Anil Kumar 1, K. Rama Koteswara Rao 1, T.S. Mahesh 2 and V.S.Manu 1 Indian Institute of Science, Bangalore Indian Institute of Science Education and Research, Pune Quantum Simulations by NMR: Recent Developments QIPA-December 2013, HRI-Allahabad

Simulating a Molecule: Using Aspuru-Guzik Adiabatic Algorithm (i) J.Du, et. al, Phys. Rev. letters 104, (2010). Used a 2-qubit NMR System ( 13 CHCl 3 ) 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. Recent Developments In 1982, Feynman suggested that it might be possible to simulate the evolution of quantum systems efficiently, using a quantum simulator

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

0 1. Nuclear spins have small magnetic moments and behave as tiny quantum magnets. 2. When placed in a magnetic field (B 0 ), spin ½ nuclei orient either along the field (|0  state) or opposite to the field (|1  state). 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 Magnetic Resonance (NMR) 3. A transverse radio-frequency field (B 1 ) tuned at the Larmor frequency of spins can cause transition from |0  to |1  (NOT Gate by a pulse). Or put them in coherent superposition (Hadamard Gate by a 90 0 pulse). Single qubit gates. NUCLEAR SPINS ARE QUBITS B1B1

DSX Tesla AMX Tesla AV Tesla AV Tesla DRX Tesla NMR Research Centre, IISc 1 PPB Field/ Frequency stability = 1:10 9

NMR Facility in IISc 800 MHz NMR spectrometer equipped with cryogenic probe Installed in April Tesla

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.

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

NMR Hamiltonian H = H Zeeman + H J-coupling =  i  zi  J ij I i  I j ii < j Weak coupling Approximation  i  j  J ij Two Spin System (AM) A2 A1 M2 M1 AA MM M 1 =  A  M 2 =  A  A 1 =  M  A 2 =  M     H =  i  zi  J ij I zi I zj Under this approximation spins having same Larmor Frequency can be treated as one Qubit ii < j Spin States are eigenstates

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

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

Pseudo-Pure State Pure State Tr(ρ ) = Tr ( ρ 2 ) = 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 We create a state in which all levels except one have EQUAL populations. Such a state mimics a pure state. ρ = 1/N ( α1 - Δρ ) Under High Temperature Approximation Here α = 10 5 and U 1 U -1 = 1 0  1  2  1  0  0  4  0 

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 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           Maximum number of qubits achieved in our lab: 8  Also performed in our Lab.  In other labs.: 12 qubits; Negrevergne, Mahesh, Cory, Laflamme et al., Phys. Rev. Letters, 96, (2006). 

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, (25 Feb., 2011) 2.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, (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 A 88, (2013). 4. Use of Genetic Algorithm for QIP. V.S. Manu and Anil Kumar, Phys. Rev. A 86, (2012)

No-Hiding Theorem S.L. Braunstein & A.K. Pati, Phys.Rev.Lett. 98, (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.

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, (2007).

NMR Pulse sequence for the Proof of No-Hiding Theorem The initial State ψ is 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, (R) (2012)

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

Use of nearest neighbor Heisenberg-XY interaction.

Solution Use Nearest Neighbor Interactions 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.

Koteswara Rao et al, IJQI, 10 (4) (2012) 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. In a first study towards this goal, we took a linear chain of 3-qubits Entangle end qubits as well as Entangle all the 3 qubits And Using only the Nearest-Neighbor Heisenberg XY Interaction,

Nearest-Neighbour Heisenberg XY Interaction

σ j x/y are the Pauli spin matrices and J is the coupling constant between two spins. 213 JJ Divide the H XY into two commuting parts Consider a linear Chain of 3 spins with equal couplings Jingfu Zhang et al., Physical Review A, 72, (2005)

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

Two and three qubit Entangling Operators Bell States Phase Gate on 2 nd spin W - State

Sample Equilibrium spectra Experiment Pulse sequence to prepare PPS (using only the nearest neighbour couplings) 1H1H 13 C 19 F Gz Tomography of ‌ 010 > PPS

Pulse sequence to simulate XY Hamiltonian 1H1H 13 C 19 F

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, (2012). Experimental results

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

Quantum simulation of frustrated Ising spins by NMR K. Rama Koteswara Rao 1, Hemant Katiyar 3, T.S. Mahesh 3, Aditi Sen (De) 2, Ujjwal Sen 2 and Anil Kumar 1 : Phys. Rev A 88, (2013). 1 Indian Institute of Science, Bangalore 2 Harish-Chandra Research Institute, Allahabad 3 Indian Institute of Science Education and Research, Pune

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. If J is negative Ferromagnetic If J is positive Anti-ferromagnetic The system is frustrated 3-spin transverse Ising system The system is non-frustrated

This rotation was realized by a numerically optimized amplitude and phase modulated radio frequency (RF) pulse using GRadient Ascent Pulse Engineering (GRAPE) technique 1. 1 N. Khaneja and S. J. Glaser et al., J. Magn. Reson. 172, 296 (2005). Diagonal elements are chemical shifts and off- diagonal elements are couplings.  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

Non-frustratedFrustrated

Multipartite quantum correlations Non-frustrated regime: Higher correlations Frustrated regime: Lower correlations Entanglement Score using deviation Density matrix Quantum Discord Score using full density matrix Ground State GHZ State (J >> h) ( ׀ 000> - ׀ 111>)/√2 Fidelity =.984 Initial State: Equal Coherent Superposition State. Fidelity =.99 Koteswara Rao et al. Phys. Rev A 88, (2013).

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

Mirror Inversion of quantum states in an XY spin chain* Entangled states of multiple qubits can be transferred from one end of the chain to the other end J1J1 J2J2 J N-1 N N *Albanese et al., Phys. Rev. Lett. 93, (2004) *P Karbach, and J Stolze et al., Phys. Rev. A 72, (R) (2005) The above XY spin chain generates the mirror image of any input state up to a phase difference.

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

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

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

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).

Molecular structure and Hamiltonian parameters 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-trifluorobenzene is partially oriented in a liquid- crystal medium MBBA The Hamiltonian of the spin system in the doubly rotating frame: 5-spin system Experiment

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

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

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

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

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

The effective transverse relaxation times (T 2 *): ms ms Fluorine transitions Proton transitions 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.

The Genetic Algorithm Directed search algorithms based on the mechanics of biological evolution Developed by John Holland, University of Michigan (1970’s) John Holland Charles Darwin

“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” Genetic Algorithm Quantum Information Processing 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, (2012)

Representation Scheme Representation scheme is the method used for encoding the solution of the problem to individual genetic evolution. Designing a good genetic representation is a hard problem in evolutionary computation. Defining proper representation scheme is the first step in GA Optimization*. In our representation scheme we have selected the gene as a combination of (i) an array of pulses, which are applied to each channel with amplitude (θ) and phase (φ), (ii) An arbitrary delay (d). It can be shown that the repeated application of above gene forms the most general pulse sequence in NMR * Whitley, Stat. Compt. 4, 65 (1994)

The Individual, which represents a valid solution can be represented as a matrix of size (n+1)x2m. Here ‘m’ is the number of genes in each individual and ‘n’ is the number of channels (or spins/qubits). So the problem is to find an optimized matrix, in which the optimality condition is imposed by a “Fitness Function”

Fitness function In operator optimization GA tries to reach a preferred target Unitary Operator (U tar ) from an initial random guess pulse sequence operator (U pul ). Maximizing the Fitness function F pul = Trace (U pul Χ U tar ) In State-to-State optimization F pul = Trace { U pul (ρ in ) U pul (-1) ρ tar † }

Single Qubit Rotation in a Two-qubit Homonuclear case H = 2π δ (I 1z – 1 2z ) + 2π J 12 (I 1z I 2z ) δ = 500 Hz, J= 3.56 Hz φ 1 = 2π, φ 2 = π, Θ = π/2, φ = π/2 φ 1 = π, φ 2 = 0 Θ = π/2, φ = π/2 π/2 Simulated using J = 0 Hamiltonian used Non-Selective (Hard) Pulses applied at the centre Sequence obtained by Genetic Algorithm By changing φ 1 and φ 2 one can excite either spin 1 or 2

Fidelity for finite J/δ

C-NOT Equilibrium Sequence obtained by Genetic Algorithm

Pseudo Pure State (PPS) creation All unfilled rectangles represent 90 0 pulse The filled rectangle is pulse. Phases are given on the top of each pulse. Fidelity w.r.t. to J/δ Sequence obtained by Genetic Algorithm 00 and 11 PPS are obtained by -/+ on 180 pulse. Other PPS are obtained by using an additional 180 SQR pulse

Bell state creation: From Equilibrium (No need of PPS) The Singlet Bell State Experimental Fidelity > 99.5 % Shortest Pulse Sequence for creation of Bell States directly from Equilibrium All blank pulses are 90 0 pulses. Filled pulse is a pulse. Phases and delays Optimized for best fidelity. T 1 =8.7 s T s =11.2 s V.S. Manu et al. Phys. Rev. A 86, (2012) Sequence obtained by Genetic Algorithm

(b) Quantum Simulation of Dzyaloshinsky-Moriya (DM) interaction ( H DM ) in presence of Heisenberg XY interaction (H XY ) for study of Entanglement Dynamics Manu et al. (Under Review)

DM Interaction 1,2  Anisotropic antisymmetric exchange interaction arising from spin-orbit coupling.  Proposed by Dzyaloshinski to explain the weak ferromagnetism of antiferromagnetic crystals (Fe 2 O 3, MnCO 3 ). DM Interaction 1,2  Anisotropic antisymmetric exchange interaction arising from spin-orbit coupling.  Proposed by Dzyaloshinski to explain the weak ferromagnetism of antiferromagnetic crystals (Fe 2 O 3, MnCO 3 ). Quantum simulation of a Hamiltonian H requires unitary operator decomposition (UOD) of its evolution operator, (U = e -iHt ) in terms of experimentally preferable unitaries. Using Genetic Algorithm optimization, we numerically evaluate the most generic UOD for DM interaction in the presence of Heisenberg XY interaction. 1. I. Dzyaloshinsky, J. Phys & Chem of Solids, 4, 241 (1958). 2. T. Moriya, Phys. Rev. Letters, 4, 228 (1960).

Decomposing the U in terms of Single Qubit Rotations (SQR) and ZZ- evolutions. SQR by Hard pulse ZZ evolutions by Delays The Hamiltonian Heisenberg XY interaction DM interaction Evolution Operator:

Fidelity is defined as. A. The Decomposition for ϒ = 0 -1: 1 Manu et al. (under Review). Fidelity > % Period of U(ϒ,τ) This has a maximum value of Optimization is performed for τ -> 0-15, which includes one complete period.

2 When ϒ > 1 -> ϒ’ < 1 ϒ’ = 1/ ϒ Using above decomposition, we studied entanglement preservation in a two qubit system.

Hou et al. 1 demonstrated a mechanism for entanglement preservation using H(J,D). They showed that preservation of initial entanglement is performed by free evolution interrupted with a certain operator O, which makes the state to go back to its initial state. 1 Hou et al. Annals of Physics, (2012) Entanglement Preservation Without Operator O With Operator O concurrence µ i are eigen values of the operator ρSρ*S, where S= σ 1y ⊗ σ 2y Entanglement (concurrence) oscillates during Evolution. Entanglement (concurrence) is preserved during Evolution. This confirms the Entanglement preservation method of Hou et al. (Manu et al. Under Review)

Other IISc Collaborators Prof. Apoorva Patel Prof. K.V. Ramanathan Prof. N. Suryaprakash Prof. Malcolm H. Levitt - UK Prof. P.Panigrahi IISER Kolkata Prof. Arun K. Pati HRI-Allahabad Prof. Aditi Sen HRI-Allahabad Prof. Ujjwal Sen HRI-Allahabad Mr. Ashok Ajoy BITS-Goa-MIT Prof. Arvind Prof. Kavita Dorai Prof. T.S. Mahesh Dr. Neeraj Sinha Dr. K.V.R.M.Murali Dr. Ranabir Das Dr. Rangeet Bhattacharyya - IISER Mohali - IISER Pune - CBMR Lucknow - IBM, Bangalore - NCIF/NIH USA - IISER Kolkata Dr.Arindam Ghosh - NISER Bhubaneswar Dr. Avik Mitra - Philips Bangalore Dr. T. Gopinath - Univ. Minnesota Dr. Pranaw Rungta - IISER Mohali Dr. Tathagat Tulsi – IIT Bombay Acknowledgements Other Collaborators Funding: DST/DAE/DBT Former QC- IISc-Associates/Students This lecture is dedicated to the memory of Ms. Jharana Rani Samal* (*Deceased, Nov., 12, 2009) Current QC IISc - Students Mr. K. R. Koteswara Rao Mr. V.S. Manu Thanks: NMR Research Centres at IISc, Bangalore and IISER-Pune for spectrometer time

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Quantum State Transfer Two qubit Entangling Operator Three qubit Entangling Operator