Introduction to the GRAPE Algorithm - of Michael Goerz

Introduction to the GRAPE Algorithm
Michael Goerz
June 8, 2010
Michael Goerz Introduction to the GRAPE Algorithm

Reference: J. Mag. Res. 172, 296 (2005)
Abstract
Journal of Magnetic Resonance 172 (2005) 296–305
Optimal control of coupled spin dynamics: design of NMR
pulse sequences by gradient ascent algorithms
Navin Khaneja a, *, Timo Reiss b , Cindie Kehlet b , Thomas Schulte-Herbrüggen b ,
Steffen J. Glaser b, *
a Division of Applied Sciences, Harvard University, Cambridge, MA 02138, USA
b Department of Chemistry, Technische Universität München, 85747 Garching, Germany
Received 27 June 2004; revised 23 October 2004
Available online 2 December 2005
www.elsevier.com/locate/jmr
In this paper, we introduce optimal control algorithm for the design of pulse sequences in NMR spectroscopy. This methodology
is used for designing pulse sequences that maximize the coherence transfer between coupled spins in a given specified time, minimize
the relaxation effects in a given coherence transfer step or minimize the time required to produce a given unitary propagator, as
desired. The application of these pulse engineering methods to design pulse sequences that are robust to experimentally important
parameter variations, such as chemical shift dispersion or radiofrequency (rf) variations due to imperfections such as rf inhomogeneity
is also explained.
Michael Goerz Introduction to the GRAPE Algorithm

lse debsenceracterion
of
5]
The Basic Idea
ð1Þ
re the
to the
,um(t))
anged
oblem
ds that
a specimumermiasured
ð2Þ
rators,
of the
kj is the backward propagated target operator C at the
same time t = jDt. Let us see how the performance U0 changes when we perturb the control amplitude uk (j)
at time step j to uk(j) +duk(j). From Eq. (4), the change
in Uj to first order in duk(j) is given by
dU j ¼ iDtdukðjÞHkU j ð8Þ
Acronym
with
Z Dt
HkDt ¼ U jðsÞH kU jð sÞds ð9Þ
0
GRAPE: Gradient Ascent Pulse Engineering
Fig. 1. Schematic representation of a control amplitude u k(t),
consisting of N steps of duration Dt = T/N. During each step j, the
control amplitude uk(j) is constant. The vertical arrows represent
gradients dU0=dukðjÞ, indicating how each amplitude u k(j) should be
modified in the next iteration to improve the performance function U0.
Pulse Update
Φ0
uk(j) −→ uk(j) + ɛ ∂Φ0
∂uk(j)
original value
uk(j)
at time index j: go in direction of gradient
Michael Goerz Introduction to the GRAPE Algorithm

Working in Liouville Space
Density Matrix
Liouville-von Neumann Equation
Time Propagation
˙ρ(t) = −i [H, ρ(t)] − = −i
Uj = exp
(
|Ψ〉 −→ ρ = |Ψ〉〈Ψ|
−i∆t
"
H0 +
H0 +
mX
! #
uk(t)Hk , ρ
k=1
mX
!)
uk(j)Hk
k=1
ρ(T ) = UN . . . U1 ρ(0) U †
1 . . . U†
N
= |Ψ(T )〉〈Ψ(T )| with Ψ(T ) = Un . . . U1Ψ(0)
Michael Goerz Introduction to the GRAPE Algorithm
−

Definition of Fidelity
Fidelity in Liouville space is defined in analogy to fidelity in Hilbert space: as the
overlap between the propagated state with the optimal state.
Fidelity
“
Φ0 = 〈C|ρ(T )〉 ≡ tr C † ”
ρ(T )
C = O |Ψ(0)〉〈Ψ(0)| O †
ρ(T ) = U |Ψ(0)〉〈Ψ(0)| U †
Michael Goerz Introduction to the GRAPE Algorithm

Definition of Fidelity
Fidelity in Liouville space is defined in analogy to fidelity in Hilbert space: as the
overlap between the propagated state with the optimal state.
Fidelity
“
Φ0 = 〈C|ρ(T )〉 ≡ tr C † ”
ρ(T )
C = O |Ψ(0)〉〈Ψ(0)| O †
Equivalence to “normal” fidelity
“
tr C † ”
ρ(T )
ρ(T ) = U |Ψ(0)〉〈Ψ(0)| U †
= X D ˛ ˛ E D ˛
˛ ˛
˛
n ˛O ˛ Ψ(0) Ψ(0) ˛O
n
† ˛ E D ˛
˛
˛
U˛
Ψ(0) Ψ(0) ˛U †˛ E
˛ n
D ˛
= Ψ(0) ˛U † X˛ E D ˛ ˛ E D ˛
˛˛ ˛ ˛
˛
n n ˛ O ˛ Ψ(0) Ψ(0) ˛O † ˛ E
˛
U˛
Ψ(0)
D ˛
= Ψ(0) ˛U † ˛ E D ˛
˛
˛
O˛
Ψ(0) Ψ(0) ˛O † ˛ E
˛
U˛
Ψ(0)
˛D
˛
˛ ˛
= ˛ Ψ(0) ˛O † ˛ E˛
˛ ˛˛ 2
U˛
Ψ(0)
Michael Goerz Introduction to the GRAPE Algorithm

Fidelity through Backward- and Forward-Propagation
A trace is invariant under cyclic permutation of its factors!
Fidelity at T
D
Φ0 = 〈C|ρ(T )〉 = C|UN . . . U1ρ(0)U †
E
1 . . . U†
N
D
= U †
j+1 . . . U†
NCUN . . . Uj+1|Uj . . . U1ρ(0)U †
E
1 . . . U†
j
Propagated States → Fidelity at tj
λj ≡ U †
j+1 . . . U†
NCUN . . . Uj+1 bw. propagated optimal state
ρj ≡ Uj . . . U1ρ(0)U †
1 . . . U†
j fw. propagated initial state
Φ0 = 〈C|ρ(T )〉 = ˙ ¸
λj |ρj
Note: all propagations with guess pulse!
Michael Goerz Introduction to the GRAPE Algorithm

Calculation of Pulse Update
Pulse Update
Two steps:
uk(j) −→ uk(j) + ɛ ∂Φ0
∂uk(j)
For a variation δuk(j), calculate δUj
Use δUj to calculate ∂Φ0
∂u k (j)
Calculations are not completely trivial.
Solution:
Gradient
We need to calculate ∂Φ0
∂uk(j)
∂Φ0
∂uk(j) = − ˙ ¸
λj |i∆t[Hk, ρj ]−
Michael Goerz Introduction to the GRAPE Algorithm

Grape Algorithm
Pulse Update
uk(j) −→ uk(j) + ɛ ∂Φ0
∂uk(j) ;
Guess initial controls uk(j)
Update pulse according to gradient:
∂Φ0
∂uk(j) = − ˙ ¸
λj |i∆t[Hk, ρj ]−
Forward propagation of ρ(0): calculate and store all ρj = Uj . . . U1ρ(0)U †
1
for j ∈ [1, N]
. . . U†
j
Backward propagation of C: calculate and store all λj = U †
j+1 . . . U†
NCUN . . . Uj+1
for j ∈ [1, N]
Evaluate ∂Φ0 and update the m × N control amplitudes uk(j)
∂uk (j)
Done if fidelity converges
Michael Goerz Introduction to the GRAPE Algorithm

Variations
Non-Hermitian Operators
Φ1 = ℜ[Φ0];
Φ2 = |Φ0| 2 ;
Unitary Transformations
Φ3 = ℜ 〈UF |U(T )〉 = ℜ
D
∂Φ1
= − λ
∂uk(j) x j |i∆t[Hk, ρ x E D
j − λ y
j |i∆t[Hk, ρ y
E
j
∂Φ2
∂uk(j) = −2ℜ ˘˙ ¸ ˙ y
λj |i∆t[Hk, ρj ρN |C¸¯
D
U †
j+1 . . . U†
NUF E
|Uj . . . U1 = ℜ ˙ ¸
Pj |Xj
∂Φ3
∂uk(j) = −ℜ ˙ ¸
Pj |i∆tHkXj
Φ4 = |〈UF |U(T )〉| 2 = ˙ Pj |Xj
¸ ˙ ¸
Xj |Pj
∂Φ4
∂uk(j) = −2ℜ ˘˙ ¸ ˙ ¸¯
Pj |i∆tHkXj Xj |Pj
Also works with Lindbladt-Operators. Additional energy constraints are possible.
Michael Goerz Introduction to the GRAPE Algorithm

5]
Comparison with with OCT
ð1Þ
re the
to the
,um(t))
anged
oblem
ds that
a specimumermiasured
ð2Þ
rators,
of the
j k k j
Z Dt
HkDt ¼ U jðsÞH kU jð sÞds ð9Þ
0
Fig. 1. Schematic representation of a control amplitude u k(t),
consisting of N steps of duration Dt = T/N. During each step j, the
control amplitude uk(j) is constant. The vertical arrows represent
gradients dU0=dukðjÞ, indicating how each amplitude u k(j) should be
modified in the next iteration to improve the performance function U0.
Ψin
ɛ (1)
ɛ (0)
∆u(j) ∼ ˙ Ψbw (tj ) |µ| Ψfw (tj ) ¸
GRAPE also needs forward- and backward-propagation, but only with old pulse.
Propagated states also need to be stored.
Pulse update at point j in the current iteration does not depend on other
updated pulse values (non-sequential update)
All updates in GRAPE can in principle be calculated in parallel.
Convergence tends to be pretty lousy (so I’m told)
What about the choice of ɛ?
Michael Goerz Introduction to the GRAPE Algorithm
tj
Ψtgt

Thank You!
Michael Goerz Introduction to the GRAPE Algorithm