Robustness and Risk-Sensitive Control of Quantum Systems. Matt James Australian National University

Transcription

1 Robustness and Risk-Sensitive Control of Quantum Systems Matt James Australian National University 1

2 Outline Introduction and motivation Feedback control of quantum systems Optimal control of quantum systems Risk-neutral Risk-sensitive Robustness properties of risk-sensitive optimal control Example References 2

3 Introduction and Motivation The issue: how to cope with model uncertainty and disturbances? Robustness: good performance under nominal conditions, and acceptable performance in other than nominal conditions A systematic approach to robust control design might include quantification of performance specification of a nominal model specification of a class of other than nominal models quantification of allowed deviations from nominal bound on performance in terms of quantified deviation 3

4 Some background: Late 70 s - G. Zames initiates so-called H approach to robust control Intensive development of robust control for linear and nonlinear systems in 80 s and 90 s Links established among H control, dissipative systems, Riccati equations, dynamic games, stochastic risk-sensitive optimal control Typical H robust performance bound: T 0 z(t) 2 dt β(x 0 ) + T 0 w(t) 2 dt performance initial size of quantity bias uncertainty 4

5 Some robust/quantum papers: A.C. Doherty, J. Doyle, H. Mabuchi, K. Jacobs, and S. Habib, Robust control in the quantum domain, Proc. IEEE CDC J.K. Stockton, J.M Geremia, A.C. Doherty, and H. Mabuchi, Robust quantum parameter estimation: coherent magnetometry with feedback to appear in Phys. Rev. A (2004). V. Protopopescu, R. Perez, C. D Helon and J. Schmulen, Robust control of decoherence in realistic one-qubit quantum gates, 2003 J. Phys. A: Math. Gen J. Beumee and H. Rabitz, Robust optimal control theory for selective vibrational excitation in molecules: A worst case analysis, J. Chem. Physics, 97(2)

6 Collaborators on this topic area: Ian Petersen (ADFA/UNSW) Stuart Wilson (ANU) Andrew Doherty (UQ) 6

7 Feedback Control of Quantum Systems input u physical system output y K 7

8 Discrete-time model of quantum system: (controlled stochastic master equation) ω k+1 = Λ Γ (u k, y k+1 )ω k, where (conditional state) ω = density operator (instrument) Γ(u, y) = quantum operation p Γ (y u, ω) = Γ(u, y)ω, I Λ Γ (u, y)ω = Γ(u, y)ω p(y u, ω) 8

9 Feedback controller: where and u = K(y) K = {K 0, K 1,...,K M 1 } u 0 = K 0 u 1 = K 1 (y 1 ) u 2 = K 2 (y 1, y 2 ) etc. Probability distributions: (determined by instrument process) P K ω 0,0(y 1,...,y M ) = = M 1 k=0 M 1, k=0 p Γ (y k+1 u k, ω k ) Γ(u k, y k+1 )ω 0, I 9

10 Optimal Control - Risk-Neutral Minimize: M 1 J ω,0 (K) = E K ω,0[ Dynamic programming: i=0 ω i, L(u i ) + ω M, N ] V (ω, k) = inf { ω, L(u) + V (Λ Γ (u, y)ω, k + 1)p(y u, ω)} u U y Y V (ω, M) = ω, N Optimal controller: K u ω 0 u (ω, k) argmin{ ω, L(u) + V (Λ Γ (u, y)ω, k + 1)p(y u, ω))} u U y Y 10

11 Optimal risk-neutral controller has a separation structure, with SME as filter: input u physical system state ω k SME output y control u (ω k, k) filter state ω k copy SME feedback controller K u ω 0 11

12 Optimal Control - Risk-Sensitive Two generalizations of classical risk-sensitive criteria are: ( M 1 ) J 1,µ ω,0 (K) = EK ω,0[exp ω k, µl(u k ) + ω M, µn ] General framework: k=0 M 1 J 2,µ ω,0 (K) = EK ω,0[ k=0 J µˆω,0 (K) = ω k, e µl(u k) ω M, e µn ] y 1,M Y M ˆω, G 0 G k = R (u k )Γ (u k, y k+1 )G k+1 G M = F (for suitable operator valued cost R(u)) 12

13 Operator valued costs for above criteria are: R 1 (u)ˆω = e ˆω,µL(u) / ˆω,1 ˆω and R 2 (u)ˆω = ˆω, eµl(u) ˆω, 1 ˆω [Another choice: R 3 (u)ˆω = c Z c (u)ˆωz c (u) (arises from a dilation of the quantum operation model)] 13

14 Solution is in terms of a modified SME dynamics: ˆω k+1 = Λ Γ,R (u k, y k+1 )ˆω k where (modified conditional state) ˆω = unnormalized density operator Γ R (u, y) = Γ(u, y)r(u) (unnormalized) p Γ,R (y u, ˆω) = Γ R(u, y)ˆω, I R(u)ˆω, I Λ Γ,R (u, y)ˆω = Γ(u, y)ˆω p Γ,R (y u, ˆω) 14

15 Alternate representation of cost criterion in terms of modified conditional state: Dynamic programming: J µˆω,0 (K) = EKˆω,0[ ˆω M, F ] W(ˆω, k) = inf { W(Λ Γ,R (u, y)ˆω, k + 1)p Γ,R (y u, ˆω)} u U y Y W(ˆω, M) = ˆω, F Optimal control: K u,µ ω 0 û (ˆω, k) argmin{ W(Λ Γ,R (u, y)ˆω, k + 1)p Γ,R (y u, ˆω)} u U y Y 15

16 Optimal risk-sensitive controller has a separation structure, with modified SME as filter: input u physical system state ω k SME output y control û (ˆω k, k) filter state ˆω k mod. SME feedback controller Kû,µ ˆω 0 16

17 Robustness Properties of Risk-Sensitive Control Consider risk-sensitive criteria no. 2, and use to obtain controller. M 1 J 2,µ ω,0 (K) = EK ω,0[ k=0 ω k, e µl(u k) ω M, e µn ] Let Kû,µ ˆω 0 be the optimal risk-sensitive controller obtained using a nominal model (Kraus form) Γ nom (u, y)ω = a γ nom,a (u, y)ωγ nom,a(u, y) Let the true model be among the other than nominal models given by (0 λ a (u, y) c(u, y)) (abs. cts.) Γ true (u, y)ω = a λ a (u, y)γ nom,a (u, y)ωγ nom,a(u, y) 17

18 Corresponding probability distributions: (P true << Pnom) P true (y 1,...,y M ) = M 1 k=0 p Γtrue (y k+1 u k, ω true k ) Lemma P nom (y 1,...,y M ) = M 1 k=0 p Γnom (y k+1 u k, ω nom k ) dp true (y 1,...,y M ) = P M 1 true(y 1,...,y M ) dp nom P nom (y 1,...,y M ) = where k=0 f k+1 (y k+1 y 1,...,y k ) f k+1 (y k+1 y 1,...,y k ) = p Γ true (y k+1 u k, ωk true ) p Γnom (y k+1 u k, ωk nom ) 0 18

19 Duality formula: (free energy-relative entropy) loge P [e Z ] = sup {E Q [Z] R(Q P)} Q<<P Lemma log ω, e µx = sup {µ ω, Xg(X u, ω) C ω ( X X)} g( X,ω) where g( X, ω) 0, g(x X, ω) ω, P X (dx) = 1, and C ω ( X X) = ω, g(x X, ω) log g(x X, ω). 19

20 Theorem (robustness bound) E true ω,0 [ M 1 k=0 ω k, L k (u k ) + ω M, ÑM ] 1 µ log J2,µ ω,0 (Kû,µ ˆω 0 ) + 1 µ R(P true P nom ) + 1 µ C( L L) where L k (u k ) = L(u k )g k (L(u k ) L(u k ), ω k ), Ñ M = Ng M (N N, ω M ) M 1 R(P true P nom ) = E true ω,0 [ M 1 C( L L) = E true ω,0 [ k=0 k=0 log f k+1 (y k+1 y 1,...,y k )] C ωk ( L(u k ) L(u k )) + C ωm (Ñ N)]. (Perturbed cost and cost entropy arise naturally in this quantum setup.) 20

21 Example 2-level system, noisy measurement of spin observable A = want state to be spin up 1 = 0 1, or 1 1 = through a series of measurements and feedback control actions (u = 0 do nothing, or u = 1 flip) 21

22 Nominal model: Γ nom (u, y)ω = q(y 1)P 1 T u ωt u P 1 + q(y 1)P 1 T u ωt u P 1, 1 0 if u = T u = 0 1 if u = 1, 1 0 P 1 = 1 0, P 1 = 0 0, and (0 < α < 1) q( 1 1) = q(1 1) = 1 α q( 1 1) = q(1 1) = α. (measurement error prob.) 22

23 Cost observable: L(u) = X 2 + c(u)i and N = X 2 where X = 1 2 (A I) and The expected value of X 2 is c(0) = 0, c(1) = p, with p 0 1 X 2 1 = tr[x ] = 0 1 X 2 1 = tr[x ] = 1 23

24 Robustness interpretation Perturbed instrument and probability distribution: γ nom,a (u, y) = q(y a)p a T u True value of meas. error 0 < α < 1. (RN deriv.) λ a (u, y) = q(y a) q(y a) Class of true models: Γ true (u, y)ω = q(y 1)P 1 T u ωt u P 1 + q(y 1)P 1 T u ωt u P 1 24

25 Perturbed cost: ω, L(1) = (1 + p)ω 11 + pω 22. ω, L(1) = (1 + p)ω 11 + p(1 1 + p 1 + p ω 11). 25

26 References V.P. Belavkin. On the theory of controlling observable quantum systems. Automation and Remote Control, 44(2): , V.P. Belavkin and P. Staszewski. A Radon-Nikodym theorem for completely positive maps. Reports on Mathematical Physics, 24(1):49 55, August M.R. James. Risk-sensitive optimal control of quantum systems. Phys. Rev. A, 69:032108, M.R. James and I.R. Petersen. Robustness Properties of a Class of Optimal Risk-Sensitive Controllers for Quantum Systems M. Raginsky. Radon-Nikodym derivatives of quantum operations. Journal of Mathematical Physics, 44(11): , November