Quantum Control - Indian Institute of Science

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Oct 23, 2013 (3 years and 5 months ago)

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Quantum Control

Synthesizing Robust Gates


T. S. Mahesh


Indian Institute of Science Education and Research,
Pune

1.
DiVincenzo

Criteria

2.
Quantum Control

3.
Single and Two
-
qubit

control

4.
Control via Time
-
dependent Hamiltonians


Progressive Optimization


Gradient Ascent

5.
Practical Aspects


Bounding within hardware limits


Robustness


Nonlinearity

6.
Summary

Contents

Criteria for Physical Realization of QIP

1.
Scalable physical system with mapping of qubits

2.
A method to initialize the system

3.
Big decoherence time to gate time

4.
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

Given a quantum system,

how best can we control its dynamics?

Quantum Control



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


General unitary is state independent:

Example: NOT, CNOT,
Hadamard
, etc.

General Unitary

U
TG



1


Hilbert Space

Fidelity =


Tr
{U
EXP
∙U
TG
} / N


2



0


U
EXP

obtained by

simulation or

process

tomography

A particular input state is transferred to a particular output state

Eg
.

000




(

000


+

111


) /

2

State to State Transfer


Initial



Target


Hilbert Space


Final


Fidelity =

Final

Target


2

obtained by

tomography

Universal Gates



Local gates (
eg
.
R
y
(

),
R
z
(

)) and CNOT gates together form a universal set

Example: Error Correction Circuit

Chiaverini

et al,

Nature

2004

Degree of control



For
fault tolerant computation: Fidelity ~
0.999


-

E.
Knill

et al, Science 1998.


Quantum gates need not be
perfect


Error correction can take care of imperfections


Fault
-
tolerant computation

Single Qubit (spin
-
1/2) Control

(up to a global phase)

Bloch sphere

~

Sample

resonance at

0

=

B
0


RF coil

Pulse/Detect

Superconducting

coil

B
0

B
1
cos(

rf
t
)

NMR spectrometer

Control Parameters

~




1

=

B
1



rf



B
0

B
1
cos(

rf
t
)




RF duration



1

RF amplitude




RF phase





RF offset

RF offset =


=

rf

-


ref



(kHz
rad
)

Chemical Shift


01

=

0
-


ref

All frequencies are

measured
w.r.t
.

ref

time

Single Qubit (spin
-
1/2) Control

(up to a global phase)

Bloch sphere

(in RF frame)

(in REF frame)

A general state:

90
-
x

90
x


y


x


y

Single Qubit (spin
-
1/2) Control

(in RF frame)

(in REF frame)

Single Qubit (spin
-
1/2) Control

x

y


01

Turning OFF

0

: Refocusing

X

Refocus Chemical Shift





time

(in RF frame)

(in REF frame)

Two Qubit Control

Local Gates

Qubit Selective Rotations
-

Homonuclear

Band
-
width


1/


1

2

1

2

dibromothiophene



=

1


non
-
selective

selective



=

1


Not a good method: ignores the time evolution

Qubit Selective Rotations
-

Heteronuclear



Larmor

frequencies are separated by MHz




Usually irradiated by different coils in the probe




No overlap in bandwidths at all




Easy to rotate selectively

13
CHCl
3

1
H (500 MHz @ 11T)

13
C (125 MHz @ 11T)

~

~

Two Qubit Control

Local Gates

CNOT Gate

Two Qubit Control

Chemical shift

Coupling constant

Chemical shift

X

X

Refocus Chemical Shifts

1

2

Refocussing
:





X

Refocus

0

&
J
-
coupling

1

2





Z

R
z
(
90
)

R
z
(
90
)



R
z
(

0

)





= 1/(4J)

time

time

Two Qubit Control

Chemical shift

Coupling constant

Chemical shift

Z

H

H

=


1/(4J)

1/(4J)

R
-
z
(
90
)

R
-
z
(
90
)

time

X

X

X

R
-
y
(
90
)

R
-
y
(
90
)

=

Control via Time
-
dependent Hamiltonians

H  H


(t)
,


(t)

,


(t)
,




NOT EASY !!
(exception: periodic dependence)


(t)

t

Control via Piecewise Continuous Hamiltonians


3


3


3

H
3


1


1


1

H
1


2


2


2

H
2


4


4


4

H
4

Time

Gradient Ascent

Navin

Khaneja

et al, JMR 2005

Numerical Approaches for Control


Progressive Optimization

D. G.
Cory & co
-
workers,
JCP
2002


Mahesh &
Suter
, PRA 2006

1.
Generate piecewise continuous Hamiltonians

2.
Start from a random guess, iteratively proceed

3.
Good solution not guaranteed

4.
Multiple solutions may exist

5.
No global optimization

Common features


1
,
1
1
,
1
,
1



2
,
1
2
,
2
,
2



3
,
1
3
,
3
,
3




Piecewise Continuous Control

D. G. Cory, JCP 2002

Strongly Modulated Pulse (SMP)

Progressive Optimization

D. G. Cory, JCP 2002

Random Guess

Maximize Fidelity

Split

Maximize Fidelity

Split

Maximize Fidelity

simplex

simplex

simplex

Example

Fidelity : 0.99

Shifts:

500 Hz,
-

500 Hz

Coupling:


20 Hz

Target Operator :

(

/2
)
y
1

Shifts:

500 Hz,
-

500 Hz

Coupling:


20 Hz

Target Operator :

(

/2
)
y
1

SMPs are

not limited

by bandwidth

Initial state

Iz1+Iz2

Shifts:

500 Hz,
-

500 Hz

Coupling:


20 Hz

Target Operator :

(

/2
)
y
1

SMPs are

not limited

by bandwidth

Initial state

Iz1+Iz2

Shifts:

500 Hz,
-

500 Hz

Coupling:


20 Hz

Target Operator :

(

/2
)
y
1

1

2

3

Time (
m
s)

Amp (kHz)

Pha (deg)

Amp (kHz)

Pha (deg)

Amp (kHz)

Pha (deg)

0.99

0.99

0.99

C
H
3

C

C

NH
3


O


O

H

3

1

2

13C
Alanine

AB

1

2

3

4

5

6

7

8

9

10

11

12

AB

-
1423

134

6.6

1

-
13874

52

35.2

4.1

2.0

1.8

5.3

2

1444

2.2

74

11.5

4.4

11.5

2.2

4.4

3

-
9688

53.6

147

6.1

4

0

201

11.5

2.2

4.4

5

8233

5.3

6

998

3.6

4.3

6.7

7

-
998

8

4421

16.2

5.3

9

4279

16.2

5.3

10

2455

221.8

11

1756

12

-
3878

Shifts and J
-
couplings



Benchmarking circuit

AA’

1

2

3

4

5

6

7

8

9

10

11

Time

Qubits

A

A’

1

2

3

4

5

6

7

9

8

10

11

Benchmarking 12
-
qubits

PRL, 2006

Fidelity: 0.8

Quantum Algorithm for NGE (QNGE) :

PRA, 2006

in liquid

crystal

Quantum Algorithm for NGE (QNGE) :

Quantum Algorithm for NGE (QNGE) :

C
rob
: 0.98

PRA, 2006

Progressive Optimization

D. G. Cory, JCP 2002

1.
Works well for small number of qubits ( < 5 )

2.
Can be combined with other optimizations (genetic algorithm etc)

3.
Solutions consist of small number of segments


easy to analyze

Advantages

Disadvantage

1. Maximization is usually via Simplex algorithms



Takes a long time

SMPs : Calculation Time

2 x 2

Single ½ :
H
eff

=

4 x 4

Two spins :
H
eff

=


2
10

x 2
10


~ Million

10 spins :
H
eff

=

.

.

.

During SMP calculation: U = exp(
-
iH
eff

t) calculated typically over 10
3

times

Qubits


Calc. time

1
-

3


minutes

4
-

6


Hours


> 7


Days (estimation
)

Matrix Exponentiation

is a difficult job


-

Several dubious ways !!

Gradient Ascent

Navin

Khaneja

et al, JMR 2005

Control
parameters

Liouville

von
-
Neuman

eqn

Final density matrix:

Gradient Ascent

Navin

Khaneja

et al, JMR 2005

Correlation:

Backward
propagated
opeartor


at t =
j

t

Forward
propagated
opeartor


at t =
j

t

Gradient Ascent

Navin

Khaneja

et al, JMR 2005

?



=




t







(up to 1
st

order in

t
)

Gradient Ascent

Navin

Khaneja

et al, JMR 2005

Step
-
size

Gradient Ascent

Navin

Khaneja

et al, JMR 2005






Guess
u
k











No

Yes

Stop

Correlation
> 0.99?

GRAPE Algorithm

Practical Aspects

1.
Bounding within hardware limits


2.
Robustness


3.
Nonlinearity



Bounding the control parameters

Quality factor = Fidelity + Penalty function

Shoots
-
up if any
control parameter
exceeds the limit

To be maximized

Practical Aspects

1.
Bounding within hardware limits


2.
Robustness


3.
Nonlinearity



Spatial inhomogeneities in RF / Static field


Initial



Final


Hilbert Space

Incoherent Processes

U
EXP
k
(

)


Final



Final


Coherent control in the presence of incoherence:

Robust Control


Initial


Hilbert Space


Target


U
EXP
k
(

)

Inhomogeneities

SFI Analysis of spectral line shapes







RFI Analysis of nutation decay

f

f

Ideal

SFI

x

y

z

x

y

z

Ideal

RFI

RFI: Spatial

nonuniformity

in RF power

RF Power

Desired RF Power

0

1

In practice

Ideal

Probability

of

distribution

RF inhomogeneity

RF inhomogeneity

Bruker PAQXI probe (500 MHz)

Example

Shifts:

500 Hz,
-

500 Hz

Coupling:


20 Hz

Target Operator :

(

/2
)
y
1

Shifts:

500 Hz,
-

500 Hz

Coupling:


20 Hz

Target Operator :

(

/2
)
y
1

Shifts:

500 Hz,
-

500 Hz

Coupling:


20 Hz

Target Operator :

(

/2
)
y
1

Shifts:

500 Hz,
-

500 Hz

Coupling:


20 Hz

Target Operator :

(

/2
)
y
1

Initial state

Iz1+Iz2

Shifts:

500 Hz,
-

500 Hz

Coupling:


20 Hz

Target Operator :

(

/2
)
y
1

Initial state

Iz1+Iz2

Robust Control

Eg
. Two
-
qubit

system

Shifts: 500 Hz,
-
500 Hz

J = 50 Hz

Fidelity = 0.99

Target Operator : (

)
y
1

-

Initial state

Ix1+Ix2

Robust Control

Eg
. Two
-
qubit

system

Shifts: 500 Hz,
-
500 Hz

J = 50 Hz

Fidelity = 0.99

-

Target Operator : (

)
y
1

Initial state

Ix1+Ix2

Practical Aspects

1.
Bounding within hardware limits


2.
Robustness


3.
Nonlinearity



Spectrometer non
-
linearities

Computer
:

“This is what I sent”

Spectrometer non
-
linearities

Computer
:

“This is what I sent”

Spins
: “This is what we got”



~

Multi
-
channel probes:

Target coil

Spy coil

-

D. G. Cory et al, PRA 2003.

Spectrometer non
-
linearities

F



Feedback correction

F

F
-
1

F

-

D. G. Cory et al, PRA 2003.



hardware

hardware

Feedback correction:

Spins
: “This is what we got”

Computer
:

“This is what I sent”

Compensated

Shape

-

D. G. Cory et al, PRA 2003.

Summary

1.
DiVincenzo

Criteria

2.
Quantum Control

3.
Single and Two
-
qubit

control

4.
Control via Time
-
dependent Hamiltonians


Progressive Optimization


Gradient Ascent

5.
Practical Aspects


Bounding within hardware limits


Robustness


Nonlinearity