali T Bombay This lecture is dedicated to the memory of Ms. Jharana Rani Samal* (*Deceased, Nov.,, 009) Current QC Sc - Students Mr. Rama K. Koteswara Rao Mr. V.S. Manu Other Sc Collaborators Prof. Apoorva Patel Prof. K.V. Ramanathan Prof. N. Suryaprakash Other Collaborators Prof. Malcolm H. Levitt - UK Prof. P.Panigrahi SER Kolkata Dr. Arun K. Pati OP +HR Mr. Ashok Ajoy BTS-Goa-MT Funding: DST/DAE/DBT

9 DSX 300 AV 700 NMR Research Centre, Sc Field/Fr equency stability = :0 9 ppb DRX 500 AV 500 AMX 400

10 ntroduction to QC/QP

11 All present day computers (classical computers) use binary (0,) logic, and all computations follow this Yes/No answer. Feynman (98) suggested that it might be possible to simulate the evolution of quantum systems efficiently, using a quantum simulator What is Special about Quantum Systems? Coherent Superposition Classical bit 0 Qubit 0 c 0 + c New Possibilities Entanglement EPR States: (-/) ( 00 + ) NOT resolvable into tensor product of individual particles ( 0 + ) ( 0 + ) Quantum Computation Teleportation

12 Quantum Algorithms. PRME FACTORZATON Classically : exp [(ln c) /3 (ln ln c) /3 ] Shor s algorithm : (994) (ln c) digit 0 0 years (Age of the Universe) 3 years. SEARCHNG UNSORTED DATA-BASE Classically : N/ operations Grover s Search Algorithm : (997) N operations 3. DSTNGUSH CONSTANT AND BALANCED FUNCTONS: Classically : ( N- + ) steps Deutsch-Jozsa(DJ) Algorithm : (99). step 4. Quantum Algorithm for Linear System of Equation: A.W. Harrow, A. Hassidim and Seth Lloyd; PRL, 03, 5050 (009). Exponential speed-up

13 Recent Developments 5. Simulating a Molecule: Using Aspuru-Guzik Algorithm (i) J.Du, et. al, Phys. Rev. letters 04, (00). Used a -qubit NMR System ( 3 CHCl 3 ) to calculated the ground state energy of Hydrogen Molecule up to 45 bit accuracy. (ii) Lanyon et. al, Nature Chemistry, 06 (00). Used Photonic system to calculate the energies of the ground and a few excited states up to 0 bit precision.

14 Experimental Techniques for Quantum Computation:. Trapped ons. Polarized Photons Lasers 3. Cavity Quantum Electrodynamics (QED) on Trap: Linear Paul-Trap kv Laser Cooled ons T ~ mk ~ 6 MHz kv 4. Quantum Dots 5. Cold Atoms 6. NMR 7. Josephson junction qubits 8. Fullerence based ESR quantum computer

15 ntroduction to NMR QP

16 Nuclear Magnetic Resonance (NMR). Nuclear spins have small magnetic moments () and behave as tiny quantum magnets.. When placed in a large magnetic field B 0, they oriented either along the field ( 0 state) or opposite to the field ( state). 3. A transverse radiofrequency field (B ) tuned at the Larmor frequency of spins can cause transition from 0 to (NOT Gate by a 80 0 pulse). Or put them in coherent superposition (Hadamard Gate by a 90 0 pulse). Single qubit gates. 4. Spins are coupled to other spins by indirect spin-spin (J) coupling, and controlled (C-NOT) operations can be performed using J-coupling. Multi-qubit gates SPNS ARE QUBTS 0 B

17 NMR sample has ~ 0 8 spins. Do we have 0 8 qubits? No - because, all the spins can t be individually addressed. Progress so far Spins having different Larmor frequencies can be individually addressed in the Frequency Space. As many Larmor Frequencies as many qubits One needs resolved couplings between the spins in order to encode information as qubits.

18 NMR Hamiltonian H = H Zeeman + H J-coupling = w i zi + J ij i j i i < j Weak coupling Approximation w i - w j >> J ij Two Spin System (AM) bb = A = M M = A ab = 0 ba = 0 H = w i zi + J ij zi zj i i < j Spin Product States are Eigenstates Under this approximation all spins having same Larmor Frequency can be treated as one Qubit. M = 0 A A A aa = 00 A = 0 M M M Different Larmor frequencies = Different Qubits w A w M

19 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 3 CHFBr H = 500 MHz 3 C = 5 MHz 9 F = 470 MHz

20 Homo-nuclear spins having different Chemical shifts (Larmor frequencies) also form multi-qubit systems Qubit Qubits 3 Qubits CHCl

21 . Preparation of Pseudo-Pure States. 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 Achievements of NMR - QP 0. Quantum State Tomography. Geometric Phase in QC. Adiabatic Algorithms 3. Bell-State discrimination 4. Error correction 5. Teleportation 6. Quantum Simulation 7. Quantum Cloning 8. Shor s Algorithm 9. No-Hiding Theorem Also performed in our Lab. Maximum number of qubits achieved in our lab: 8

22 The two methods Coupling (J) Evolution Examples XOR y x z + z /4J /4J z + z z y z + x (/J) z + z y x Transition-selective Pulses NOT p 0 0 p p 0 0

23 y x -y y x -y SWAP 0 p 0 p p 3 00 y y -x -x Toffoli p y y -x x non-selective p pulse + a p on OR/NOR p

24 Pure States: Tr(ρ ) = Tr ( ρ ) = 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 = /N ( α - Δρ ) Here α = 0 5 and U U - = We create a state in which all levels except one have EQUAL populations. Such a state mimics a pure state.

25 Pseudo-Pure State n a two-qubit Homo-nuclear system: (i) Equilibrium: ρ = Δρ = {,,, 0} (ii) Pseudo-Pure Δρ = {4, 0, 0, 0}

26 Preparation of Pseudo-pure states Spatial Averaging Logical Labeling Temporal Averaging Cory, Price, Havel, PNAS, 94, 634 (997) N. Gershenfeld et al, Science, 75, 350 (997) Kavita, Arvind, Anil Kumar, PRA 6, (000) E. Knill et al., PRA, 57, 3348 (998) Pairs of Pure States (POPS) B.M. Fung, Phys. Rev. A 63, 0304 (00) Spatially Averaged Logical Labeling Technique (SALLT) T. S. Mahesh and Anil Kumar, PRA 64, 0307 (00) Using long lived Singlet States S.S. Roy and T.S. Mahesh, PRA, (in press) 00.

27 Spatial Averaging: Cory, Price, Havel, PNAS, 94, 634 (997) z = / z = / z z = / z + z + z z = / Pseudo-pure state Eq.= z + z (p/3) X () (p/4) X () p (p/4) Y () z + z + z z /J G x

28 . Logical Labeling N. Gershenfeld et al, Science, 997, 75, 350 Kavita, Arvind, and Anil Kumar Phys. Rev. A, 000, 6, DRX-500 SF

29 3. Temporal Averaging p p 0 0 = Pseudo-pure state E. Knill et al., PRA, 57, 3348 (998)

30 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 qubits T. S. Mahesh and Anil Kumar, PRA 64, 0307 (00)

31 Relaxation of Pseudopure states Cross Correlations retard the relaxation of some PPS Open circles 00; Filled circles PPS Arindam Ghosh and Anil Kumar, J. Magn. Reson., 73, 5 (005).

32 Logic Gates using NMR

33 Single qubit gates NOT gate Hadamard gate Phase shift gate Composit z-pulse Single qubit gates can be obtained by applying qubit selective pulses (spin selective pulses)

34 Multi-qubit gates Quantum computing also requires the implementation of unitary operators on a qubit cont to the states of other qubits, such gates are called multi-qubit gates. These gates can be implemented by, (i) J-evolution method (ii) Trnasition selective pulses J-evolution method: The evolution of J-couplings and RF pulses are used. The Hamiltonian of J-coupling, x -y

35 Logic Gates XOR (Exclusive OR or C-NOT) NOT e, e U XOR e, e e e, e U NOT e, e e e e e e e N e e NPUT OUTPUT p p NPUT OUTPUT 0 p p p 0 0 p 00

36 Logic Gates Using D NMR NOT( ) XOR XOR Kavita Dorai, PhD Thesis, Sc, 000.

37 Logical SWAP e, e e e 0 p 0 p 00 p e, e Kavita, Arvind, and Anil Kumar Phys. Rev. A, 000, 6, p 3 NPUT p OUTPUT XOR+SWAP

38 Toffoli Gate = C -NOT e, e, e 3 e, e, e 3 (e e ) AND NAND ^ nput Output p e e e 3 Eqlbm. Toffoli e e e 3 Kavita Dorai, PhD Thesis, Sc, 000.

39 CNOT GATE r + eq z z p r = + z x r = z z y r = + 4 z z z p y p z p x N OUT 00> ( + ) + 00> 0> 0> req r4 ( - ) ( ) ( - ) x y 0> - - > ( ) - ( - ) > 0> - ( + ) - ( + )

40 p y p y p p p x p x p p p p SWAP GATE - y - y 0 00 ( + ) 0 ( - ) - ( - ) - ( + ) r + eq z z 3 4 r = + r r x x = y z y z = z y z y r = +, p x J 4 x x, 00> 0> 0> > ( + ) p -x 5 r 5 = z + z Cory et al., ~997 - ( - ) ( - ) - ( + ) ( + ) - - ( - ) ( - ) ( + )

41 Logic Gates By D NMR

42 D NMR Quantum Computing Scheme Preparation Evolution Mixing Detection y -y y 0, etc. t t Computation G z Creation of nitial States Labeling of the initial states Computation Reading output states Ernst & co-workers, J. Chem. Phys., 09, 0603 (998).

43 Three-spin system: 0 H H Br C C COOH H Br Transitions of 0 are labeled by the states of and

44 Two-Dimensional Gates A complete set of 4 Reversible, One-to-one, -qubit Gates T. S. Mahesh, Kavita Dorai, Arvind and Anil Kumar, JMR, 48, 95, (00)

45 NPUT Three-qubit D-Gates: NOP NOT TOFFOL OR/NOR w T. S. Mahesh,et al, JMR,48, 95, w OUTPUT

46 Quantum Algorithms (a) DJ (b) Grover s Search

47 Operator DJ algorithm on ONE qubit with one work bit: Constant Balanced x f (x) f (x) f 3 (x) f 4 (x) U f x y x y f(x) Cleve Version x is input qubit and y is work qubit U f /P O/P Constant Balanced x y f (x) y f (x) f (x) y f (x) f 3 (x) y f 3 (x) f 4 (x) y f 4 (x) U f NOT- C-NOT-, C-NOT-,

48 Deutsch-Jozsa Algorithm Experiment One qubit DJ Two qubit DJ Eq Kavita, Arvind, Anil Kumar, Phys. Rev. A 6, (000)

49 Steps: Pure State Grover s Algorithm Superposition Selective Phase nversion Avg Grover iteration nversion about Average N times Measure

50 Grover s search algorithm by Tomography of the Density Matrix using D NMR -qubitcomputer (N,N-dimethyl formamide) H 3 C N 3 C H H 3 C O 3 C : T =5s ; H: T = 5s (eq)-equilibrium spectra. (a) - PPS (00). (c),(d) -Uniform superposition. (f),(g) Conditional state flip. (i) Searched state (). Das et al, Phys. Rev A 67, (003) Das et al, Chem Phys. Lett., 369, 8 (003)

51 Grover search algorithm using D NMR Ranabir Das and Anil Kumar, J. Chem. Phys., 7603(004)

52 Typical systems used for NMR-QP using J-Couplings qubits Chuang et al, quant-ph/ qubits 3 qubits qubits 4 qubits Marx et al, Phys. Rev. A 6, 030 (000) 7 qubits Knill et al, Nature, 404, 368 (000)

53 How to increase the number of qubits? Use Molecules Partially Oriented (~ 0-3 ) in Liquid Crystal Matrices (i) Quadrupolar Nuclei (spin >/). Reduced Quadrupolar Couplings (ii) Spin =/ Nuclei. Reduced ntramolecular Dipolar Couplings

54 Quadrupolar Systems

55 Using spin-3/ ( 7 Li) oriented system as -qubit system Pseudo-pure states Neeraj Sinha,T. S. Mahesh, K. V. Ramanathan, and Anil Kumar, JCP, 4, 445 (00).

56 -qubit Gates using 7 Li oriented system Neeraj Sinha, T. S. Mahesh, K. V. Ramanathan, and Anil Kumar, JCP, 4, 445 (00).

57 33 Cs system spin 7/ system Equilibrium Half-Adder [Cs pentadeca-fluorooctonate + D O] Subtractor -7/ -5/ -3/ -/ / 3/ 5/ 7/ Conventional Labeling Half-Adder p(5,6,5,7,6) 5-pulses Subtractor p(3,,3,5,4,3,,5,7) 9-pulses -7/ -5/ -3/ -/ / 3/ 5/ 7/ Optimal Labeling Half-Adder p(,3,) 3-pulses Subtractor p(6,4,) 3-pulses Murali et.al. Phys. Rev. A. 033 (003) R. Das et al, PRA 034 (004)

58 Collins version of 3-qubit DJ implemented on the 7/ spin of Cs-33. C B B B B B There are constant and 70 Balanced functions. Half differ in phase of the Unitary transform. B B are shown here. B B B B -Constant and - Balanced Gopinath and Anil Kumar, JMR, 93, 63 (008)

59 Dipolar Systems

60 Advantages of Oriented Molecules Large Dipolar coupling - ease of selectivity - smaller Gate time Long-range coupling - more qubits Disadvantage For Homo-nuclear spin system Weak coupling Approximation w i - w j >> D ij Spins become Strongly coupled A spin can not be identified as a qubit Solution N energy levels are collectively treated as an N-qubit system

61 3-Qubit Strongly Dipolar Coupled Spin System Bromo-di-chloro-benzene C -NOT gate ( 0 ) GHZ state ( ) POPS ( ) Z-COSY (90-t -0-τ m -0-t ) was used to label the various transitions populations coherences T.S. Mahesh et.al., Current Science 85, 93 (003).

62 5-qubit system HET-Z-COSY spectrum for labeling (in liquid crystal) Eqlbm E-level diagram Proton transitions Fluorine transitions

63 Transfer Entanglement transfer Starting from 4-qubit PPS prepared by SAALT: H Entanglement between nd and 3rd qubit Entanglement between st and 4th qubit (p) 7 (p) 40 (p) 7 w + w 7 w + w 40 Ranabir Das, Rangeet Bhattacharya and Anil Kumar, JMR. 70, 30-3 (004).

64 8-qubit system Equilibrium spectrum Molecule: -floro naphthalene (in liquid crystal ZL3) H spectrum H H H H F H H H (-5) 9 F spectrum (53-630)

65 R. Das, R Bhattacharyya and Anil Kumar, Quantum Computing Back Action, AP Proc., 864, 33 (006) HET-Z-COSY spectrum

66 Energy-level diagram (56 levels) 9 F transitions

67 Proton transitions in B domain

68 C 7 -NOT [p ] POPS() [Eq- p ] POPS(40) [Eq- p 40 ] POPS()+C 6 SWAP POPS(40) R. Das, R Bhattacharyya and Anil Kumar, Quantum Computing Back Action, AP Proceedings 864, 33 (006)

69 Geometric Quantum Computing Geometrical phase is robust, since it depends only on the solid-angle enclosed by the path and is independent of the details. This fact can be used to perform fault-tolerant Quantum nformation Processing

70 What is Geometric Phase? When a vector is parallel transported on a curved surface, it acquires a phase. A part of the acquired phase depends on the geometry of the path. A state in a two-level quantum system is a vector which can be transported on a Bloch sphere. The geometrical part of the acquired phase depends on the solid angle subtended by the path at the centre of the Bloch sphere and not on the details of the path.

71 Adiabatic Geometric Phase M. V. Berry, Quantum phase factors accompanying adiabatic changes, Proc. R. Soc. Lond. A, 39, 45 (984) When a quantum system, prepared in one of the eigenstates of the Hamiltonian, is subjected to a change adiabatically, the system remains in the eigenstate of the instantaneous Hamiltonian and acquires a phase at the end of the cycle. The phase has two parts. The dynamical and the geometric part. D = exp( -iet exp( i ) g = / ) (Dynamical Phase) (Geometric Phase) = D + (Total Phase) g Pancharatnam, S. Proc. nd. Acad. Sci. A 44, pp (956).

72 Non-Adiabatic Geometric Phase Aharovov and Anandan later showed that adiabaticity is not a necessary condition. When the density matrix corresponding to a quantum system completes a cyclic path, in the density operator space through different intermediate stages, the system acquires the same geometric phase depending upon the geometry of the path. Y. Aharonov and J. Anandan, Phase change during Cyclic Quantum Evolution, Phys. Rev. Lett. 58(6), 593 (987)

73 Non-adiabatic Geometric phases in NMR QP using transition selective pulses. Geometric phases using slice & triangular circuits. - Use of above in DJ & Grover algorithms. Ranabir Das et al, J. Magn. Reson., 77, 38 (005).. Experimental measurement of mixed state geometric phase by quantum interferometry using NMR. Ghosh et al, Physics Letters A 349, 7 (005). 3. Geometric phases in strongly dipolar coupled spins. - Fictitious spin- ½ subspaces. - Geometric phase gates in dipolar coupled 3 CH 3 CN. - Collins version of -qubit DJ algorithm. - Qubit-Qutrit parity algorithm. Gopinath et al, Phys. Rev A 73, 036 (006).

74 Geometric phase acquired by a slice circuit The state vector of the two level sub space cuts a slice on the Bloch sphere. z The slice circuit can be achieved by two transition selective pulses A.B = (p) q 0 The resulting path encloses a solid angle W =.. 0 (p) q+p+ (p) x (p) -x A spin echo sequence t-p-t is applied to refocus the evolution under internal Hamiltonian (the dynamic phase). Second (p) pulse is applied to restore the state of the first qubit altered by the (p) pulse.

75 Unitary operator associated with the slice circuit: A.B = A. B ( ) ( i e ) 0 Solid angle W =. 3 C Spectra of 3 CHCl3 Ranabir Das et al, J. Magn. Reson., 77, 38 (005).

76 Grover search algorithm using geometric phases (by slice circuit): Pseudo pure state 0 0 H H H C ij H C 00 H H i j Superpo sition Selective nversion nversion about Average Measure Selective nversion: Using Geometric phase gates x = 00 x = 0 = p nversion about Average C 00 = H U 00 H x = 0 U 00 = C 00 (p) Ranabir Das et al, J. Magn. Reson., 77, 38 (005). x =

77 Deutsch-Jozsa algorithm using geometric phases (by slice circuit): H 3 C of 3 CHCl 3 Constant U f(00) is identity matrix Constant U f() is achieved by applying p pulse on the second qubit Balanced Balanced = Ranabir Das et al, J. Magn. Reson., 77, 38 (005).

78 Experimental Measurement of Mixed State Geometric Phase by NMR Creation of 00 PPS Preparation of Mixed state by the a degree pulse and a gradient. mplementation of Slice circuit to introduce Geometric Phase. Measurement

79 Results: Geometric Phase (Shift) - = - tan r W tan The shift of the interference pattern as a function of W and r. The shift directly gives the geometric phase acquired by the spin qubit. Ghosh et al, Physics Letters A 349, 7 (005).

80 Quantum Games

81 Game Theory Von Neumann and Morgenstern Theory of games and Economic behaviour (Princeton University Press, 947) John F. Nash (Princeton University) Nobel Prize in Economics (994) Robert Aumann (Hebrew University) Nobel Prize in Economics (005) Nash Equilibrium Evolutionary Games

82 Quantum Games by NMR (i) Ulam s Quantum Game Guessing a number (ii) Three player Dilamma Game- Optimum strategy for going to a Bar with two stool. Avik Mitra et al.; J. Magn. Reson., 87, (007). (iii) Battle of Sexes Game Should Bob and Alice together watch TV or go to a football game. Avik Mitra Thesis Sc 007

83 Ulam sgame Two player game Bob thinks of a number between and n Alice tries to find out the number in minimum number of queries. Classical: n queries Quantum: Query

84 Ulam sgame Classical Algorithm 0000 Alice Total number of queries = log n = n Bob

85 Quantum Version of Ulam sgame The quantum system consists of two registers. Register A consists of n qubits Register B consists of qubit Alice makes a superposition of all the qubits. Bob performs a unitary transform and stores the result in register B. Alice again interferes the qubits and then performs a measurement. Total number of queries = --S. Mancini and L. Maccone, quant-ph/050856

86 Protocol Creation of PPS Application of Hadamard gate Unitary Transform of BOB Application of Hadamard gate M E A S U R E

87 Experimental implementation Preparation of Pseudo-Pure state - PPS

88 Results 0 0 population coherence Avik et al., J. ndian nstitute of Science; 89, (009).

89 Adiabatic Quantum Algorithms

90 (s) E Adiabatic Quantum Computing Based on Adiabatic Theorem of Quantum Mechanics: A quantum system in its ground state will remain in its ground state provided that the Hamiltonian H under which it is evolved is varied slowly enough. i U f The system evolves from H i to H f with a probability (-e ) provided the evolution rate satisfies the condition max 0 s dh( s) ; s dt g min 0; s e Where e << s Farhi et al, PRA, 65,03 (00), Science, 9, 47 (00), quant-ph/00006; ; 00835

91 Adiabatic Quantum Computing Evolve the initial state under a slowly varying Hamiltonian so that it acts as though a unitary transformation occurred on the initial state, bringing it to a final state during some time T. H(s) = (-s)h B + sh F H B beginning Hamiltonian H F Final Hamiltonian nitialize register to desired input qubits. Vary the Hamiltonian towards the final Hamiltonian whose eigenstates encodes the desired final states. N f = U n i, whereu n = exp( -ih n= 0 n t)

92 Adiabatic Algorithms by NMR (i) NMR mplementation of Locally Adiabatic Algorithms (a) Grover s search Algorithm (b) D-J Algorithm Avik Mitra et al., J. Magn. Reson. A 77, (005). (ii) Adiabatic Satisfy-ability problem using Strongly Modulated Pulses Avik Mitra, T.S. Mahesh and Anil Kumar, J. Chem. Phys. 8, 40 (008).

93 Deutsch-Jozsa Algorithm CONSTANT OR BALANCED FUNCTONS: Classically : ( N- + ) steps Deutsch-Jozsa : step (DJ) Algorithm The Constant and Balanced functions of two-qubit DJ

94 a = b H : H F : 4 Adiabatic DJ Algorithm - where = F F + b - where = a f ( 00) f ( 0) f ( 0) (- ) + (- ) + (- ) + (- ) = - a F + f ( ) Hamiltonian in terms of spin operators H = x + x + x x C ( H ) F = z + z + z Constant case z B H F = - z + z + zz + xx + yy + x + x - xz ( ) ( ) ( ) ( ) 3 a= Constant function a=0 Balanced function S. Das et al, PRA, (00) Balanced case Avik Mitra et al, JMR, 77, 85 (005) z x

95 Modification of Balanced case Hamiltonian The balanced case Hamiltonian requires complicated pulse sequence due to the presence of zero and double quantum terms. Hamiltonian diagonal in the computational basis are easy to implement. ( ) ( ) ( ) ( ) x z z x x x y y x x z z z z B F H = - The terms contributing to the off diagonal elements in balanced case Hamiltonian are dropped. ( ) z z z z B F H ~ ( ) ( ) ( ) ( ) x z z x x x y y x x z z z z B F H = - Avik Mitra et al, JMR, 77, 85 (005)

96 Eigenvalues with respect to the parameter s. Plot of Parameter s as a function of t.

97 Constant Case Final Hamiltonian H = + + C F z z z z nitial Hamiltonian H = + + x x x x Balanced Case B F Final Hamiltonian ( + ) H = _ - + z z z z The Balanced case Hamiltonian differs from the Constant case in the sign of the Hamiltonian. This is sufficient to distinguish the two cases. Avik Mitra et al, JMR, 77, 85 (005)

98 NMR mplementation. Sample H Experiment carried out in DRX500 Cl 3 C Cl H and C has resonance frequency 500 MHz and 5 MHz. Cl J HC = 09 Hz

99 Pulse Scheme for the NMR mplementation CONSTANT CASE Pulse sequence for mplementation of H B NMR Hamiltonian: Beginning Hamiltonian H = - + H = + + B z - z Jzz For v = v = - J / and with two pi/ pulses with appropriate phases x x x x Avik Mitra et al, JMR, 77, 85 (005)

100 Pulse sequence for mplementation of H F CONSTANT CASE NMR Hamiltonian: Final Hamiltonian F Pulse sequence for mplementation of H F H = - + z - z Jzz For v = v = - J / H = + + z z Free evolution under the NMR Hamiltonian between two p- pulses with appropriate phases for a time t. Avik Mitra et al, JMR, 77, 85 (005 z z

101 Experimental Result The final state is 00 Average absolute deviation Avik Mitra et al, JMR, 77, 85 (005) 5.8%

102 NMR mplementation BALANCED CASE NMR Hamiltonian: Final Hamiltonian H = - + F Pulse sequence for mplementation of H F z - z Jzz For v = v = - J / ( ) H = z z Free evolution under the NMR Hamiltonian between two p- pulses with appropriate phases for a time t. z z Avik Mitra et al, JMR, 77, 85 (005

103 Experimental Result Final state is ( )/ 3 x ~ 7% Experiment does not match well with theoretical result. Carbon: Short decoherence time Significant effect of decoherence in carbon. T of carbon was measured by CPMG sequence. Simulation was repeated after including relaxation using Bloch equations. Avik Mitra et al, JMR, 77, 85 (005)

104 Experimental Result x ~ 8% Avik Mitra et al, JMR, 77, 85 (005)

105 Adiabatic SAT Algorithm by Strongly Modulated Pulses

106 n a Homonuclear spin systems spins are close (~ khz) in frequency space Pulses are of longer duration Decoherence effects cannot be ignored Strongly Modulated Pulses circumvents the above problems

107 Strongly Modulated Pulses. U SMP = l l l - -ih ( ) ( ) ( rf U e l l ) l eff w w t t l z l F = Tr U U T N SMP w rf t Nedler-Mead Simplex Algorithm (fminsearch)

108 k-sat Problem. Let B={x,x,.,x n } be a set of n Boolean variables.. Let C i be a disjunction of k elements of B C i = x x.... F is the Boolean function that is the conjunction of m such clauses. F = C... x C Cm k Find out all the assignments of Boolean variable in F that simultaneously satisfies all the clauses i.e. F=. Three variable -SAT problem B = {x,x,x 3 }, set of three variable. Each clause (C i ) has one variable. e.g. F = x x x 3. Farhi et al, quant-ph/00006

109 NMR mplementation, using a 3-qubit system. The Sample. odotrifluoroethylene(c F 3 ) F b F a C C Equilibrium Specrum. F c J J J ab ac bc = = = 68. Hz Hz Hz

110 Step. Preparation of PPS PPS: ( ) 3 z z z 3 z z 3 z z z z 3 z z z Equilibrium: 3 z z z + +

111 Step 3. mplementation of Adiabatic Evolution H B = x + x + 3 x = x H F = z + z + 3 z = z m H + M m M ( m) = - HB HF m th step of the interpolating Hamiltonian. U m e m 80 m 80 m 80 - i x - -i z - i x - M p M p M p e e m th step of evolution operator m,,3 a pulse x pulse b m,,3 z am,,3 x pulse Pulse sequence for adiabatic evolution Total number of iteration is 3 time needed = 6 ms (400 s x 5 pulses x 3 repetitions)

112 Using Concatenated SMPs Duration: Max 5.8 ms, Min. 4.7 ms Avik Mitra et al, JCP, 8, 40 (008)

113 Results for all Boolean Formulae Avik Mitra et al, JCP, 8, 40 (008)

114 Conclusions:

115 NMR QP is at a Cross -Road Many Operations and Algorithms can be mplemented A Very Good Tool for Learning and Explorations Scaling is Difficult f All Spins are to be coupled to each other with nondegenerate Transitions Possible to obtain high number of qubits f Only nearest neighbor couplings are sufficient Backbone of a C-3, N-5 labeled protein with side chain protons deuterated How to do QP?

116 Future Directions. To increase the number of Qubits in NMR. Synthesis molecules with several hetero-nuclei.. To use strongly modulated pulses (SMPs) and Control Theory for performing Unitary operations in short times and accurately. 3. To search for systems with large coherence times. For example singlet states in equivalent spins or spins systems with symmetry with symmetry preserving relaxation. 4. Develop protocols which can be carried out using spins with nearest neighbor couplings. 5. To enhance nuclear polarization by transferring Electron polarization using DNP.

117 Thank You