quantum control synthesizing robust gates t. s. mahesh indian institute of science education and...

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