1. Quantum Computing and Nuclear Magnetic Resonance Jonathan Jones EPS-12, Budapest, August 29 2002 Oxford Centre for Quantum Computation http://www.qubit.org jonathan.jones@qubit.org

2. Outline l What is quantum computing? l How to do quantum computing with NMR l What has been done? l What are we trying to do? l How far can we go? l J. A. Jones, PhysChemComm 11 (2001) l J. A. Jones, Prog. NMR Spectrosc. 38, 325 (2001)

3. Qubits & quantum registers 0 or 1 Classical Bit 0 or 1 or 0 1 Quantum Bit Classical register 101 Quantum register 000 001 010 011 100 101 110 111

4. 000 001 010 011 100 101 110 111 F(000) F(001) F(010) F(011) F(100) F(101) F(110) F(111) F (x) Quantum Processor Quantum logic gates INPUT OUTPUT Quantum parallel processing

5. What could we do with one? l Easy to do the calculations, but hard to get the answers out from the resulting superposition! l Simulate quantum mechanics l Factor numbers: Shor’s algorithm »The end of public key cryptography? »Generalises to the Abelian Hidden Subgroup problem l Speed up searches: Grover’s algorithm l QC is not the answer to everything!

6. How might we build one? l To build a quantum computer you need l quantum two level systems (qubits), l interacting strongly with one another, l isolated from the environment, but l accessible for read out of answers etc. l See Fortschritte der Physik 48, 767-1138 (2000) »Scalable Quantum Computers - Paving the Way to Realization, S. L. Braunstein and H.-K. Lo

7. Nuclear Magnetic Resonance l Conventional liquid state NMR systems (common in chemistry and biochemistry labs) l Use two spin states of spin-1/2 nuclei ( 1 H, 13 C, 15 N, 19 F, 31 P) as a qubit l Address and observe them with RF radiation l Nuclei communicate by J couplings l Conventional multi-pulse NMR techniques allow gates to be implemented l Actually have a hot thermal ensemble of computers

8. NMR experiments

9. NMR Hamiltonians l Usually written using “product operators”. For two coupled spins l Can apply RF pulses, with good control of amplitude, frequency and phase. Normally work in a frame rotating at the RF frequency. For a single spin

10. Typical frequencies (14.4T) l For a 1 H nucleus (magnets are usually described by their 1H NMR frequency) l For a 13 C nucleus l For a 19 F nucleus l J couplings lie in the range of 1Hz–1kHz

11. Single Qubit Rotations l Single spin rotations are implemented using resonant RF pulses l Different spins have different resonance frequencies and so can be selectively addressed l Frequency space can become quite crowded, especially for 1 H (other nuclei are much better)

12. Controlled- NOT ( CNOT ) gates l The controlled-NOT gate flips the target bit (T) if the control bit (C) is set to 1 C T C T C = C T = T  C (Exclusive- OR ) C0011C0011 T0101T0101 C´ 0 1 T´ 0 1 0

13. CNOT and SPI (1973) Selective Population Inversion: a 180° pulse applied to one line in a doublet is essentially equivalent to CNOT 180°

14. Hamiltonian Sculpting l Combine periods of evolution under the background Hamiltonian with periods when various RF pulses are also applied l The Average Hamiltonian over the entire pulse sequence can be manipulated l In general terms can be scaled down or removed at will, and desired forms can be implemented l Highly developed NMR technique l Ultimately equivalent to building circuits out of gates

15. The spin echo l Most NMR pulse sequences are built around spin echoes which refocus Zeeman interactions l Often thought to be unique to NMR, but in fact an entirely general quantum property l Interaction refocused up to a global phase

16. CNOT and INEPT (1979) The CNOT propagator can be expanded as a sequence of pulses and delays using a standard NMR approach… … to give a pulse sequence very similar to an INEPT coherence transfer experiment

17. Example results

18. Molecules and algorithms l A wide range of molecules have been used with 1 H, 13 C, 15 N, 19 F and 31 P nuclei l A wide range of quantum algorithms have been implemented l Some basic quantum phenomena have been demonstrated

19. State of the art l Shor’s quantum factoring algorithm to factor 15 Nature 414, 883 (2001)

20. The initialisation problem l Conventional QCs use single quantum systems which start in a well defined state l NMR QCs use an ensemble of molecules which start in a hot thermal state l Can create a “pseudo-pure” initial state from the thermal state: exponentially inefficient l Better approach: use para-hydrogen!

21. Two spin system l A homonuclear system of two spin 1/2 nuclei: four energy levels with nearly equal populations Equalise the populations of the upper states leaving a small excess in the lowest level

22. Two spin system l A homonuclear system of two spin 1/2 nuclei: four energy levels with nearly equal populations Equalise the populations of the upper states leaving a small excess in the lowest level A “pseudo-pure” state Excess population is exponentially small

23. para-hydrogen l The rotational and nuclear spin states of H 2 molecules are inextricably connected by the Pauli principle l Cooling to the J=0 state would give pure para- hydrogen with a singlet spin state l Cooling below 150K in the presence of an ortho/para catalyst gives significant enhancement of the para population l Enhancement is retained on warming if the ortho/para catalyst is removed

24. Using para-hydrogen l para-hydrogen has a pure singlet nuclear spin state Can’t do computing with this directly because the H 2 molecule is too symmetric: we need to break the symmetry so that we can address the two nuclei individually

25. Vaska’s catalyst l Add the p-H 2 to some other molecule, e.g. Vaska’s catalyst Ir PPh 3 OCOCCl + H—H Ir PPh 3 COCO Cl H H The two 1 H nuclei now have different chemical shifts and so can be separately addressed

26. Tackling systematic errors l Quantum computers are vulnerable to errors l Random errors can be tackled using advanced forms of error correcting codes l Systematic errors can be tackled using various approaches for robust coherent control l The traditional NMR approach (composite pulses) has already proved productive

27. Tycko composite 90 sequence l Tycko’s composite 90° pulse 385 0 ·320 180 ·25 0 gives good compensation for small off- resonance errors l Works well for any initial state, not just for z l Now generalised to arbitrary angles l New J. Phys. 2.6 (2000)

28. Pulse length errors l The composite 90° pulse 115 62 ·180 281 ·115 62 corrects for pulse length errors l Works well for any initial state, not just z l Generalises to arbitrary angles l More complex sequences give even better results l quant-ph/0208092

29. How far can we go with NMR? l NMR QCs with 2–3 qubits are routine l Experiments have been performed with up to 7 qubits; 10 qubits is in sight l Beyond 10 qubits it will get very tricky! »Exponentially small signal size »Selective excitation in a crowded frequency space »Decoherence (relaxation) »Lack of selective reinitialisation »Lack of projective measurements »Fortschritte der Physik 48, 909-924 (2000)

30. Thanks to…