Introduction to the GRAPE Algorithm

Transcription

1 June 8, 2010

2 Reference: J. Mag. Res. 172, 296 (2005) Journal of Magnetic Resonance 172 (2005) 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, Garching, Germany Received 27 June 2004; revised 23 October 2004 Available online 2 December 2005

3 The Basic Idea Acronym Z GRAPE: Gradient Ascent Pulse Engineering Φ 0 original value 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. u k (j) at time index j: go in direction of gradient Pulse Update u k (j) u k (j) + ɛ Φ 0 u k (j)

4 Working in Liouville Space Density Matrix Ψ ρ = Ψ Ψ Liouville-von Neumann Equation ρ(t) = i [H, ρ(t)] = i "! # mx H 0 + u k (t)h k, ρ k=1 Time Propagation (!) mx U j = exp i t H 0 + u k (j)h k k=1 ρ(t ) = U N... U 1 ρ(0) U 1... U N = Ψ(T ) Ψ(T ) with Ψ(T ) = U n... U 1 Ψ(0)

5 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 Φ 0 = C ρ(t ) tr C ρ(t ) C = O Ψ(0) Ψ(0) O ρ(t ) = U Ψ(0) Ψ(0) U

6 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 Φ 0 = C ρ(t ) tr C ρ(t ) C = O Ψ(0) Ψ(0) O ρ(t ) = U Ψ(0) Ψ(0) U Equivalence to normal fidelity tr C ρ(t ) = X D E D E D n O Ψ(0) Ψ(0) O U Ψ(0) Ψ(0) U E n n D = Ψ(0) U X E E D E n Dn O Ψ(0) Ψ(0) O U Ψ(0) D E D E = Ψ(0) U O Ψ(0) Ψ(0) O U Ψ(0) D = Ψ(0) O E 2 U Ψ(0)

7 Fidelity through Backward- and Forward-Propagation A trace is invariant under cyclic permutation of its factors! Fidelity at T D E Φ 0 = C ρ(t ) = C U N... U 1 ρ(0)u 1... U N D E = U j+1... U N CU N... U j+1 U j... U 1 ρ(0)u 1... U j Propagated States Fidelity at t j λ j U j+1... U N CU N... U j+1 bw. propagated optimal state ρ j U j... U 1 ρ(0)u 1... U j fw. propagated initial state Φ 0 = C ρ(t ) = λ j ρ j Note: all propagations with guess pulse!

8 Calculation of Pulse Update Pulse Update u k (j) u k (j) + ɛ Φ 0 u k (j) We need to calculate Φ 0 u k (j) Two steps: For a variation δu k (j), calculate δu j Use δu j to calculate Φ 0 u k (j) Calculations are not completely trivial. Solution: Gradient Φ 0 u k (j) = λ j i t[h k, ρ j ]

9 Grape Algorithm Pulse Update u k (j) u k (j) + ɛ Φ 0 u k (j) ; Φ 0 u k (j) = λ j i t[h k, ρ j ] Guess initial controls u k (j) Update pulse according to gradient: Forward propagation of ρ(0): calculate and store all ρ j = U j... U 1 ρ(0)u 1... U j for j [1, N] Backward propagation of C: calculate and store all λ j = U j+1... U N CU N... U j+1 for j [1, N] Evaluate Φ 0 u k (j) and update the m N control amplitudes u k(j) Done if fidelity converges

10 Variations Non-Hermitian Operators Φ 1 = R[Φ 0 ]; Φ 2 = Φ 0 2 ; Φ D E D E 1 u k (j) = λ x j i t[h k, ρ x j λ y j i t[h k, ρ y j Φ 2 u k (j) = 2R λ j i t[h k, ρ j ρy C N Unitary Transformations E Φ 3 = R U F U(T ) = R DU j+1... U N U F U j... U 1 = R P j X j Φ 3 u k (j) = R P j i th k X j Φ 4 = U F U(T ) 2 = P j X j Xj P j Φ 4 u k (j) = 2R P j i th k X j Xj P j Also works with Lindbladt-Operators. Additional energy constraints are possible.

11 Comparison with OCT Z t j ɛ (1) ɛ (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. Ψ in u(j) Ψ bw (t j ) µ Ψ fw (t j ) Ψ tgt 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?

12 Thank You!