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

UTG

1

Hilbert Space

Fidelity = Tr{UEXP·UTG} / N 2

0UEXP obtained by

simulation orprocess 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 = FinalTarget 2

obtained bytomography

Universal Gates

• Local gates (eg. Ry(), Rz()) and CNOT gates together form a universal set

Example: Error Correction CircuitChiaverini 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

)2/ˆexp()(ˆ niRn kji zyx

ˆˆˆ

niI ˆ

2sin

2cos

12

sin02

cos0)()(

eRR yz

(up to a global phase)

Bloch sphere

~

Sampleresonance at 0 =B0

RF coilPulse/Detect

Superconductingcoil

B0

B1cos(wrft)

NMR spectrometer

Control Parameters

~

1 = B1

rf

B0

B1cos(wrft)

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

2/)sin(2/)cos(2/),, 101101 yxz H(

12

sin02

cos0)()(

eRR yz

(up to a global phase)Bloch sphere

(in RF frame)

zz iiiU exp),,expexp),,,, 101101 H( (

(in REF frame)

2

)(2

)( xyxz RRRR

)/,,0,0,0)( 11 (URx

)/,,90,0,0)( 11 (URy

A general state:

90-x90x y

x

y

Single Qubit (spin-1/2) Control

)()(01

10:NOT yz RRX

)2/()(11

11

2

1:Hadamard yz RRH

2/)sin(2/)cos(2/),, 101101 yxz H( (in RF frame)

zz iiiU exp),,expexp),,,, 101101 H( (

(in REF frame)

2

)(2

)( xyxz RRRR

)/,,0,0,0)( 11 (URx

)/,,90,0,0)( 11 (URy

Single Qubit (spin-1/2) Control

x

y

w01

Turning OFF 0 : Refocusing

X

Refocus Chemical Shift

time

2/)sin(2/)cos(2/),, 101101 yxz H( (in RF frame)

zz iiiU exp),,expexp),,,, 101101 H( (

(in REF frame)

Two Qubit Control

)2/ˆexp()(1 1ˆ miR m

Local Gates

)2/ˆexp()(2 2ˆ niR n

)(2)(2:NOT2 2 yz RRX

)2/(1)(1:Hadamard 1 yz RRH

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

13CHCl3

1H (500 MHz @ 11T)

13C (125 MHz @ 11T)

~~

Two Qubit Control

)2/ˆexp()(1 1ˆ miR m

Local Gates

)2/ˆexp()(2 2ˆ niR n

221 1100NOTC X 1CNOT Gate

)(2)(2:NOT2 2 yz RRX

)2/(1)(1:Hadamard 1 yz RRH

0100

1000

0010

0001

Two Qubit Control

2/2/2/ 11202101 zzzz J intH

Chemical shift Coupling constantChemical shift

X

X

Refocus Chemical Shifts

1

2

Refocussing:

X

Refocus 0 & J-coupling

1

2

ZRz(90)

Rz(90)

Rz(0)

= 1/(4J)

time

time

Two Qubit Control

2/2/2/ 11202101 zzzz J intH

Chemical shift Coupling constantChemical shift

Z HH=

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 ( a (t), b (t) , g (t) , )

dttHiTU )(exp

NOT EASY !! (exception: periodic dependence)

a (t)

t

Control via Piecewise Continuous Hamiltonians

11223344 expexpexpexp iHiHiHiHU

a3

b3

g3

H 3

a1

b1

g1

H 1

a2

b2

g2

H 2

a4

b4

g4

H 4Time

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

(t1,w11,f1,w1)

(t2,w12,f2,w2)

(t3,w13,f3,w3)

…

Piecewise Continuous ControlD. G. Cory, JCP 2002

Strongly Modulated Pulse (SMP)

Progressive OptimizationD. G. Cory, JCP 2002

Random Guess

Maximize Fidelity

Split

Maximize Fidelity

Split

Maximize Fidelity

11,H

2, 11

H

11,H

2, 11

H

11,H22 ,H

2, 22

H

11,H2, 22

H

11,H22 ,H

33,H

simplex

simplex

simplex

Example

Fidelity : 0.99

Shifts:

500 Hz, - 500 Hz

Coupling:

20 Hz

Target Operator :

(/2)y1

Shifts:

500 Hz, - 500 Hz

Coupling:

20 Hz

Target Operator :

(/2)y1

SMPs arenot limited

by bandwidth

Initial stateIz1+Iz2

Shifts:

500 Hz, - 500 Hz

Coupling:

20 Hz

Target Operator :

(/2)y1

SMPs arenot limited

by bandwidth

Initial stateIz1+Iz2

Shifts:

500 Hz, - 500 Hz

Coupling:

20 Hz

Target Operator :

(/2)y1

1 2 3Time (ms)

Am

p (k

Hz)

Pha

(deg)

Am

p (k

Hz)

Pha

(deg)

Am

p (k

Hz)

Pha

(deg)

0.99

0.99

0.99

CH3

CC

NH3+

O

-O

H

312

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

N

N

CC

C

CC

C

ND2

O

OHH

D

H H

H

H

1

A B

35

2

4

6

79

8

1011

12

N

N

CC

C

CC

C

ND2

O

OHH

D

H H

H

H

1

A B

35

2

4

6

79

8

1011

12

Shifts and J-couplings

Benchmarking circuit

AA’

1

2

3

4

5

6

7

8

9

10

11

Time

Qubits

AA’

12

34 5

6

79

8

10

11

Benchmarking 12-qubits

PRL, 2006Fidelity: 0.8

Quantum Algorithm for NGE (QNGE) :

PRA, 2006

in liquid crystal

Quantum Algorithm for NGE (QNGE) : Quantum Algorithm for NGE (QNGE) :

Crob: 0.98

PRA, 2006

Progressive OptimizationD. 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 2Single ½ : Heff =

4 x 4Two spins : Heff =

210 x 210

~ Million10 spins : Heff =

.

.

.

During SMP calculation: U = exp(-iHeff t) calculated typically over 103 times

Qubits Calc. time

1 - 3 minutes

4 - 6 Hours

> 7 Days (estimation)

Matrix Exponentiationis a difficult job

- Several dubious ways !!

Gradient Ascent Navin Khaneja et al, JMR 2005

Control parameters

)](),([)( ttit H

Liouville von-Neuman eqn

Final density matrix:

])0([ 111

11111

NjjjjNF UUUUUUUUtr

Gradient Ascent Navin Khaneja et al, JMR 2005

)]([ Ttr FCorrelation:

])0([ 11111

111

jjjNFNj UUUUUUUUtr

])0([ 11111

111

jjjNFNj UUUUUUUUtr

][ jjtr

Backward propagated opeartor at t = jt

Forward propagated opeartor at t = jt

Gradient Ascent Navin Khaneja et al, JMR 2005

? = ’ t

’’ ’

(up to 1st order in t)

Gradient Ascent Navin Khaneja et al, JMR 2005

][ jjtr ])0([ 1111

jjj UUUUtr

])0()(

[ 11111

jj

k

jj UUUU

ju

Utr

])(

)0([1

11

111 ju

UUUUUtr

k

jjjj

Step-size

Gradient Ascent Navin Khaneja et al, JMR 2005

][

])0([

111

1

1111

jNFNjj

jjj

UUUUtr

UUUUtr

Guess uk

No

Yes

StopCorrelation > 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

UEXPk(t)

Final

Final

Coherent control in the presence of incoherence:

Robust Control

Initial

Hilbert Space

Target

UEXPk(t)

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: Spatialnonuniformityin RF power RF Power

Desired RF Power

0

1

In practice

Ideal

Probabilityof

distribution

RF inhomogeneity

RF inhomogeneity

Bruker PAQXI probe (500 MHz)

Example

Shifts:

500 Hz, - 500 Hz

Coupling:

20 Hz

Target Operator :

(/2)y1

Shifts:

500 Hz, - 500 Hz

Coupling:

20 Hz

Target Operator :

(/2)y1

Shifts:

500 Hz, - 500 Hz

Coupling:

20 Hz

Target Operator :

(/2)y1

Shifts:

500 Hz, - 500 Hz

Coupling:

20 Hz

Target Operator :

(/2)y1

Initial stateIz1+Iz2

Shifts:

500 Hz, - 500 Hz

Coupling:

20 Hz

Target Operator :

(/2)y1

Initial stateIz1+Iz2

Robust Control

Eg. Two-qubit system

Shifts: 500 Hz, -500 HzJ = 50 HzFidelity = 0.99

Target Operator : ()y1

-

Initial stateIx1+Ix2

Robust Control

Eg. Two-qubit system

Shifts: 500 Hz, -500 HzJ = 50 HzFidelity = 0.99

-

Target Operator : ()y1

Initial stateIx1+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”

CompensatedShape

- 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