Anil Kumar 1 , K. Rama Koteswara Rao 1 , T.S. Mahesh 2 and V.S.Manu 1

1. Quantum Simulations by NMR: Recent Developments Anil Kumar1, K. Rama Koteswara Rao1, T.S. Mahesh2 and V.S.Manu1 Indian Institute of Science, Bangalore-560012 Indian Institute of Science Education and Research, Pune-411 008 QIPA-December 2013, HRI-Allahabad

2. In 1982, Feynman suggested that it might be possible to simulate the evolution of quantum systems efficiently, using a quantum simulator Recent Developments Simulating a Molecule: Using Aspuru-Guzik Adiabatic Algorithm (i) J.Du, et. al, Phys. Rev. letters 104, 030502 (2010). Used a 2-qubit NMR System ( 13CHCl3 ) to calculated the ground state energy of Hydrogen Molecule up to 45 bit accuracy. (ii) Lanyon et. al, Nature Chemistry 2, 106 (2010). Used Photonic system to calculate the energies of the ground and a few excited states up to 20 bit precision.

3. Experimental Techniques for Quantum Computation: 3. Cavity Quantum Electrodynamics (QED) 2. Polarized Photons Lasers 1. Trapped Ions 4. Quantum Dots 5. Cold Atoms 6. NMR 7. Josephson junction qubits 8. Fullerence based ESR quantum computer

4. 0 Nuclear Magnetic Resonance (NMR) 1.Nuclear spins have small magnetic moments and behave as tiny quantum magnets. B1 2.When placed in a magnetic field (B0), spin ½ nuclei orient either along the field (|0 state) or opposite to the field (|1 state) . 3.A transverse radio-frequency field (B1) tuned at the Larmor frequency of spins can cause transition from |0 to |1 (NOT Gate by a 1800 pulse). Or put them in coherent superposition (Hadamard Gate by a 900 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 NUCLEAR SPINS ARE QUBITS

5. DSX 300 7.0 Tesla Field/ Frequency stability = 1:10 9 AV 700 16.5 Tesla NMR Research Centre, IISc 1 PPB DRX 500 11.7 Tesla AV 500 11.7 Tesla AMX 400 9.4 Tesla

6. NMR Facility in IISc 800 MHz NMR spectrometer equipped with cryogenic probe Installed in April 2013 18.8 Tesla

7. 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.

8. Addressability in NMR 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 addressed in the frequency domain resulting-in as many “qubits” as Larmor frequencies, each having 1018 spins. (ensemble computing) One needs resolved couplings between the spins in order to encode information as qubits.

9. 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 spins having same Larmor Frequency can be treated as one Qubit wM wA

10. NMR Qubits An example of a three qubit system. JCH = 225 Hz JCF = -311 Hz JHF = 50 Hz 13CHFBr2 13C 1H = 500 MHz 13C = 125 MHz 19F = 470 MHz Br (spin 3/2) is a quadrupolar nucleus, is decoupled from the rest of the spin system and can be ignored.

11. 111 011 110 101 010 001 100 000 Homo-nuclear spins having different Chemical shifts (Larmor frequencies) also form multi-qubit systems 3 Qubits 2 Qubits 1 Qubit CHCl3 11 1 10 01 0 00

12. 0 1 1 4 0 0 0 2  00  01  11  00  01  11  10  10 Pure State 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 room temperature liquid-state NMR Pseudo-Pure State Under High Temperature Approximation ρ = 1/N ( α1 - Δρ ) Here α = 105 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.

13. Achievements of NMR - QIP 10. Quantum State Tomography 11. Geometric Phase in QC 12. Adiabatic Algorithms 13. Bell-State discrimination 14. Error correction 15. Teleportation 16. Quantum Simulation 17. Quantum Cloning 18. Shor’s Algorithm 19. No-Hiding Theorem  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                Also performed in our Lab. Maximum number of qubits achieved in our lab: 8 In other labs.: 12 qubits; Negrevergne, Mahesh, Cory, Laflamme et al., Phys. Rev. Letters, 96, 170501 (2006).

14. Recent Publications from our Laboratory • 1. Experimental Test of Quantum of No-Hiding theorem. • Jharana Rani Samal, Arun K. Pati and Anil Kumar, • Phys. Rev. Letters, 106, 080401 (25 Feb., 2011) • Use of Nearest Neighbor Heisenberg XY interaction for creation of entanglement on end qubits in a linear chain of 3-qubit system. • K. Rama Koteswara Rao and Anil Kumar IJQI, 10, 1250039 (2012). • 3.Multi-Partite Quantum Correlations Revel Frustration in a Quantum Ising Spin System. • K. Rama Koteswara Rao, Hemant Katiyar, T.S. Mahesh, Aditi Sen (De), Ujjwal Sen and Anil Kumar: Phys. Rev A88, 022312 (2013). • 4.Use of Genetic Algorithm for QIP. • V.S. Manu and Anil Kumar, Phys. Rev. A86, 022324 (2012)

15. No-Hiding Theorem S.L. Braunstein & A.K. Pati, Phys.Rev.Lett. 98, 080502 (2007). Any physical process that bleaches out the original information is called “Hiding”. If we start with a “pure state”, this bleaching process will yield a “mixed state” and hence the bleaching process is “Non-Unitary”. However, in an enlarged Hilbert space, this process can be represented as a “unitary”. The No-Hiding Theorem demonstrates that the initial pure state, after the bleaching process, resides in the ancilla qubits from which, under local unitary operations, is completely transformed to one of the ancilla qubits.

16. Quantum Circuit for Test of No-Hiding Theorem using State Randomization (operator U). H represents Hadamard Gate and dot and circle represent CNOT gates. After randomization the state |ψ> is transferred to the second Ancilla qubit proving the No-Hiding Theorem. (S.L. Braunstein, A.K. Pati,PRL 98, 080502 (2007).

17. NMR Pulse sequence for the Proof of No-Hiding Theorem The initial State ψis prepared for different values of θ and φ Jharana Rani et al Pulse Sequence for U obtained by using the method out lined in A Ajoy, RK Rao, A Kumar, and P Rungta, Phys. Rev. A 85, 030303(R) (2012)

18. Experimental Result for the No-Hiding Theorem. The state ψ is completely transferred from first qubit to the third qubit Input State s Output State s S = Integral of real part of the signal for each spin 325 experiments have been performed by varying θand φin steps of 15o All Experiments were carried out by Jharana (Dedicated to her memory) Jharana Rani Samal, Arun K. Pati and Anil Kumar, Phys. Rev. Letters, 106, 080401 (25 Feb., 2011)

20. Use of nearest neighbor Heisenberg-XY interaction.

21. Until recently we had been looking for qubit systems, in which all qubits are coupled to each other with unequal couplings, so that all transitions are resolved and we have a complete access to the full Hilbert space. However it is clear that such systems are not scalable, since remote spins will not be coupled. Solution Use Nearest Neighbor Interactions

22. In a first study towards this goal, we took a linear chain of 3-qubits And Using only the Nearest-Neighbor Heisenberg XY Interaction, Entangle end qubits as well as Entangle all the 3 qubits Heisenberg XY interaction is normally not present in liquid-state NMR: We have only ZZ interaction. We create the XY interaction by transforming the ZZ interaction by the use of RF pulses. Koteswara Rao et al, IJQI, 10 (4) 1250039 (2012)

23. Nearest-NeighbourHeisenberg XY Interaction

24. 1 2 J J 3 Consider a linear Chain of 3 spins with equal couplings σjx/y are the Pauli spin matrices and J is the coupling constant between two spins. Divide the HXY into two commuting parts Jingfu Zhang et al., Physical Review A, 72, 012331(2005)

25. The operator U in Matrix form for the 3-spin system where φ = Jt/√2 Jingfu Zhang et al., Physical Review A, 72, 012331(2005)

26. Phase Gate on 2ndspin Two and three qubit Entangling Operators Bell States W - State

27. Experiment Equilibrium spectra Sample Pulse sequence to prepare PPS (using only the nearest neighbour couplings) Tomography of ‌ 010 > PPS 1H 13C 19F Gz

28. Pulse sequence to simulate XY Hamiltonian 1H 13C 19F

29. Experimental results Bell state on end qubits Attenuated Correlation(c)=0.86 Attenuated Correlation(c)=0.88 K R K Rao, and Anil Kumar, Int. J. Quant. Inf. 10, 1250039 (2012).

30. Experimental results W-state Attenuated Correlation(c)=0.89 GHZ-state ( |000> + |111> )/√2 Attenuated Correlation(c)=0.81 K R K Rao, and Anil Kumar, Int. J. Quant. Inf. 10, 1250039 (2012).

31. Quantum simulation of frustrated Ising spins by NMR K. Rama Koteswara Rao1, Hemant Katiyar3, T.S. Mahesh3, Aditi Sen (De)2, Ujjwal Sen2 and Anil Kumar1: Phys. Rev A88 , 022312 (2013). 1 Indian Institute of Science, Bangalore 2 Harish-Chandra Research Institute, Allahabad 3 Indian Institute of Science Education and Research, Pune

32. A spin system is frustrated when the minimum of the system energy does not correspond to the minimum of all local interactions. Frustration in electronic spin systems leads to exotic materials such as spin glasses and spin ice materials. 3-spin transverse Ising system The system is non-frustrated If J is positive Anti-ferromagnetic The system is frustrated If J is negative Ferromagnetic

33. Here, we simulate experimentally the ground state of a 3-spin system in both the frustrated and non-frustrated regimes using NMR. Experiments at 290 K in a 500 MHz NMR Spectrometer of IISER-Pune Diagonal elements are chemical shifts and off-diagonal elements are couplings. This rotation was realized by a numerically optimized amplitude and phase modulated radio frequency (RF) pulse using GRadient Ascent Pulse Engineering (GRAPE) technique1. 1N. Khaneja and S. J. Glaser et al., J. Magn. Reson. 172, 296 (2005).

34. Non-frustrated Frustrated

35. Multipartite quantum correlations Entanglement Score using deviation Density matrix Initial State: Equal Coherent Superposition State. Fidelity = .99 Non-frustrated regime: Higher correlations Frustrated regime: Lower correlations Ground State GHZ State (J >> h) (׀000> -׀111>)/√2 Fidelity = .984 Quantum Discord Score using full density matrix Koteswara Rao et al.Phys. Rev A88 , 022312 (2013).

36. Recent Developments in our Laboratory: (To be published) • 1. Simulation of Mirror Inversion of Quantum States in an XY spin-chain. • 2. Quantum Simulation of Dzyaloshinsky-Moriya (DM) interactionin presence of Heisenberg XY interaction for study of Entanglement Dynamics

37. Mirror Inversion of quantum states in an XY spin chain* JN-1 J1 J2 3 N N-1 1 2 • The above XY spin chain generates the mirror image of any input state up to a phase difference. • Entangled states of multiple qubits can be transferred from one end of the chain to the other end *Albanese et al., Phys. Rev. Lett. 93, 230502 (2004) *P Karbach, and J Stolze et al., Phys. Rev. A 72, 030301(R) (2005)

38. NMR Hamiltonian of a weakly coupled spin system Control Hamiltonian Simulation In practice

39. Simulation GRAPE algorithm 2) An algorithm by A Ajoy et al. Phys. Rev. A 85, 030303(R) (2012) • Here, we use a combination of these two algorithms to simulate the unitary evolution of the XY spin chain

40. 4-spin chain 5-spin chain In the experiments, each of these decomposed operators are simulated using GRAPE technique

41. N-spin chains For odd N For even N The number of operators in the decomposition increases only linearly with the number of spins (N).

42. Experiment Molecular structure and Hamiltonian parameters 5-spin system The dipolar couplings of the spin system get scaled down by the order parameter (~ 0.1) of the liquid-crystal medium. The sample 1-bromo-2,4,5-trifluorobenzeneis partially oriented in a liquid-crystal medium MBBA The Hamiltonian of the spin system in the doubly rotating frame:

43. Quantum State Transfer: 4-spin pseudo-pure initial states Diagonal part of the deviation density matrices (traceless) The x-axis represents the standard computational basis in decimal form

44. Coherence Transfer: 5-spin initial states Spectra of Fluorine spins Proton spins K R K Rao, T S Mahesh, and A Kumar, (under review), Phys. Rev. A

45. Coherence Transfer: 5-spin initial states Proton spins Spectra of Fluorine spins

46. Entanglement Transfer Initial States Final States Reconstructed deviation density matrices (trace less) of spins 1 and 2, and spins 4 and 5. K R K Rao, T S Mahesh, and A Kumar, (under review), Phys. Rev. A

47. Entanglement Transfer Initial States Final States Reconstructed deviation density matrices (trace less) of spins 1 and 2, and spins 4 and 5.

48. Fluorine transitions Proton transitions The effective transverse relaxation times (T2*): 40-100ms 110-150ms The Fidelity of the Final state is high. This means that the state is reproduced with high fidelity. But the attenuated correlations are low. This means the intensities of the signal in the final state are low compared to initial state. This is mainly due to decoherence.

49. The Genetic Algorithm John Holland Charles Darwin 1866 1809-1882

50. Genetic Algorithm “Genetic Algorithms are good at taking large, potentially huge, search spaces and navigating them, looking for optimal combinations of things, solutions one might not otherwise find in a lifetime” Here we apply Genetic Algorithm to Quantum Information Processing We have used GA for (1) Quantum Logic Gates (operator optimization) and (2) Quantum State preparation (state-to-state optimization) V.S. Manu et al. Phys. Rev. A 86, 022324 (2012)