1. Quantum Control Synthesizing Robust Gates T. S. Mahesh
Indian Institute of Science Education and Research, Pune

2. Contents DiVincenzo Criteria Quantum Control
Single and Two-qubit control
Control via Time-dependent Hamiltonians
Progressive Optimization
Gradient Ascent
Practical Aspects
Bounding within hardware limits
Robustness
Nonlinearity
Summary

3. Criteria for Physical Realization of QIP
Scalable physical system with mapping of qubits
A method to initialize the system
Big decoherence time to gate time
Sufficient control of the system via time-dependent Hamiltonians
(availability of a universal set of gates).
5. Efficient measurement of qubits
DiVincenzo, Phys. Rev. A 1998

4. how best can we control its dynamics?
Quantum Control
Given a quantum system,
how best can we control its dynamics?
Control can be a general unitary or a state to state transfer
(can also involve non-unitary processes: eg. changing purity)
Control parameters must be within the hardware limits
Control must be robust against the hardware errors
Fast enough to minimize decoherence effects
or combined with dynamical decoupling to suppress decoherence

5. General Unitary Hilbert Space 1 UTG UEXP 0
General unitary is state independent:
Example: NOT, CNOT, Hadamard, etc.
Hilbert Space
1
UTG
UEXP
obtained by
simulation or
process
tomography
0
Fidelity =  Tr{UEXP·UTG} / N 2

6. State to State Transfer
A particular input state is transferred to a particular output state
Eg. 000  ( 000 + 111 ) /2
Hilbert Space
Target
Final
obtained by
tomography
Initial
Fidelity = FinalTarget 2

7. Universal Gates
Local gates (eg. Ry(), Rz()) and CNOT gates together form a universal set
Example: Error Correction Circuit
Chiaverini et al, Nature 2004

8. Degree of control Fault-tolerant computation
- E. Knill et al, Science 1998.
Quantum gates need not be perfect
Error correction can take care of imperfections
For fault tolerant computation: Fidelity ~

9. Single Qubit (spin-1/2) Control
(up to a global phase)
Bloch sphere

10. ~ NMR spectrometer B0 B1cos(wrft) RF coil Pulse/Detect Sample
resonance at 0 =B0 ~ B0
Superconducting
coil
B1cos(wrft)

11. ~ Control Parameters B0 B1cos(wrft)  01 = 0 - ref 1 = B1  rf
All frequencies are measured w.r.t. ref 
RF offset = 
= rf - ref
 (kHz rad)
Chemical Shift
01 = 0 - ref
1 = B1 
rf ~
time B0
 RF duration
1 RF amplitude
 RF phase
 RF offset
B1cos(wrft)

12. Single Qubit (spin-1/2) Control
(in RF frame) x
(in REF frame) y
Bloch sphere
90-x
90x y
A general state:
(up to a global phase)

13. Single Qubit (spin-1/2) Control
(in RF frame)
(in REF frame)

14. Single Qubit (spin-1/2) Control
(in RF frame)
(in REF frame)
Turning OFF 0 : Refocusing y X
Refocus Chemical Shift 
time x
w01

15. Two Qubit Control
Local Gates

16. Qubit Selective Rotations - Homonuclear
Band-width  1/ 
 = 1 1 2
dibromothiophene
non-selective
 = 1
selective
Not a good method: ignores the time evolution

17. ~ Qubit Selective Rotations - Heteronuclear 13CHCl3 1H (500 MHz @ 11T)
13C (125 MHz @ 11T)
Larmor frequencies are separated by MHz
Usually irradiated by different coils in the probe
No overlap in bandwidths at all
Easy to rotate selectively ~

18. Two Qubit Control
Local Gates
CNOT Gate

19. Two Qubit Control   Refocussing: X 1 2   Z X 1 2  Chemical shift
Coupling constant
Refocussing: X
Refocus Chemical Shifts 1 2   Z
Rz(90) 
 = 1/(4J)
time X
Refocus 0 & J-coupling 1 2 
Rz(0) 
time

20. Two Qubit Control Z H = = Chemical shift Chemical shift
Coupling constant Z H =
1/(4J)
R-z(90)
time X
R-y(90) =

21. Control via Time-dependent Hamiltonians
H = H (a (t), b (t) , g (t) , )
a (t) t
NOT EASY !! (exception: periodic dependence)

22. Control via Piecewise Continuous Hamiltoniansb3 g3
H 3 a1 b1 g1
H 1 a2 b2 g2
H 2 a4 b4 g4
H 4
Time

23. Numerical Approaches for Control
Progressive Optimization
D. G. Cory & co-workers, JCP 2002
Mahesh & Suter, PRA 2006
Gradient Ascent
Navin Khaneja et al, JMR 2005
Common features
Generate piecewise continuous Hamiltonians
Start from a random guess, iteratively proceed
Good solution not guaranteed
Multiple solutions may exist
No global optimization

24. Piecewise Continuous Control
D. G. Cory, JCP 2002
Strongly Modulated Pulse (SMP)
(t3,w13,f3,w3) …
(t1,w11,f1,w1)
(t2,w12,f2,w2)

25. Progressive Optimization
D. G. Cory, JCP 2002
Random Guess
Maximize Fidelity
simplex
Split
Maximize Fidelity
simplex
Split
Maximize Fidelity
simplex

26. Example Fidelity : 0.99 Shifts: 500 Hz, - 500 Hz Coupling: 20 Hz
Target Operator :
(/2)y1
Fidelity : 0.99

27. Shifts:
500 Hz, Hz
Coupling:
20 Hz
Target Operator :
(/2)y1

28. Initial state Iz1+Iz2 SMPs are not limited by bandwidth Shifts:
500 Hz, Hz
Coupling:
20 Hz
Target Operator :
(/2)y1
Initial state
Iz1+Iz2
SMPs are
not limited
by bandwidth

29. Initial state Iz1+Iz2 SMPs are not limited by bandwidth Shifts:
500 Hz, Hz
Coupling:
20 Hz
Target Operator :
(/2)y1
Initial state
Iz1+Iz2
SMPs are
not limited
by bandwidth

30. 1 2 3 0.99 0.99 0.99 CH3 C NH3+ O -O H 3 1 2 13C Alanine Amp (kHz)
Pha (deg)
0.99
Amp (kHz)
Pha (deg)
0.99
Amp (kHz)
Pha (deg)
CH3 C
NH3+ O -O H 3 1 2
Time (ms)
13C Alanine

31. Shifts and J-couplingsAB 1 2 3 4 5 6 7 8 9 10 11 12
-1423
134
6.6
-13874 52
35.2
4.1
2.0
1.8
5.3
1444
2.2 74
11.5
4.4
-9688
53.6
147
6.1
201
8233
998
3.6
4.3
6.7
-998
4421
16.2
4279
2455
221.8
1756
-3878

32. Benchmarking 12-qubits A 8 A’ 2 11 1 3 10 5 4 9 7 6 Qubits Time
Benchmarking circuit
AA’ 1 2 3 4 5 6 7 8 9 10 11
Qubits
Time
Fidelity: 0.8
PRL, 2006

33. Quantum Algorithm for NGE (QNGE) :
in liquid
crystal
PRA, 2006

34. Quantum Algorithm for NGE (QNGE) : Quantum Algorithm for NGE (QNGE) :
Crob: 0.98
PRA, 2006

35. Progressive Optimization
D. G. Cory, JCP 2002
Advantages
Works well for small number of qubits ( < 5 )
Can be combined with other optimizations (genetic algorithm etc)
Solutions consist of small number of segments – easy to analyze
Disadvantage
1. Maximization is usually via Simplex algorithms
Takes a long time

36. SMPs : Calculation Time
During SMP calculation: U = exp(-iHeff t) calculated typically over 103 times
Qubits Calc. time
minutes
Hours
> 7 Days (estimation)
Single ½ : Heff =
2 x 2
Two spins : Heff =
4 x 4 .
Matrix Exponentiation
is a difficult job
- Several dubious ways !!
210 x 210
~ Million
10 spins : Heff =

37. Gradient Ascent Final density matrix: Navin Khaneja et al, JMR 2005
Liouville von-Neuman eqn
Control parameters
Final density matrix:

38. Gradient Ascent Navin Khaneja et al, JMR 2005 Correlation:
Backward propagated opeartor
at t = jt
Forward propagated opeartor
at t = jt

39. Gradient Ascent ’  = ’ t ? (up to 1st order in t)
Navin Khaneja et al, JMR 2005
 = ’ t ’ ?
(up to 1st order in t)

40. Gradient Ascent
Navin Khaneja et al, JMR 2005
Step-size

41. Gradient Ascent GRAPE Algorithm Stop Guess uk
Navin Khaneja et al, JMR 2005
GRAPE Algorithm
Guess uk No
Correlation > 0.99?
Yes
Stop

42. Practical Aspects Bounding within hardware limits Robustness
Nonlinearity

43. Shoots-up if any control parameter exceeds the limit
Bounding the control parameters
Quality factor = Fidelity + Penalty function
Shoots-up if any control parameter exceeds the limit
To be maximized

44. Practical Aspects Bounding within hardware limits Robustness
Nonlinearity

45. Incoherent Processes Spatial inhomogeneities in RF / Static field
Hilbert Space
Final
Final
Final
Initial
UEXPk(t)

46. Robust Control Coherent control in the presence of incoherence:
Hilbert Space
Target
Initial
UEXPk(t)

47. Inhomogeneities SFI Analysis of spectral line shapes
RFI Analysis of nutation decay
Ideal
SFI f f z z
Ideal
RFI y y x x

48. RF inhomogeneity 1 Ideal Probability of distribution In practice
RFI: Spatial
nonuniformity
in RF power
RF Power
Desired RF Power

49. RF inhomogeneity
Bruker PAQXI probe (500 MHz)

50. Example Shifts: 500 Hz, - 500 Hz Coupling: 20 Hz Target Operator :
(/2)y1

51. Shifts:
500 Hz, Hz
Coupling:
20 Hz
Target Operator :
(/2)y1

52. Shifts:
500 Hz, Hz
Coupling:
20 Hz
Target Operator :
(/2)y1

53. Initial state Iz1+Iz2 Shifts: 500 Hz, - 500 Hz Coupling: 20 Hz
Target Operator :
(/2)y1
Initial state
Iz1+Iz2

54. Initial state Iz1+Iz2 Shifts: 500 Hz, - 500 Hz Coupling: 20 Hz
Target Operator :
(/2)y1
Initial state
Iz1+Iz2

55. Robust Control Initial state Ix1+Ix2 - Eg. Two-qubit system
Shifts: 500 Hz, -500 Hz
J = 50 Hz
Fidelity = 0.99
Target Operator : ()y1
Initial state
Ix1+Ix2 -

56. Robust Control - Initial state Ix1+Ix2 Eg. Two-qubit system
Shifts: 500 Hz, -500 Hz
J = 50 Hz
Fidelity = 0.99
Target Operator : ()y1 -
Initial state
Ix1+Ix2

57. Practical Aspects Bounding within hardware limits Robustness
Nonlinearity

58. Spectrometer non-linearities
Computer:
“This is what I sent”

59. Spectrometer non-linearities
Spins: “This is what we got”
Computer:
“This is what I sent”

60. ~ Multi-channel probes: Target coil Spy coil
- D. G. Cory et al, PRA 2003.

61. Spectrometer non-linearitiesF

62. Feedback correction F hardware F-1 F hardware
- D. G. Cory et al, PRA 2003.

63. Feedback correction: Computer: Spins: “This is what we got”
“This is what I sent”
Spins: “This is what we got”
Compensated
Shape
- D. G. Cory et al, PRA 2003.

64. Summary DiVincenzo Criteria Quantum Control
Single and Two-qubit control
Control via Time-dependent Hamiltonians
Progressive Optimization
Gradient Ascent
Practical Aspects
Bounding within hardware limits
Robustness
Nonlinearity