Controllability and Time Optimal Control in Spin Systems

k1

k2

G/K

G

K

k1 k2

Interactions

SI

νSJ

ν I

B

(D)

Spin Hamiltonian: H + H (t)

B (t)rf

0

0 rf

Navin Khaneja, Harvard

KITP Quantum Control, June 10, 2009

Bilinear Control Systems in Quantum Control

dU (t)dt

= −i[Hd + uj Hjj=1

m

∑ ]U(t); U(0) = I

| ( ) ( ) | ( )d t iH t tdt

ψ ψ⟩ = − ⟩

Can the state of a quantum mechanical system be steered between points of interest with available Hamiltonians.

What possible Unitary Transformations can be produced in a given time with available Hamiltonians.

Controllability, Lie Algebras and Chows Theorem

2 1 2 12

1 2

( ) exp( )exp( )exp( )exp( )

( ) [ , ]new generator

U t iH t iH t iH t iH t

I t iH iH

Δ = Δ Δ − Δ − Δ

≈ + Δ − −1442443

( )j jj

dU i u H Udt

= − ∑

{ }j LAiH−If the Lie Algebra generated by span{ }jiH−the Lie algebra of the unitary group, then the system is controllable

1 2

1 01 0

1 01 1 0 0

1 0 0 10 1

u u

−⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟−

Θ = + Θ⎜ ⎟ ⎜ ⎟−⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟−⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

O O

O O

If the control amplitude is unbounded then the points that can be reached can be reached in no time

Brockett, Sussman, Jurdjevic

Controllable Linear Systems with unbounded controls can be steered between points in arbitrary small time

dX AX Budt

= +

( )( ) (0) ( ) ( )At A tX t e X e B u dτ τ τ τ−= + ∫

Even if drift is required, it takes arbitrary small time to steer the system between points of interest, if system

is controllable.

j jj

dX AX u bdt

= +∑

Controllability with Drift

0[ ]j jj

dU i H u H Udt

= − +∑0,{ }j LAH HIf the Lie Algebra span

the Lie algebra of the unitary group, then the system is controllable

0exp( )iH tΔThe backward evolution

is obtained arbitrarily well by waiting long enough on a compact group

Inspite of the Unbounded Controls there is a minimum time to reach anywhere

Time Optimal Control of Quantum Systems

U(0)

UF

G

K

dU (t)dt

= −i[Hd + uj Hjj=1

m

∑ ]U(t); U(0) = I

k = {−iHj}LA

K = exp(k)

Physical Review A , 63, 032308, 2001

{ }( ) inf ( )tT U U R t∗ = ∈

( ) 0T K∗ =

1 1 2 1( ) ( )T K U K T U∗ ∗=

Manipulation of Coupled Spin Dynamics

1, 2 1 2( ) cos( )cos( )s Is t t t tη ω ω=

Time Optimal Control of Quantum Systems

U(0)

UF

G

K

dU (t)dt

= −i[Hd + uj Hjj=1

m

∑ ]U(t); U(0) = I

k = {−iHj}LA

K = exp(k)

Physical Review A , 63, 032308, 2001

{ }( ) inf ( )tT U U R t∗ = ∈

( ) 0T K∗ =

1 1 2 1( ) ( )T K U K T U∗ ∗=

Cartan Decomposition of Lie Algebra

G/K is a Riemannian Symmetric Space

; [ , ] ; [ , ] ; [ , ] ;( , ) ( );X Y

g p k p p k k k k p k pB X Y tr ad ad p k= ⊕ ⊆ ⊆ ⊆

= ⊥

The velocities of the shortest pathsin G/K always commute!

dU (t)dt

= −i[Hd + uj Hjj=1

m

∑ ]U(t); U(0) = I

1 2 2 1 1exp( ) exp( ) exp( )n d n n d dK iH t K iH t K iH t K+ − − −K

1

† † †1 2 2 2 1 1 1

( )

exp( ) exp( ) exp( )K

n n d n n d d

Ad H

K i K H K t i K H K t i K H K t+ − − −K14243

Cartan Decomposition of Lie algebra and Lie Group.

;[ , ] ; [ , ] ; [ , ] ;g p k p kp p k k k k p k p= ⊕ ⊥

⊆ ⊆ ⊆

G = KAK

p iS= −( )k so n=( )g su n= S traceless Symmetric

1

2

n

h i a

λλ

λ

⎡ ⎤⎢ ⎥⎢ ⎥= − ∈⎢ ⎥⎢ ⎥⎣ ⎦

O

1

21 2exp( )

n

G K i K

λλ

λ

⎡ ⎤⎢ ⎥⎢ ⎥= −⎢ ⎥⎢ ⎥⎣ ⎦

O

K = SO(n){ }1 2 3 na λ λ λ λ+ = ≥ ≥ ≥ ≥K

g be a real semisimple lie algebra( )KAd p p=

( )KK

Ad a p=Uexp( )G p K=

Cartan Decomposition

;[ , ] ; [ , ] ; [ , ] ;g p kp p k k k k p k p= ⊕

⊆ ⊆ ⊆

( ) ( ) (1)k su m su n u= × ×( )g su m n= +

{ }1 2 3 0mD diag λ λ λ λ= ≥ ≥ ≥ ≥ ≥K

0 0exp( )

0 0k

a AK i

b B⎡ ⎤ ⎡ ⎤

= − =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦123

†

00c

p ic⎡ ⎤

= − ⎢ ⎥⎣ ⎦14243

0 00 0

0 0 0

Da D+

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

Cartan Decompositions in Two-Spin Systems and Canonical Decomposition of SU(4)

G = SU(4); K = SU(2)⊗SU (2)

Iα = σα ⊗ I ;Sα = I ⊗σα ;IαSβ = σα ⊗σβ ;

Interactions

SI

νSJ

ν I

B

(D)

Spin Hamiltonian: H + H (t)

B (t)rf

0

0 rf

Khaneja, Brockett, Glaser, Physical Review A , 63, 032308, 2001

( , , )x y zα α α | |x y zα α α≥ ≥

Another Canonical Decomposition of SU(4):Electron Nuclear Spin Dynamics

(4); (2) (2) (1)G SU K SU SU U= = × ×

{ }, ,z zk i S S I Iα β= −0 0

exp( )0 0

k

a AK i

b B⎡ ⎤ ⎡ ⎤

= − =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦123

{ },z x xa i S I I= −1

2

1

2

0 0 00 0 0

0 0 00 0 0

i

λλ

λλ

⎡ ⎤⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥⎣ ⎦

;c z z S IH JI S J= Ω Ω Interactions

SI

νSJ

ν I

B

(D)

Spin Hamiltonian: H + H (t)

B (t)rf

0

0 rf

1 1 2 2

1 0 0 0exp( )

0 0 0 1x xK i Kλ σ λ σ⎛ ⎞ ⎛ ⎞

− ⊗ + ⊗⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Zeier, Yuan, Khaneja, PRA

(2008)

{ }1 2 0λ λ≥ ≥

Controllability and Cartan Decomposition

1

( ) [ ] ( ); (0)m

d j jj

dU t i H u H U t U Idt =

= − + =∑ ( )k so n=

exp( ) ( )K k SO n= =1

2d

n

H

λλ

λ

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

O

G = KAK

1 2 2exp( ) exp( ) exp(n d n n d dK iH t K iH t K iH t+ − − −K

1

† † †2 2 2 1 1 1

( )

1

2

exp( ) exp( )exp( )

exp( )

P d

b n d n n d d a

Ad H

b a

n

K i P H P t i P H P t i P H P t K

K i K

μμ

μ

− − −

⎛ ⎞⎜ ⎟⎜ ⎟= −⎜ ⎟⎜ ⎟⎝ ⎠

K14243

O

Reachable Set

1

( ) [ ] ( ); (0)m

d j jj

dU t i H u H U t U Idt =

= − + =∑exp( ) ( )K k SO n= =

1

2d

n

H

λλ

λ

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

O

G = KAK

1 2( ) ( ) ( ) ( )K t A t K t U t=

† †1 2( ) ( ) ( ) ( )A t K t U t K t=

( ) ( ( )) ( )K ddA t diag Ad iH A t

dt= −

Schur Convexity

1

2

1 22 2

2 1

0cos sin cos sin0sin cos sin cos

cos sin

diagλθ θ θ θ

λθ θ θ θ

λ λθ θ

λ λ

− ⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠⎝ ⎠

⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

1 11

2 22†

n nn

aa

K K

a

λλ

λ

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

O O

O O

O O O O

O O

{ {

11 1

22 2 ; ( )j jj

nn n

a

aa

a P

aλ

λλ

α λ

λ

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∑pM M

Diagonal of a Symmetric Matrix is Majorized (lies in the convex hull)

of its eigenvalues

a+

Reachable Set

1

( ) [ ] ( ); (0)m

d j jj

dU t i H u H U t U Idt =

= − + =∑ ( )k so n=

G = KAK

1 2( ) ( ) ( ) ( )K t A t K t U t=

( ) ( ( )) ( )K ddA t diag Ad iH A t

dt= −

( ( )) ( )K d j jj

diag Ad iH i Pα λ− = − ∑

0

( ( )) exp( ( ) ( ) )T

j jj

diag A T i t P dt

μ

α λ= − ∑∫14243

Reachable Set

1

( ) [ ] ( ); (0)m

d j jj

dU t i H u H U t U Idt =

= − + =∑

1

2d

n

H

λλ

λ

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

O

1

21 2( ) exp( )

n

R T K i K

μμ

μ

⎡ ⎤⎢ ⎥⎢ ⎥= −⎢ ⎥⎢ ⎥⎣ ⎦

O

1

2

n

T

μμ

λ

μ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

pM

Schur Convexity

1 11

2 22†

n nn

aa

K K

a

λλ

λ

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

O O

O O

O O O O

O O

11 1

22 2

nn n

aa

a

λλ

λ

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

pM M

1

2 †( )

n

tr Z K K

λλ

λ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

O

1

2

n

Z

μμ

μ

⎛ ⎞⎜ ⎟⎜ ⎟=⎜ ⎟⎜ ⎟⎝ ⎠

O

([ , ( )] )K dtr Z Ad H kexp( )K k K→[ , ]p p k∈

Kostant Convexity

, max lg( ) ( ) ; ( ( ))K X

a p imal abelian suba ebraX Ad X a c X is the convex hull of⊆

Δ = ∩ Δ Δ

:: ( ) ( ( ))K

T p a orthogonal projectionT Ad X c X

→= Δ

( ) ( ) ( ) (1)g su p q k su p su q u= + = × ×

1 2

( ( ))( ) ( ) ( ) ( )

KA Ad X AU t K t A t K t= Γ

=

( ) exp( ( ))

1

jj kj

jj

A t T Ad Xα

α

=

≤

∑

∑

; [ , ] ; [ , ] ; [ , ] ;( , ) ( );X Y

g p k p p k k k k p k pB X Y tr ad ad p k= ⊕ ⊆ ⊆ ⊆

= ⊥

Time Optimal Tori Theorem

g = p + k; p⊥k[ p, p]⊆ k; [k,k]⊆ k ; [p,k]⊆ p;

, max lg( ) ( ) ; ( ( )) ( )K

a p imal abelian suba ebraX Ad X a c X is the convex hull of X⊆

Δ = ∩ Δ Δ

a+

dX iH= −

1 2exp( ( ( )) ) ;K c X t KΔ

dU (t)dt

= −i[Hd + uj Hjj=1

m

∑ ]U(t); U(0) = I

1 2exp( ( )) ;ii K

i

K t Ad X Kα∑

Two-Spin Systems and Canonical Decomposition of SU(4)

G = SU(4); K = SU(2)⊗SU (2)Iα = σα ⊗ I ;Sα = I ⊗σα ;IαSβ = σα ⊗σβ ;

Interactions

SI

νSJ

ν I

B

(D)

Spin Hamiltonian: H + H (t)

B (t)rf

0

0 rf

g = p + k; p⊥k[ p, p]⊆ k; [k,k]⊆ k ; [p,k]⊆ p;

(4)g su=

| |x y zα α α≥ ≥

Cartan Decompositions , Two-Spin Systems and Canonical Decomposition of SU(4)

Interactions

SI

νSJ

ν I

B

(D)

Spin Hamiltonian: H + H (t)

B (t)rf

0

0 rf

1

†( )l l x x x y y y z z z

H

U J I S U I S I S I Sαβ α βαβ

α α α→ + +∑ 14444244443

( , , )x y zα α α | |x y zα α α≥ ≥

2 4 4 3 3 2 2 1 1 1exp( )exp( )exp( )exp( )U iH t iH t iH t iH t U− − − −

1 2e x p ( ( ( ) ) ) ;K i c X t K− Δ

( ( ) ){ ; | | | |}x x x y z x y z

c Xq q q qα α α αΔ =≤ + ± ≤ + ±

3 3[ ] XJαβ

Computations

1 ( 01 10 ) ;2

3 ( 00 11 ) ;212 ( 00 11 ) ;2

14 ( 01 10 ) ;2

i

i

−= +

−= −

= +

= −

1 2 1 2U AU D→Θ Θ

21 1

T TD UUΘ Θ =

Eigenvalue Problem

†

exp( )exp( )x y y yP i I S i I S

U PVP

π π= − −

=

Reachable set under time varying drift

1

( ) [ ( ) ] ( ); (0)m

d j jj

dU t i H t u H U t U Idt =

= − + =∑k = {−iHj}LA

K = exp(k)

;( )d

g p k p kiH t p= + ⊥

− ∈

1 2exp( ( ( )) )K c Y t KΔ

0

( ) ( ( ))

( )

K dT

X t Ad iH t a

Y X dτ τ

+ +

+

= − ∩

= ∫

a+

H. Yuan and N. KhanejaSystem and Control Letters (2006)

Reachable Set

1

( ) [ ( ) ] ( ); (0)m

d j jj

dU t i H t u H U t U Idt =

= − + =∑ ( )k so n=

exp( ) ( )K k SO n= =1

2

( )( )

( )

( )

d

n

tt

H t

t

λλ

λ

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

O

1

21 2exp( )

n

K i K

μμ

μ

⎡ ⎤⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥⎣ ⎦

O

1

2

n

Y

μμ

μ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

pM

0( )

TY dλ τ τ↓= ∫

Another K+P Problem

1

0

;

T

X UX U p

U U dtη

•

= ∈

= ∫

0 1 0

0 1

( ) exp( ) exp( );

U t M t M M tM k M p

= −

∈ ∈

0 0 0 00 0 ; exp( 0 0 1 )0 0 0 1 0

u vuv

•− −⎡ ⎤ ⎡ ⎤

⎢ ⎥ ⎢ ⎥Θ = Θ −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

cos( ); sin( )u A t v A tω θ ω θ= + = +

Dynamics of n-coupled Spins

The dynamics of coupled spin ½ particles is described by an element of

A basis of Lie algebra of can be expressed as tensor product of pauli spin matrices

k1

k2

G/K

G

K

k1 k2

[ , ]p p k∉

1 1 2 2 3

2 1 2 2 3

3 1 2 3 22 / 2

z x x z

z y y z

z z z z

H I I I IH I I I I

H I I I I

= +

= +

= +

Time Optimal Quantum Information Processing

x-x -y

I1

I2

I3

I1

I2

I3

I1

I2

I3

-y

x-y y-xy -yx

x x

-y -y -y -x x y y y

x -x

12J

12J

κ2J

κ2J

14J

14J

τ*

conventional experiment (with decoupling)

improved experiment (without decoupling)

OPTIMAL experiment (without decoupling)

SU(8)SU (2)⊗ SU(2)⊗SU (2)

κ = 1

1 2 3

J J

τ∗ =κ (4 −κ )

2J

U = exp −iHeff( )Heff = 2πκ I1α I2β I3γ( )

1 1 2 2 3

2 1 2 2 3

3 1 2 3 22 / 2

z x x z

z y y z

z z z z

H I I I IH I I I I

H I I I I

= += +

= +

Khaneja, Glaser, Brockett

Reiss, Khaneja, GlaserJ. Mag. Reson. 165 (2003)

Geometry, Control and NMR

0 cos 0cos 0 sin

0 sin 0

x xd y ydt

z z

θθ θ

θ

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

3

2

tan xx

θ =

2 2

2

( ) ( )dx dzy+

tθ ω=

1 1

1 2 1 2

1 2 1 2

1 2 3 1 2 3

0 12 21 0

0 12 21 04 4

x x

y z y z

y x y x

y y z y y z

I I

I I I Iududt I I I I

I I I I I I

⎡ ⎤ ⎡ ⎤−⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥−⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥−⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

1 2 3

J J

1 2 3(4 )

exp( 2 ),2z z zi I I I

Jκ κ

πκ−

−

Khaneja, et.al PRA 2007

The problem of manipulating quantum systems with uncertainities or inhomogeneities in parameters govering the system dynamics is ubiquitous in coherent spectroscopy and quantum information processing.

a) Understanding controllability of quantum dynamics with inhomogeneities.b) Understanding what aspect of system dynamics makes compensation possible.c) What kind of inhomogeneities or errors can or cannot be corrected.

Widespread use of composite pulse sequences and pulse shaping first to correct for errors or compensate for inhomogeneties

Typical settings includea) Resonance offsetsb) Inhomogeneities in the strength of excitation field (systematic errors)c) Time dependent noise (nonsystematic errors)d) Addressing errors or cross talk

Ensemble Controllability