1. David G. Cory Department of Nuclear Engineering Massachusetts Institute of Technology Using Nuclear Spins for Quantum Information Processing and Quantum Computing

2. Quantum Information Processing The precise control of a set of coupled 2-level systems. H int =  I I z interaction with B field E |0 > |1 > E | > Qubit can be in a continuum of states Most states are superpositions of 0 and 1 0 1 “0 and 1” qubit spin

3. Addressable Qubits Chemically distinct spins H int =  I I z +  S S z H int =  I I z +  S S z +2πJI z S z interaction with B field IS 2-3 Dibromothiophene coupling between spins J IS

4. External Hamiltonian –Experimentally Controlled Hamiltonian: –Total Hamiltonian: H ext (t) =  RFx (t)·(I x +S x )+  RFy (t)·(I y +S y ) H total (t) controlled via H ext (t) I S J IS 9.6 T RF wave spins couple to RF field H total (t) = H int + H ext (t)

5. Single Qubit Gates IS qubit selective inversion pulse

6. Conditional Qubit Gates I S selective π/2 coupling selective π/2

7. Quantum & Classical Channel         Partial Trace System Environment 0.950.99 0.981.000.96 0.99

8. Decoherence Free Subspace phase noise: Collective phase noise: Encode Logical Qubit

9. DFS for Memory Information Noise strength (Hz) Encoded Un-Encoded EngineeredNoise    EncodeDecode

10. Samples and Hamiltonians Alanine (3 Qubits) Crotonic Acid (4 Qubits) C4 C1 C2 C3 J 12 J 23 J 34 C3 C1 C2 J 12 J 23 J 13

11. C N N C C C C C O–O– ND3+ND3+ D H H H H O H

12. Control of Entanglement X Measurement C=0.92 Z Measurement C=0.89 GHZ State C=0.88 Traced state C=0.71 Traced state Z Measurement C=0.80 X Measurement C=0.77 W State C=0.73

13. Pseudopure state - Product of two Singlet states (real part of the density matrix) Strong Measurement Entangle bits 1 & 2, and bits 3 & 4 Map bits 2 & 3 onto the Bell basis H Measurement of bits 2 & 3 |01> + |10>

14. Final Results – After Selective Strong Measurement in the Bell basis

15. n (number of H-CNot pairs) Final Correlation C C H H H () n H |000  GHZ | 000  for n = 0, 8, 16, … |100  for n = 4, 12, 20,... Output for n = 128 Correlation: 96.65% Correlation: 90.89% Output for n = 64 16

16. Output state Quantum Fourier Transform Shor’s algorithm Quantum simulations Quantum chaos Input state QFT Superoperator Fidelity =.99 Fidelity =.80

17. QFT Superoperator Theoretical QFT Superoperator Experimental QFT Superoperator

18. Statistical Verification of Control = number of qubits  reduced density matrix for qubit i m = 16 (+) m = 24 (.) m = 32 (x) m = 40 (o) Inset: average Q approaches CUE average exponentially. Random Circuit on n q = 8 qubits = 4, 6, 8, 10. P(Q)

19. 0.999 0.99 0.9 Gate Fidelity 0.9999 RF Power (Hz) Strongly Modulated Pulses modulation frequency addressability

20. Why is quantum noise so bad? Consider an entangled state: If any are disturbed, they all collapse How do we protect our information? To make matters worse: Can’t Copy Can’t Look No Majority Coding

21. Quantum Error Correction If the noise is sufficiently weak compared with your control rates then you correct a subset of errors with only a finite accuracy requirements. Encoder Memory Extra Bits Regularly Correct Error Decoder Discarded Bits

22. Arbitrary Collective Noise Expand your state space in the joint eigenstates of J 2 and J z J  can’t distinguish between the 2 paths to the (1/2) state. (1/2) 1  (1/2) 2 (0) 12  (1/2) 3 (1) 12  (1/2) 3 (1/2) 123 (3/2) 123

23. Noise Strength (Hz) Encoded, Y, Z Noise No Encoding, Y Noise Information Weak Noise Experimental Results Strong Noise Limit Z-X Noise 0.24 Un-Encoded 0.70 NS-Encoded No Noise 0.70 Z-X Noise Z-Y Noise Info

24. Nuclear Spins in the Solid State M = -N/2 N/2 N/2 -1 N/2 -2 -N/2 +1 -N/2 +2 0 Liquid state is a good test-bed for QIP, not a scalable approach to QC. Solid State appears to be scalable.

25. Spin Hamiltonian + Zeeman Hamiltonian +  =  B 0 Secular Dipolar Hamiltonian H tot = H Z + H D  >>  D ij ) use chemistry locally for error correction use spatial addressing to define qubits (magnetic field gradients) + + dipolar to nearest neighbor coupling single spin detection + +

26. The selective decoupling problem Consider a system consisting of pairs of spins. nearest neighbor coupling is well defined (d), there is a quasi-continuous broadening that arises from coupling to distant spins (D ) Typically d >> D. increasing D decreasing D d is fixed

27. Dipolar -> nn

28. Status of NMR QIP The zero-order average Hamiltonian is, where J 0 (x) and H 0 (x) are the zeroth order Bessel and Struve functions

29. Starting from I x A without control (left) and with control (right) BLUE: I x A ; BROWN: I x B ; BLACK: I x C BC coupling = 1/8 AB coupling

30. Starting from I x A without control (left) and with control (right)

31. Quantum Simulation, Spin Diffusion spin-spin correlation time ~ 6 µs diffusion time 10 -> 100 s steps ~ 10^8 mean displacement ~ 1 µm # spins involved 10^11 To connect with theorists combine this with nn coupling scheme.

32. Diffusion measurements k 2  s/cm 2 ) k 2  s  cm 2 ) (~10 -12 cm 2 /s) k 2  s/cm 2 )

33. 1 single spin: Result of a quantum computation Set of N spins: Collective measurement Transfer of polarization Transfer of polarization single spin transducer spins. single spin transducer spins. The final state of transducer spins is determined by the state of the controlling (single) spin The final state of transducer spins is determined by the state of the controlling (single) spin Spin Transducer 1H1H 19 F

34. Global CNOT Ideal behavior: |0> 0 |00000…0> |0> 0 |00000…0> 1-N |1> 0 |00000…0> 1-N |1> 0 |11111…1> 1-N (initial state) (final state) |0>|0> |2>|2> |1>|1> |N>|N> … … Trace Measurement

35. Series of Cnots Requisites on Control Addressable spins. Interactions w/ single spin Control operator Entanglementnone Gates or basic steps # for max Contrast Gate: CNOT n Final Contrast2

36. Maximum Entanglement Scheme    |0 00..0> |  >(|1 00..>+|0 00..>)/  2 |  >(|1 10..>+|0 00..>)/  2 |  > (|1 11..>+|0 00..>)/  2 |1>(|0 11..>+|1 00..>)/  2 |1>(|0 11..>+|1 01..>)/  2 |1> (|0 11..>+|1 11..>)/  2 |1>(|0 11..>+ |1 11..>- |1 11..>+|0 11..>)/2 =  fin  |1>  |0 111111...1> vs. |0>  |0 00000..0> U 1 =U HAD U 2 =U CNOT 12 *… U CNOT 1N U 3 = U CNOT 01 U 4 =U 2 -1 U 5 =U 1 -1 =U HAD |0>|0> |2>|2> |1>|1> |N>|N> H …… … … H Measurement  U 1  U 2  U 3  U 4  U 5

37. Series of Cnots Entanglement w/ Cnots Entanglement w/ MQC Perturbative approach Requisites on Control Addressable spins. Interactions w/ single spin Addressable spins. Only one interaction w/ single spin Collective control Refocusing of the control operator Collective control Control operator EntanglementnoneCat State Ground State of Heisenberg Hamiltonian Gates or basic steps # for max Contrast Gate: CNOT n Gate:CNOT 2n-1 Sequence: DQ & Dipolar Ham: 2n +1CNOT Sequence: DQ1 & Dipolar Hamiltonian: ~n Final Contrast22~1

38. Creation of Cat State With the N-quantum Grade Raising operator: the pure state is transformed into the cat state:

39. Series of Cnots Entanglement w/ Cnots Entanglement w/ MQC Perturbative approach Requisites on Control Addressable spins. Interactions w/ single spin Addressable spins. Only one interaction w/ single spin Collective control Refocusing of the control operator Collective control Control operator EntanglementnoneCat State Ground State of Heisenberg Hamiltonian Gates or basic steps # for max Contrast Gate: CNOT n Gate:CNOT 2n-1 Sequence: DQ & Dipolar Ham: 2n +1CNOT Sequence: DQ1 & Dipolar Hamiltonian: ~n Final Contrast22~1

40. Limited number of spins. Simulated all the pulse sequences needed for the wanted propagators, varying the # of repetitions of the cycle. Quantum Transducer Model: –linear chain of spins. –all the couplings are taken into account.

41. Results: Entanglement Scheme Simulations (8 spins) Global Entanglement Entanglement of 1 st spin 1

42. Conclusions Liquid state NMR is a useful QIP test-bed ~ 8 qubits. Systems of dipole coupled spins are universal for QIP. Solid state, nuclear spin approaches appear to be scalable. For selected problems we are already beyond the capabilities of classical computing. Introduced a quantum transducer that uses entanglement to make a classical measurement that could not otherwise be realized.

43. Dr. Timothy Havel Professor Seth Lloyd Professor Raymond Laflamme Dr. Chandrasekhar Ramanathan Dr. Joseph Emerson Dr. Grum Teklemariam Dr. Marco Pravia Dr. Evan Fortunato Dr. Greg Boutis Dr. Yaakov Weinstein Nicolas Boulant Paola Cappellaro Zhiying (Debra) Chen Hyung Joon Cho Daniel Greenbaum Jonathan Hodges Suddhasattwa Sinha