Quantum Information Processing by NMR Anil Kumar Department

Quantum Information Processing by NMR Anil Kumar Department of Physics and NMR Research Centre Indian Institute of Science, Bangalore-560012 “International Conference on Quantum Frontiers and Fundamentals” 30 th April to 4 th May, 2018 Raman Research Institute, Bangalore 1

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

Nuclear Magnetic Resonance (NMR) 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). B 1 0 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 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 3

DSX 300 7. 0 Tesla AV 700 16. 5 Tesla NMR Research Centre, IISc DRX 500 11. 7 Tesla AV 500 11. 7 Tesla Field/ Frequency stability = 1: 10 9 1 PPB AMX 400 9. 4 Tesla 4

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. 5

Addressability in 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 un-equal couplings between the spins, yielding resolved transitions in a multiplet, in order to encode information as qubits. 6

NMR Qubits An example of a Hetero-nuclear three qubit system. 13 CHFBr 1 H = 500 MHz 2 13 C = 125 MHz JCH = 225 Hz JCF = -311 Hz JHF = 50 Hz 19 F = 470 MHz Br (spin 3/2) is a quadrupolar nucleus, is self-decoupled from the rest of the spins, and can be ignored. 7

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

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 ( α 1 + 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. 9

Pseudo-Pure State In a two-qubit Homo-nuclear system: (Under High Field Approximation) (i) Equilibrium: ρ = 105 + Δρ = {2, 1, 1, 0} Δρ ~ Iz 1+Iz 2 = { 1, 0, 0, -1} 0 11 1 01 2 00 (ii) Pseudo-Pure 0 11 Δ ρ = {4, 0, 0, 0} Δρ ~ Iz 1+Iz 2 + 2 Iz 1 Iz 2 = { 3/2, -1/2, -1/2} 1 10 0 01 4 00 0 10

Preparation of Pseudo-Pure States • Spatial Averaging • Logical Labeling Cory, Price, Havel, PNAS, 94, 1634 (1997) N. Gershenfeld et al, Science, 275, 350 (1997) Kavita, Arvind, Anil Kumar, Phy. Rev. A 61, 042306 (2000) • Temporal Averaging E. Knill et al. , Phy. Rev. A 57, 3348 (1998) • Pairs of Pure States (POPS) B. M. Fung, Phys. Rev. A 63, 022304 (2001) • Spatially Averaged Logical Labeling Technique (SALLT) T. S. Mahesh and Anil Kumar, Phys. Rev. A 64, 012307 (2001) • Using long lived Singlet States S. S. Roy and T. S. Mahesh, Phys. Rev. A 82, 052302 (2010). 11

Spatial Averaging: Cory, Price, Havel, PNAS, 94, 1634 (1997) Most commonly used method I 1 z = 1/2 1 0 0 0 0 -1 1 0 0 0 0 -1 0 0 0 0 -1 I 2 z = 1/2 (2) (p/3)X Eq. = I 1 z+I 2 z 1 (1) (p/4)X 2 Pseudo-pure state (1) p 3 (p/4)Y 4 1/2 J Gx 2 I 1 z I 2 z = 1/2 3 0 0 0 0 -1 I 1 z + I 2 z + 2 I 1 z. I 2 z = 1/2 1 0 0 0 0 -1 0 0 1 5 I 1 z + I 2 z + 2 I 1 z. I 2 z 6 Gradient Pulses make this a non 12 unitary operation

Achievements of NMR - QIP 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 10. Quantum State Tomography 12. Adiabatic Algorithms 11. Geometric Phase in QC 13. Bell-State discrimination 14. Error correction 15. Teleportation 16. Quantum Simulation 17. Quantum Cloning 18. Shor’s Algorithm 8. Creation of EPR and GHZ states 9. Entanglement transfer 19. No-Hiding Theorem Also performed in our Lab. Maximum number of qubits achieved in our lab: 8 In other labs. : 12 qubits; 13 Negrevergne, Mahesh, Cory, Laflamme et al. , Phys. Rev. Letters, 96, 170501 (2006).

Our own contributions are distributed into 8 Ph. D. theses and nearly 40 Publications. A few of these are briefly highlighted in the following. 14

Quantum Gates by NMR 15

Manipulation of Coupled Spins: Two Methods Coupling (J) Evolution Examples CNOT I 1 Transition-selective Pulses I 1 z+I 2 z p y x y 11 I 1 z+I 2 x (1/2 J) I 1 z+2 I 1 z. I 2 y I 2 00 x 1/4 J I 1 I 2 1/4 J 01 10 I 1 z+2 I 1 z. I 2 z 11 NOT 1 10 p 01

y I 1 y x x 11 -y SWAP -y 10 p 2 I 2 p 1 01 p 3 00 y y -x -x 111 I 1 Toffoli C 2 NOT I 2 010 I 3 I 2 I 3 001 100 000 y I 1 110 p 011 y -x x non-selective p pulse + a p on 000 001 011 OR/NOR 010 111 110 001 000 p 101 100

Logic Gates Using Spin (qubit) NOT(I ) 1 Selective pulses XOR 2 XOR 1 Using Transition Selective pulses Kavita Dorai, Ph. D Thesis, IISc, 2000.

Logical SWAP INPUT 0 0 OUTPUT 0 0 1 0 1 1 0 2 1 0 1 , 2 2 , 1 1 1 p 0 1 1 1 0 11 p 1 01 p 2 XOR+SWAP 1 p 1 2 00 10 p 3 Kavita, Arvind, and Anil Kumar Phys. Rev. A, 2000, 61, 042306

Toffoli Gate = C 2 -NOT 1 2 3 Using Transition Selective Pulses Eqlbm. Toffoli Kavita Dorai, Ph. D Thesis, IISc, 2000. 1 2 3

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

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

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

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

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.

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 Information Processing

Adiabatic Geometric Phase M. V. Berry, Quantum phase factors accompanying adiabatic changes, Proc. R. Soc. Lond. A, 392, 45 (1984) Non-Adiabatic Geometric Phase Y. Aharonov and J. Anandan, Phase change during Cyclic Quantum Evolution, Phys. Rev. Lett. 58(16), 1593 (1987) Cyclic path is a sufficient condition. We have used Non-Adiabatic Geometric Phase in several experiment

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 10 A. B = (p)q 11 . 10 (p) 11 q+p+f The resulting path encloses a solid angle W = 2 f. (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.

Unitary operator associated with the slice circuit: A. B = Solid angle W = 2 f. 13 C Spectra of 13 CHCl 3 Ranabir Das et al, J. Magn. Reson. , 177, 318 (2005).

Deutsch-Jozsa algorithm using geometric phases (by slice circuit): 1 H 13 C of 13 CHCl 3 Constant Uf(00) is identity matrix Constant Uf(11) is achieved by applying p pulse on the second qubit Balanced = Ranabir Das et al, J. Magn. Reson. , 177, 318 (2005).

Grover search algorithm using geometric phases (by slice circuit): Pseudo pure state |0 H Superpo sition H Cij H Selective Inversion: C 00 H | i H | j Inversion about Average Using Geometric phase gates f=p Inversion about Average C 00 = H U 00 = C 00(p) Ranabir Das et al, J. Magn. Reson. , 177, 318 (2005). Measure

Recent Developments in our Laboratory (i) Multipartite quantum correlations reveal frustration in quantum Ising spin systems: Experimental demonstration. K. Rama Koteswara Rao, Hemant Katiyar, T. S. Mahesh, Aditi Sen(De), Ujjwal Sen and Anil Kumar; Phys. Rev. A 88, 022312 (2013). (ii) An NMR simulation of Mirror inversion propagator of an XY spin Chain. K. R. Koteswara Rao, T. S. Mahesh and Anil Kumar, Phys. Rev. A 90, 012306 (2014). (iii) Quantum simulation of 3 -spin Heisenberg XY Hamiltonian in presence of DM interaction- entanglement preservation using initialization operator. V. S. Manu and Anil Kumar, Phys. Rev. A 89, 052331 (2014). 32

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 , 022312 (2013). Indian Institute of Science, Bangalore 2 Harish-Chandra Research Institute, Allahabad 3 Indian Institute of Science Education and Research, Pune 1

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 If J is negative Ferromagnetic The system is non-frustrated If J is positive Anti-ferromagnetic The system is frustrated

ØHere, 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 IISERPune 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) technique 1. 1 N. Khaneja and S. J. Glaser et al. , J. Magn. Reson. 172, 296 (2005).

Non-frustrated Frustrated

Multipartite quantum correlations Entanglement Score using deviation Density matrix Non-frustrated regime: Higher correlations Initial State: Equal Coherent Superposition State. Fidelity =. 99 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 A 88 , 022312 (2013).

(ii) Quantum simulation of 3 -spin Heisenberg XY Hamiltonian in presence of DM interaction. and Entanglement preservation using initialization operator. V. S. Manu and Anil Kumar, Phys. Rev. A 89, 052331 (2014). 38

Manu et al. Phys. Rev. A 89, 0523331 (2014) Using Genetic Algorithm, Quantum Simulation of Dzyaloshinsky-Moriya (DM) interaction (HDM) in presence of Heisenberg XY interaction (HXY) for study of Entanglement Dynamics and Entanglement preservation. Hou et al. 1 demonstrated a mechanism for entanglement preservation. 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, 327 292 (2012) Similar to Quantum Zeno Effect 39

The Hamiltonian Heisenberg XY interaction DM interaction Evolution Operator: Decomposing the U in terms of Single Qubit Rotations (SQR) and ZZ- evolutions. SQR by Hard pulse ZZ evolutions by Delays 40

Entanglement Preservation 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. concurrence µi are eigen values of the operator ρSρ*S, where S= σ1 y ⊗ σ2 y Without Operator O With Operator O Similar to Quantum Zeno Effect Entanglement (concurrence) oscillates during Evolution. Entanglement (concurrence) is preserved during Evolution. This confirms the Entanglement preservation method of Hou et al. 1 Manu et al. Phys. Rev. A 89, 052331 (2014). 1 Hou et al. Annals 41 of Physics, 327 292 (2012)

Some Earlier Developments in our Laboratory 1. Non-destructive discrimination of Bell States. Jharana Rani Samal, Manu Gupta, P. K. Panigrahi and Anil Kumar, J. Phys. B, 43, 095508 (2010) 2. Non-destructive discrimination of arbitrary set of orthogonal quantum States by phase estimation. V. S. Manu and Anil Kumar (75 years of Entanglement, Foundations and Information Theoretic Applications, Koltata Jan. , 2011, AIP conf. Proceedings; 1384, 229 -240 (2011). 3. 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) 4. Use of Nearest Neighbour 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 (Int. Journal of Quantum Information -10, 1250039 (2012).

Non-destructive discrimination of Bell States are Maximally Entangled 2 -qubit states. There are 4 Bell States |Φ+> = (|00> + |11>)/√ 2 |Φ-> = (|00> - |11>)/√ 2 |ψ+> = (|01> + |10>) √ 2 |ψ-> = (|01> - |10>)√ 2 Bell states play an important role in teleportation protocols

Protocol for Non-destructive Discrimination of Bell States Manu Gupta and P. Panigrahi (quant-ph/0504183 v); Int. J. of Quantum Information 5, 627 (2007) Theory Jharana Rani Samal*, Manu Gupta, P. Panigrahi and Anil Kumar, J. Phys. B, 43, 095508 (2010). Experimental verification by NMR

Panigrahi Circuit Needs two Ancilla Qubits Jharana Circuits Needs one Ancilla but two measurements Phase Measurement Parity Measurement

NMR Pulse Sequence for Discrimination of Bell States using one Ancilla Qubit For Parity measurement the Hadamard gates are removed and the CNOT Gates are reversed Jharana et al, J. Phys. B. , 43, 095508 (2010)

Created Bell States (|00> + |11>)HF |0>C (|00> - |11>)HF |0>C |Φ-> |Φ+> |ψ-> (|01> + |10>)HF |0>C (|01> - |10>)HF |0>C 1 = |000>; 7 = |110>; 3 = |010>; 5 = |100>

Population Spectra of 13 C |Φ+> |Φ-> |ψ+> |ψ->

Tomograph of the real part of the Density matrix confirming the Phase and Parity measurement. Jharna et al J. Phys. B 43, 095508 (2010)

Non-Destructive Discrimination of Arbitrary set of Orthogonal Quantum states by NMR using Quantum Phase Estimation. For this algorithm, the states need not have definite PARITY (and can even be in a coherent superposition state). This algorithm is thus more general than the just described Bell-State Discrimination. V. S. Manu and Anil Kumar (75 years of Entanglement, Foundations and Information Theoretic Applications, Koltata Jan. , 2011, AIP conf. Proceedings; 1384, 229 -240 (2011).

For a given eigen-vector |φ> of a Unitary Operator U, Phase Estimation Circuit, can be used for finding the eigen-value of |φ>. Conversely, with defined eigen-values, the Phase Estimation can be used for discriminating eigenvectors. By logically defining the operators with preferred eigen-values, the discrimination, as shown here, can be done with certainty. Quantum Phase Estimation ØSuppose a unitary operation U has a eigen vector |u> with eigen value e-iφ. Ø The goal of the Phase Estimation Algorithm is to estimate φ. As the state is the eigen-state, the evolution under the Hamiltonian during phase estimation will preserve the state.

Finding the n Operators Uj Let Mj be the diagonal matrix formed by eigen-value array {ei}j of Uj. And V is the matrix formed by the column vectors {|φk>}, Uj = V-1 × Mj × V Forming Eigen-value arrays 1. Eigen-value arrays { ei } should contain equal number of +1 and -1 2. 1 st eigen value array can have any order of +1 and -1. 3. 2 nd onwards should also contain equal number of +1 and -1, but should not be equal to earlier arrays or their complements.

Two Qubit Case Consider a set A complete set of orthogonal States, which are not Bell states. They have the 1 st qubit in state |0> or 1> and the 2 nd qubit in a superposed State ( 0> ± 1>) U 1 and U 2 can be shown as, ……… (3) Experimental implementation of this case is performed here by NMR Quantum state Discrimination Using NMR 55

For the operators U 1 and U 2 described in Eqn. (3) In terms of NMR Product Operators The Hamiltonians are given by Since various terms in H 1 and H 2 commute each other, we can write, Quantum state Discrimination Using NMR 56

Thin pulses are π/2 and broad pulses are π pulses. Phase of pulses on top Quantum state Discrimination Using NMR 57

Non-destructive Discrimination of two-qubit orthonormal states. Original Circuit Needing 2 -ancilla qubits Split Circuit needing 1 ancilla qubit Quantum state Discrimination Using NMR 58

Quantum state Discrimination Using NMR 60

Conclusions of the State Discrimination Ø A general scalable method for quantum state discrimination using quantum phase estimation algorithm is discussed, and experimentally implemented for a two qubit case by NMR. Ø As the direct measurements are performed only on the ancilla, the discriminated states are preserved. V. S. Manu and Anil Kumar (75 years of Entanglement, Foundations and Information Theoretic Applications, Koltata Jan. , 2011, AIP conf. Proceedings; 1384, 229 -240 (2011). Quantum state Discrimination Using NMR 61

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 in 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, 080502 (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 et al

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 θ and φ in steps of 15 o Jharana Rani Samal, Arun K. Pati and Anil Kumar, Phys. Rev. Letters, 106, 080401 (25 Feb. , 2011).

These three experiments have been recently (last summer) repeated* using IBM’s “Quantum Experience”, a 5 -Qubit. Superconductivity based quantum computer accessible to one-and-all via the “Cloud”. * By my 2017 summer students namely 1. Ayan Majumdar, IISER-Mohali 2. Santanu Mohapatra, IIT Khrgpur 3. Porvika Bala, NIT, Trichy

Hardware of the IBM quantum computer q http: //research. ibm. com/ibm-q/learn/what-is-quantum-computing/

v Superconducting coaxial lines q https: //www. youtube. com/watch? v=S 52 rx. ZG-zi 0

Old version v Coupling map = {0: [2], 1: [2], 3: [2], 4: [2]} where, a: [b] means a CNOT with qubit a as control and b as target can be implemented. chip layout

New version q This device went online January 24 th, 2017 v Coupling map = {0: [1, 2], 1: [2], 3: [2, 4], 4: [2]} where, a: [b] means a CNOT with qubit a as control and b as target can be implemented. NEW v The connectivity is provided by two coplanar NEW waveguide (CPW) resonators with resonances around 6. 0 GHz (coupling Q 2, Q 3 and Q 4) and 6. 5 GHz (coupling Q 0, Q 1 and Q 2). Each qubit has a dedicated CPW for control and readout. This picture shows the chip layout. IBM Quantum Experience ibmqx 2 device

To get No. of Shots low error in result, you can increase the number of shots(experi ments) from here Quantum score ----> time progresses from left to right Quantum gates Quantum Gates Measurement operator Measurement Operator qubits freq. of qubit relaxation time decoherence time

Bell State 0. 510 Example Number of shots 100 Probability of finding the system in state 11 0. 490 Probability of finding the system in state 00

Fidelity improves as the number of shots is increased. 0. 502 0. 505 Number of shots 8192 Numberofofshots 4000 400 0. 508 0. 497 0. 495 0. 492 Number of shots 400

Nondestructive discrimination of Bell states using phase & parity checking circuit v. This experiment already verified by NMR Fidelity 4. 0% v Recently this experiment was also implemented in IBM quantum experience by v Mitali Sisodia, Abhishek Shukla, Anirban Pathak, ar. Xiv: 1705. 00670 [quant-ph]) v This group used the old version of the ibmqx 2 device And got low fidelity 13. 7%

Anirban Phatak et al Actual circuit v Fidelity Modified circuit

Ayan used the New version of the ibmqx 2 device, had fewer gates, and got high Fidelity (0. 9%) q Measurement in x-basis v High Fidelity = 0. 9%

Nondestructive discrimination of arbitrary set of orthogonal quantum states v This protocol already verified by a NMR v Manu V S & Anil Kumar, AIP Conf. Proc. 1384, 229 -240(2011).

Also Verified here by using the New version of the ibmqx 2 device (Ayan) v Max. Abs. Deviation: NMR = 7. 2%, ibmqx 2 = 2. 0 %

Experimental Test of Quantum of No-Hiding theorem by NMR. Jharana Rani Samal, Arun K. Pati and Anil Kumar, Phys. Rev. Letters, 106, 080401 (25 Feb. , 2011) Has now been implemented by new version of ibmqx 2 by Santanu Mohapatra of IIT Kolkata, in my lab.

IMPLEMENTING IT IN IBM QUANTUM COMPUTER In order to implement the above pulse sequences in this quantum computer, We need to convert these into quantum gates. We already know that So, the sequence of quantum gates for the randomization operator U is:

Initial state U Measuring 3 rd qubit Number of shots = 8129 0. 503 Probabilities Of the 3 rd qubit state 0. 497 Local unitary Operation for Extraction of

Extracting the Bell state Measuring 1 And 2 nd qub X basis Bell state Number of shots = 8192 0. 503 0. 497

IBM 16 Qubit Quantum Processor (ibmqx 3) IBM has recently placed a 16 -qubit quantum processor(ibmqx 3) on the cloud but it is still not accessible for all users. We first describe Secure multiparty summation using the 5 -qubit system. We later extend it to Secure summation of bigger number (n=> 2). We also simulate the summation of squares and cubes of numbers, and extend it to multiplication of numbers. For the later parts we need the 16 -Qubit system and therefore we have only simulations 83

EXPERIMENTAL VERIFICATION of Secure summation. For three party (m = 3) and one qubit (n = 1) secret state, assume that secret integers for P 1, P 2 and P 3 are 0, 1 and 0 respectively For fifteen party (m = 15) and one qubit (n = 1) secret integer Needing ibmqx 3 Figure showing the experimental result of the protocol for m = 3 and n = 1. The above result is obtained by taking an average of 8192 number of shots. The state 1 is obtained with a probability 0: 882. Result of Simulation. Figure showing the simulation of the protocol for m = 15 and n = 1. The y is are taken to be 1, 0, 1, 1, 0, 0, 0, 1, 1, and 0 respectively and q[1] is the ancilla qubit. Experiment will be done as soon as ibmqx 3 will 84 become available.

Circuit diagram for the square summation Circuit diagram for the cubic summation We are ready for the 16 qubit system and will implement the above protocols and many more as soon as this machine is made available to us. 85

• Thanks to the IBM for developing such a wonderful experimental setup and making it available to one and all

Summary NMR is continuing to provide a test bed for many quantum Phenomenon and Quantum Algorithms. 87

Acknowledgements Former QC- IISc-Associates/Students Prof. Arvind Prof. Kavita Dorai Prof. 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 Recent QC IISc - Students - IISER Mohali Dr. R. Koteswara Rao - Dortmund - IISER Mohali Dr. V. S. Manu - Univ. Minnesota - IISER Pune - CBMR Lucknow Other IISc Collaborators - IBM, Bangalore Prof. Apoorva Patel - NCIF/NIH USA Prof. K. V. Ramanathan - IISER Kolkata Prof. N. Suryaprakash - NISER Bhubaneswar Other Collaborators - Philips Bangalore - Univ. Minnesota Prof. Malcolm H. Levitt - UK Prof. P. Panigrahi IISER Kolkata - IISER Mohali Prof. Arun K. Pati HRI-Allahabad – IIT Bombay This lecture is dedicated to the memory of Ms. Jharana Rani Samal* (*Deceased, Nov. , 12, 2009) Prof. Aditi Sen Prof. Ujjwal Sen Mr. Ashok Ajoy HRI-Allahabad BITS-Goa-MIT-UCB Funding: DST/DAE/DBT 88 Thanks: NMR Research Centres at IISc, Bangalore for spectrometer time

Thank You 89