1. Liquid State NMR Quantum Computing
Financial supports from Kinki Univ.,
MEXT and JSPS
Liquid State NMR Quantum Computing
Mikio Nakahara,
Research Centre for Quantum Computing, Kinki University, Japan
Physical Realizations of Tehran, Jan. 2009

2. Physical Realizations of QC @ Tehran, Jan. 2009
Plan of Talk
1. Introduction
2. NMR
3. NMR Hamiltonian
4. Gate Operations
5. Pseudopure State
6. Measurement
7. DiVincenzo Criteria
8. Summary
Physical Realizations of Tehran, Jan. 2009

3. Physical Realizations of QC @ Tehran, Jan. 2009
1. Introduction
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4. Physical Realizations of QC @ Tehran, Jan. 2009
Qubits in NMR Molecule
Trichloroethylene
Physical Realizations of Tehran, Jan. 2009

5. Physical Realizations of QC @ Tehran, Jan. 2009
Plan of Talk
1. Introduction
2. NMR
3. NMR Hamiltonian
4. Gate Operations
5. Pseudo-Pure State
6. Measurement
7. DiVincenzo Criteria
8. Summary
Physical Realizations of Tehran, Jan. 2009

6. NMR (Nuclear Magnetic Resonance ) =MRI (Magnetic Resonance Imaging)
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7. Physical Realizations of QC @ Tehran, Jan. 2009
NMR
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8. Physical Realizations of QC @ Tehran, Jan. 2009
Schematic of NMR
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9. Molecules used in NMR QC
Physical Realizations of Tehran, Jan. 2009

10. Physical Realizations of QC @ Tehran, Jan. 2009
Plan of Talk
1. Introduction
2. NMR
3. NMR Hamiltonian
4. Gate Operations
5. Pseudopure State
6. Measurement
7. DiVincenzo Criteria
8. Summary
Physical Realizations of Tehran, Jan. 2009

11. 3.1 Single-Qubit Hamiltonian
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12. Hamiltonian in Rotating Frame
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13. Physical Realizations of QC @ Tehran, Jan. 2009

14. Physical Realizations of QC @ Tehran, Jan. 2009
2-Qubit Hamiltonian
Physical Realizations of Tehran, Jan. 2009

15. Physical Realizations of QC @ Tehran, Jan. 2009

16. Physical Realizations of QC @ Tehran, Jan. 2009

17. Physical Realizations of QC @ Tehran, Jan. 2009
Plan of Talk
1. Introduction
2. NMR
3. NMR Hamiltonian
4. Gate Operations
5. Pseudopure State
6. Measurement
7. DiVincenzo Criteria
8. Summary
Physical Realizations of Tehran, Jan. 2009

18. Physical Realizations of QC @ Tehran, Jan. 2009
1-Qubit Gates
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19. Example: Hadamard gate
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20. Example: Hadamard gate 2
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21. Physical Realizations of QC @ Tehran, Jan. 2009
Selective addressing
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22. Physical Realizations of QC @ Tehran, Jan. 2009
In resonance:
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23. Physical Realizations of QC @ Tehran, Jan. 2009

24. Physical Realizations of QC @ Tehran, Jan. 2009

25. Physical Realizations of QC @ Tehran, Jan. 2009
2-Qubit Gates: CNOT
Physical Realizations of Tehran, Jan. 2009

26. Physical Realizations of QC @ Tehran, Jan. 2009
Plan of Talk
1. Introduction
2. NMR
3. NMR Hamiltonian
4. Gate Operations
5. Pseudopure State
6. Measurement
7. DiVincenzo Criteria
8. Summary
Physical Realizations of Tehran, Jan. 2009

27. Spins are in mixed state!
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28. Preparation of a pseudopure state in terms of temporal average method
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29. Temporal average method
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30. Averaging three contributions
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31. Physical Realizations of QC @ Tehran, Jan. 2009
Plan of Talk
1. Introduction
2. NMR
3. NMR Hamiltonian
4. Gate Operations
5. Pseudopure State
6. Measurement
7. DiVincenzo Criteria
8. Summary
Physical Realizations of Tehran, Jan. 2009

32. 6.1 Free Induction Decay (FID)
|00〉
|01〉
|10〉
|11〉
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33. Free Induction Decay (FID)
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34. Physical Realizations of QC @ Tehran, Jan. 2009

35. 6.2 Quantum State Tomography
We want to “measure” the density matrix.
Measure observable such as magnetizations to find linear combinations of the matrix elements of the density matrix.
Not enough equations are obtained.
Deform the density matrix with pulses to obtain enough number of equations.
Physical Realizations of Tehran, Jan. 2009

36. Physical Realizations of QC @ Tehran, Jan. 2009
2-Qubit QST
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37. Physical Realizations of QC @ Tehran, Jan. 2009

38. Physical Realizations of QC @ Tehran, Jan. 2009

39. Physical Realizations of QC @ Tehran, Jan. 2009
Plan of Talk
1. Introduction
2. NMR
3. NMR Hamiltonian
4. Gate Operations
5. Pseudopure State
6. Measurement
7. DiVincenzo Criteria
8. Summary
Physical Realizations of Tehran, Jan. 2009

40. DiVincenzo Criteria for NMR QC
A scalable physical system with well characterized qubits.
The ability to initialize the state of the qubits to a simple fiducial state, such as |00…0>.
Long decoherence times, much longer than the gate operation time.
A “universal” set of quantum gates.
A qubit-specific measurement capability.
Physical Realizations of Tehran, Jan. 2009

41. Physical Realizations of QC @ Tehran, Jan. 2009
Scalability
Selective addressing to each qubit becomes harder and hader as the # of qubits increases. Limited # of nuclear spices and overlap of resonance freqs.
Signal strength is suppressed as the # of qubits increases. Readout problem.
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42. Initialization (pseudopure state)
# of steps required to prepare a pseudopure state increases exponentially as the # of qubits increases.
No real entanglement
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43. Physical Realizations of QC @ Tehran, Jan. 2009
Long decoherence time
Decoherence time
Single-qubit gate operation time
Two-qubit gate op. time
May execute Shor’s algorithm for 21=3X7.
Physical Realizations of Tehran, Jan. 2009

44. A “universal” set of quantum gates.
One-qubit gates by Rabi oscillation.
Two-qubit gates by J-coupling.
Cannot turn off interactions; reforcusing technique becomes complicated as the # of qubits increases.
Physical Realizations of Tehran, Jan. 2009

45. Measurement capability.
FID is a well-established techunique.
Quantum State Tomograpy and Quantum Process Tomography are OK.
S/N scales as , which limits the # of qubits to ~ 10.
Physical Realizations of Tehran, Jan. 2009

46. Physical Realizations of QC @ Tehran, Jan. 2009
Still…
NMR QC is commercially available.
It can execute small scale quantum algorithms.
It serves as a test bed for a real QC to come.
May ideas in other realizations are inspired from NMR.
We use NMR QC to demonstrate theoretical ideas, such as decoherence suppression, optimal control of a Hamiltonian etc.
Physical Realizations of Tehran, Jan. 2009

47. Physical Realizations of QC @ Tehran, Jan. 2009
Plan of Talk
1. Introduction
2. NMR
3. NMR Hamiltonian
4. Gate Operations
5. Pseudopure State
6. Measurement
7. DiVincenzo Criteria
8. Summary
Physical Realizations of Tehran, Jan. 2009

48. Physical Realizations of QC @ Tehran, Jan. 2009
Liquid state NMR QC is based on a well-established technology. Most of the materials introduced here have been already known in the NMR community for decades.
There are still many papers on NMR QC.
It is required to find a breakthrogh for a liquid state NMR to be a candidate of a working QC.
ENDOR, Solid state NMR…
Thank you for your attention.
Physical Realizations of Tehran, Jan. 2009