Read [2ex]Telling optimal control how to maximize the entangling power of two-qubit gates text version

Telling optimal control how to maximize the entangling power of two-qubit gates

Christiane P. Koch

joint work with Daniel Reich , Michael Goerz , Mamadou Ndong , Matthias M. M¨ller, u Jiri Vala, Haidong Yuan, Birgitta Whaley & Tommaso Calarco

Institut f¨r Theoretische Physik, Freie Universit¨t Berlin, Germany u a

introduction: optimal control & QIP

^+^ ^ ^ Tr O PN U(T , 0; )PN ^ desired gate operation : O ^ actual evolution : U(T , 0; ) desired fidelity : 1 - where < 10-4

introduction: optimal control & QIP

^+^ ^ ^ Tr O PN U(T , 0; )PN ^ desired gate operation : O ^ actual evolution : U(T , 0; ) desired fidelity : 1 - where < 10-4 ^ choice of O ?

outline

1

introduction: optimal control

terminology & basic concepts functionals: how to convey the physics to the algorithm

2

a functional to optimize the entangling power of two-qubit gates

local equivalence classes the new functional 2nd order Krotov algorithm

3

application to a Rydberg gate in cold atoms where can we go from here?

4

some terminology & basics of optimal control

coherent control

quantum mechanics probabilistic, but deterministic theory |(t > 0)

|(t = 0)

Schr¨dinger equation o

given an initial state, |(t = 0) or (t = 0), ^ ^ which dynamics ( which H) guarantees a particular outcome, |(t > 0) or (t > 0) ? ^

principle of coherent control

wave properties of matter (superposition principle) variation of phase between different, but indistinguishable quantum pathways constructive interference in desired channel destructive interference in all other channels

final goal: understanding the intricate workings of quantum interferences

principle of coherent control (rev'd)

wave properties of matter (superposition principle) interacting quantum (sub)systems

variation of phase between different, but indistinguishable quantum pathways constructive interference in desired channel destructive interference in all other channels

final goal: understanding the intricate workings of interferences & entanglement

coherent control & optimal control

goal: improve outcome of process vary some parameters

simple, intuitive schemes

bichromatic c., pump-dump, STIRAP

in time or in frequency domain

coherent control & optimal control

goal: improve outcome of process vary some parameters goal: obtain maximum control over process tune `all' available parameters complex outcome discovery of new schemes

not necessarily accessible by intuition

simple, intuitive schemes

bichromatic c., pump-dump, STIRAP

in time or in frequency domain

in time / frequency "phase space"

global or local in time

classical vs quantum control

classical quantum

coherent control control

classical vs quantum control

classical quantum

optimal control theory (based on variational calculus)

coherent control control

let's put philosophy aside ... work out the tools

- they will be needed anyhow -

optimal control theory

time/frequency 'phasespace' picture t=0 |i t=T |f

inverse problem: given the target and the equations of motion, calculate the field

optimal control theory

time/frequency 'phasespace' picture t=0 |i t=T |f

inverse problem: given the target and the equations of motion, calculate the field local in time

impose 2 conditions (for phase and amplitude) derive equations for field also: Lyapunov control / tracking

global in time

information from dynamics throughout time interval to reach desired target at final T OCT

optimal control theory

t=0 |i define the objective : GOAL ^+ i |U (T , 0; )|f

2

t=T |f

= -F

as a functional of the field

optimal control theory: variants

variational approach `guess' the right functional, including eqs. of motion & phasefactors do the variations to obtain eqs. of motion and eq. for the field `guess' the correct time discretization s.t. method converges

W. Zhu, J. Botina, H. Rabitz, JCP 108, 1953 (1998)

optimal control theory: variants

variational approach `guess' the right functional, including eqs. of motion & phasefactors do the variations to obtain eqs. of motion and eq. for the field `guess' the correct time discretization s.t. method converges

W. Zhu, J. Botina, H. Rabitz, JCP 108, 1953 (1998)

Krotov method ingredients: objective + constraint(s) + eqs. of motion construct auxiliary functional L with auxil. potential to guarantee monoton. convergence derive the eq. for (t) from the minimization of L

Sklarz & Tannor, PRA 66, 053619 (2002), Palao & Kosloff, PRA 68, 062308 (2003)

optimal control theory: schemes

improve the field by

j+1 (t) =

S(t) 0

Im

^ i | U (T , 0; j ) |f forward propagation (1)

+

^+ f | U (t, T ; j ) µ U(t, 0; j+1 ) |i ^ ^ backward propagation (2) forward propagation (3)

optimal control theory: schemes

improve the field by

j+1 (t) =

S(t) 0

Im

^ i | U (T , 0; j ) |f forward propagation (1)

+

^+ f | U (t, T ; j ) µ U(t, 0; j+1 ) |i ^ ^ backward propagation (2) forward propagation (3)

(t)

interference between past and future events

i 0

1 t

0

f T

optimal control theory: schemes

improve the field by

j+1 (t) =

S(t) 0

Im

^ i | U (T , 0; j ) |f forward propagation (1)

+

^+ f | U (t, T ; j ) µ U(t, 0; j+1 ) |i ^ ^ backward propagation (2) forward propagation (3)

(t)

interference between past and future events

i 0

1 t

0

f T

variational approach & Krotov method lead to similar schemes Maday & Turinici, JCP 118, 8191 (2003) & work by N.C. Nielsen group

optimal control theory: schemes

improve the field by

j+1 (t) =

S(t) 0

Im

^ i | U (T , 0; j ) |f forward propagation (1)

+

^+ f | U (t, T ; j ) µ U(t, 0; j+1 ) |i ^ ^ backward propagation (2) forward propagation (3)

(t)

interference between past and future events

i 0

1 t

0

f T

variational approach & Krotov method lead to similar schemes Maday & Turinici, JCP 118, 8191 (2003) & work by N.C. Nielsen group

to date: reduced Krotov method only (1st order variant)

functionals or

how to convey the desired physics to the OCT algorithm

objective functionals / costs

J [{k (t), (t)}, (t)] = k JT [{k (T ), (T )}] + Jt [{k (t), (t)}, (t)] k k final-time target intermediate-time target time-dependent cost state-dependent cost

functionals of the field (t) explicitly implicitly through k (t), k (T )

final-time objectives JT

(i) state-to-state transfer (ii) unitary transformation

final-time objectives JT

(i) state-to-state transfer (ii) unitary transformation

) |11

|01 , |10 |00

d

goal: perform a two-qubit gate on the logical basis

example: phasegate for two Ca atoms

qubit encoding:

example: phasegate for two Ca atoms

qubit encoding:

two-atom Hamiltonian:

example: phasegate for two Ca atoms

qubit encoding:

two-atom Hamiltonian:

example: phasegate for two Ca atoms

target: true two-qubit phase = 00 - 01 - 10 + 11 d = 5 nm

example: phasegate for two Ca atoms

target: true two-qubit phase = 00 - 01 - 10 + 11 d = 200 nm: scaling up interaction C3

example: phasegate for two Ca atoms

target: true two-qubit phase = 00 - 01 - 10 + 11

d = 200 nm: scaling up interaction C3

limiting factor for fast gate: sufficient time to resolve GS motional dynamics

(will play a role in any scheme based on resonant excitation)

example: phasegate for two Ca atoms

target: true two-qubit phase = 00 - 01 - 10 + 11

d = 200 nm: scaling up interaction C3

limiting factor for fast gate: sufficient time to resolve GS motional dynamics

(will play a role in any scheme based on resonant excitation)

what about flexibility in single-qubit phases?

final-time objectives JT

JT = - 0 ^ ^+^ ^ Re Tr O PN U(T , 0; )PN N

real-valued, phase-sensitive functional

^ O target operator 0 weight ^ N = dim{O}

^ PN projector on ^ subspace of O ^ U(T , 0; ) actual time evolution

final-time objectives JT

JT = - 0 ^ ^+^ ^ Re Tr O PN U(T , 0; )PN N

real-valued, phase-sensitive functional

^ O target operator 0 weight ^ N = dim{O}

^ PN projector on ^ subspace of O ^ U(T , 0; ) actual time evolution

^ state-to-state transfer: O = |target target |, N = 1 single-qubit gate: N = 2, two-qubit gate: N = 4

cf. Palao & Kosloff, PRA 68, 062308 (2003)

intermediate-time objectives Jt

assumption: additive costs

T

Jt =

0

{ga [(t)] + gb [(t), (t)]} dt

intermediate-time objectives Jt

assumption: additive costs

T

Jt =

0

{ga [(t)] + gb [(t), (t)]} dt examples

ga [(t)] = a S(t) [(t) - ref (t)]2

minimization of field intensity (ref (t) = 0) or change in field intensity (ref (t) = old ) choice of ref (t) determines update vs replacement rule !

intermediate-time objectives Jt

assumption: additive costs

T

Jt =

0

{ga [(t)] + gb [(t), (t)]} dt examples

ga [(t)] = a S(t) [(t) - ref (t)]2

minimization of field intensity (ref (t) = 0) or change in field intensity (ref (t) = old ) choice of ref (t) determines update vs replacement rule !

^ gb [(t), (t)] = b (t)|D(t)|(t)

^ D(t) target operator a , b weights, S(t) switch/shape function

time-dependent targets

^ gb [(t), (t)] = b (t)|D(t)|(t) prescribing a desired evolution

' '

^ D(t)

=

|6 6|(T1 - t) + |1 1|(t - T1 )(T2 - t) + |7 7|(t - T2 )(T3 - t) + |2 2|(t - T3 )(T - t)

Ndong, Tal-Ezer, Kosloff, CPK, JCP 130, 124108 (2009)

time-dependent targets

^ gb [(t), (t)] = b (t)|D(t)|(t) prescribing a desired evolution

' '

keeping the dynamics in a subspace

^ D(t)

=

|6 6|(T1 - t) + |1 1|(t - T1 )(T2 - t) + |7 7|(t - T2 )(T3 - t) + |2 2|(t - T3 )(T - t)

Ndong, Tal-Ezer, Kosloff, CPK, JCP 130, 124108 (2009)

^ ^ D(t) = Pallow

Palao, Kosloff, CPK, PRA 77, 063412 (2008)

where are we? outline!

1

introduction: optimal control

terminology & basic concepts functionals: how to convey the physics to the algorithm

2

a functional to optimize the entangling power of two-qubit gates

local equivalence classes the new functional 2nd order Krotov algorithm

3

application to a Rydberg gate in cold atoms where can we go from here?

4

local equivalence classes

classification of two-qubit gates

G = SU(4) group of all two-qubit gates K = SU(2) SU(2) local gates G /K = SU(4)/SU(2) SU(2) non-local gates

su(4) = k p

Cartan decomposition of Lie algebras

i ^ k U = ^1 e - 2

j=x,y ,z

cj 1 2 ^ ^j ^j k2

^ U2 = ^1 U2^2 k ^ k

Zhang, Vala, Sastry, Whaley, PRA 67, 042313 (2003)

Yuan & Khaneja, PRA 72, 040301 (2005)

local invariants

^T ^ m = UB UB ^ ^ ^+^ ^ UB = Q UQ

^ (i.e. U in Bell basis)

^ g1 = Re Tr[m]2 /16 det(U) ^ ^ g2 = Im Tr[m]2 /16 det(U) ^ ^ g3 = Tr[m]2 - Tr[m2 ]/4 det(U) ^ ^ ^ g1 , g2 , g3 define local equivalence class [U], i.e. a class of two-qubit gates that are equivalent up to local (single-qubit) operations

Zhang, Vala, Sastry, Whaley, PRA 67, 042313 (2003)

Weyl chamber

the new functional

^ ^ optimization target [O] instead of O

^ (old) functional to obtain O JT = - 0 ^+^ ^ ^ Re Tr O PN U(T , 0; )PN N

^ (new) functional to obtain [O] JT = g1 2 + g2 2 + g3 2

^ ^ with gi 2 = gi (O) - gi (U)

2

^ ^ and gi (O) the local invariants of O

^ ^ optimization target [O] instead of O

^ (old) functional to obtain O JT = - 0 ^+^ ^ ^ Re Tr O PN U(T , 0; )PN N

^ (new) functional to obtain [O] JT = g1 2 + g2 2 + g3 2

^ ^ with gi 2 = gi (O) - gi (U)

2

^ ^ and gi (O) the local invariants of O

remember: J = J [{k (t), (t)}, (t)] k

to carry out variations, we need to express gi in terms of k (t)

functional based on local invariants

using the definition of the invariants and of the Bell basis

functional based on local invariants

using the definition of the invariants and of the Bell basis and after quite some algebra

functional based on local invariants

using the definition of the invariants and of the Bell basis and after quite some algebra

JT = f12 + f22 + f32 + f42 + f5

functional based on local invariants

using the definition of the invariants and of the Bell basis and after quite some algebra

JT = f12 + f22 + f32 + f42 + f5

f1 = " "" " h i 1 X 2 2 2 2 2 2 ^ k l + k l - 2k l - 4 k · k l · l Re a0 det(U) - 16 k,l h i " " " " 1 X 2 2 ^ Im a0 det(U) - 4k l · l - 4k l · l 16 k,l h i " "" " " "2 1 X 2 2 2 2 2 2 2 ^ Re b0 det(U) - k l + k l - 2k l - 4 k · k l · l - (k · l ) - k · l 4 k,l " " " " +2 (k · l ) k · l + 4 (k · l ) k · l " " h i " " " " " "" " 1 X 2 2 ^ Im b0 det(U) - 4k l · l - 4k l · l - 4 (k · l ) k · l + 4 k · l k · l 4 k,l

f2

=

f3

=

f4

=

functional based on local invariants

using the definition of the invariants and of the Bell basis and after quite some algebra

JT = f12 + f22 + f32 + f42 + f5

f1 = " "" " h i 1 X 2 2 2 2 2 2 ^ k l + k l - 2k l - 4 k · k l · l Re a0 det(U) - 16 k,l h i " " " " 1 X 2 2 ^ Im a0 det(U) - 4k l · l - 4k l · l 16 k,l h i " "" " " "2 1 X 2 2 2 2 2 2 2 ^ Re b0 det(U) - k l + k l - 2k l - 4 k · k l · l - (k · l ) - k · l 4 k,l " " " " +2 (k · l ) k · l + 4 (k · l ) k · l " " h i " " " " " "" " 1 X 2 2 ^ Im b0 det(U) - 4k l · l - 4k l · l - 4 (k · l ) k · l + 4 k · l k · l 4 k,l

f2

=

f3

=

f4

=

^ with a0 = Tr2 (mO ) /16 det(O) and b0 = Tr2 (mO ) - Tr m2 ^ ^ ^O

^ /4 det(O)

(k )m = Re [ m|k (T ) ], (k )m = Im [ m|k (T ) ], m = 1, . . . , dim(H)

functional based on local invariants

JT = f12 + f22 + f32 + f42 + f5

f1 = h i " "" " 1 X 2 2 2 2 2 2 ^ Re a0 det(U - k l + k l - 2k l - 4 k · k l · l 16 k,l h i " " " " 1 X 2 2 ^ Im a0 det(U) - 4k l · l - 4k l · l 16 k,l h i " "" " " "2 1 X 2 2 2 2 2 2 2 ^ Re b0 det(U) - k l + k l - 2k l - 4 k · k l · l - (k · l ) - k · l 4 k,l " " " " +2 (k · l ) k · l + 4 (k · l ) k · l h i " " " " " " " "" " 1 X 2 2 ^ k · l Im b0 det(U) - 4k l · l - 4k l · l - 4 (k · l ) k · l + 4 k · l 4 k,l

f2

=

f3

=

f4

=

^ with a0 = Tr2 (mO ) /16 det(O) and b0 = Tr2 (mO ) - Tr m2 ^ ^ ^O

^ /4 det(O)

(k )m = Re [ m|k (T ) ], (k )m = Im [ m|k (T ) ], m = 1, . . . , dim(H)

problem:

functional based on local invariants

JT = f12 + f22 + f32 + f42 + f5

f1 = h i " "" " 1 X 2 2 2 2 2 2 ^ Re a0 det(U - k l + k l - 2k l - 4 k · k l · l 16 k,l h i " " " " 1 X 2 2 ^ Im a0 det(U) - 4k l · l - 4k l · l 16 k,l h i " "" " " "2 1 X 2 2 2 2 2 2 2 ^ Re b0 det(U) - k l + k l - 2k l - 4 k · k l · l - (k · l ) - k · l 4 k,l " " " " +2 (k · l ) k · l + 4 (k · l ) k · l h i " " " " " " " "" " 1 X 2 2 ^ k · l Im b0 det(U) - 4k l · l - 4k l · l - 4 (k · l ) k · l + 4 k · l 4 k,l

f2

=

f3

=

f4

=

^ with a0 = Tr2 (mO ) /16 det(O) and b0 = Tr2 (mO ) - Tr m2 ^ ^ ^O

^ /4 det(O)

(k )m = Re [ m|k (T ) ], (k )m = Im [ m|k (T ) ], m = 1, . . . , dim(H)

problem: JT is 8th degree polynomial in {k , k }, resp. {|k } non-convex

optimization of non-convex functionals

^ (old) functional to obtain O JT = - 0 ^+^ ^ ^ Re Tr O PN U(T , 0; )PN N quadratic (new) functional to ^ obtain [O] JT = g1 2 + g2 2 + g3 2 non-convex

for non-convex functionals local optima may exist how to ensure monotonic convergence?

2nd order Krotov algorithm

basic concept

ingredients:

final-time target JT [T , ] T time-dep. targets / costs ga [ ] + gb [(t), (t)] equations of motion ^ i t |(t) = H(t)|(t) |(t0 ) = |0

Sklarz & Tannor, PRA 66, 053619 (2002)

basic concept

ingredients:

final-time target JT [T , ] T time-dep. targets / costs ga [ ] + gb [(t), (t)] equations of motion ^ i t |(t) = H(t)|(t) |(t0 ) = |0

construction of auxiliary functional L L[, , , ] = J[, , ] choose arbitrary scalar potential [, , t] such that L[i , ,i , i , ] L[i+1 , ,i+1 ,

i+1

, ]

building in monotonic convergence

Sklarz & Tannor, PRA 66, 053619 (2002)

auxiliary functional L

L[, , , ] = G [(T ), (T )] - [(0), (0), 0]

T

-

0

R[(t), (t), (t), t]dt

final-time contribution: G [(T ), (T )] = JT [(T ), (T )] + [(T ), (T ), T ]

auxiliary functional L

L[, , , ] = G [(T ), (T )] - [(0), (0), 0]

T

-

0

R[(t), (t), (t), t]dt

final-time contribution: G [(T ), (T )] = JT [(T ), (T )] + [(T ), (T ), T ] intermediate-time contribution: R [(t), (t), (t), t] = - ga [ (t)] + gb [(t), (t)] + + + t k=1

k N k

· fk [, , , t]

· fk [, , , t]

auxiliary functional L

L[, , , ] = G [(T ), (T )] - [(0), (0), 0]

T

-

0

R[(t), (t), (t), t]dt

final-time contribution: G [(T ), (T )] = JT [(T ), (T )] + [(T ), (T ), T ] intermediate-time contribution: R [(t), (t), (t), t] = - ga [ (t)] + gb [(t), (t)] + + + t k=1

k N k

· fk [, , , t]

· fk [, , , t]

the choice of [(t), (t), t] completely determines G , R, L

central idea of Krotov's method

goal: minimization of L, resp. JT two-step solution

1

we need an extremum in i

G (i)

= 0 and

R (i)

=0

equation for backward propagation

d (t) = -JT (t) · (t) + dt (T ) = - JT (i) (T )

g t, (i) ,

(i)

central idea of Krotov's method

2

we want a minimum of L, i.e. minimum of G & maximum of R but L is changed by both changes in and changes in Krotov's solution

Konnov & Krotov, Automation Remote Control 60, 10 (1999)

central idea of Krotov's method

2

we want a minimum of L, i.e. minimum of G & maximum of R but L is changed by both changes in and changes in Krotov's solution

(i) choose at the extremum, i , such that it is the worst possible choice with respect to any change in the states maximize L when going from i to i+1 for fixed i (ii) then any change in the field from minimization of L

(i+1) (t) i

to

i+1

will lead to a or ,t < 0

(t) = arg max R (t)(i+1) , (t), t ,t = 0 , 2 R (i+1) , 2

(i+1)

R (i+1) ,

(i+1)

Konnov & Krotov, Automation Remote Control 60, 10 (1999)

central idea of Krotov's method

2

we want a minimum of L, i.e. minimum of G & maximum of R but L is changed by both changes in and changes in Krotov's solution

(i) choose at the extremum, i , such that it is the worst possible choice with respect to any change in the states maximize L when going from i to i+1 for fixed i (ii) then any change in the field from minimization of L

(i+1) (t) i

to

i+1

will lead to a or ,t < 0

(t) = arg max R (t)(i+1) , (t), t ,t = 0 , 2 R (i+1) , 2

(i+1)

R (i+1) ,

(i+1)

Konnov & Krotov, Automation Remote Control 60, 10 (1999)

central idea of Krotov's method

Krotov's solution (i) optimization with respect to change in states is translated into construction of at the extremum i (ii) convergence for step in field, R by

(i)

(i+1)

, assured globally for

R (i+1) (i+1) , ,t = 0 R = R (i+1) , (i+1) , t - R (i+1) , (i) , t 0 a global optimum would be found, if we could actually implement

(i+1)

(t) = arg max R (t)(i+1) , (t), t

(t)

Krotov's step (i) second order construction of

Krotov's ansatz

(t, (i+1) , ,(i+1) ) = 1 2 +

N

construct to second order in the states |k

k |k

k=1 N (i) (i+1)

(i+1)

+ k

(i+1)

|k

(i)

1 (i+1) (i) (i+1) (i) k - k |^ kl (t)|l - l 2 k,l=1

choose kl (t) such that ^ maximum condition for G and minimum condition for R are fulfilled

Krotov: constructive proof for global conditions (t) = e (T -t) - 1 + · 1 (t) · 1 ^ 1 1

Konnov & Krotov, Automation Remote Control 60, 10 (1999)

Krotov's ansatz

(t, (i+1) , ,(i+1) ) = 1 2 +

N

construct to second order in the states |k

k |k

k=1 N (i) (i+1)

(i+1)

+ k

(i+1)

|k

(i)

1 (i+1) (i) (i+1) (i) k - k |^ kl (t)|l - l 2 k,l=1

choose kl (t) such that ^ maximum condition for G and minimum condition for R are fulfilled

Krotov: constructive proof for global conditions (t) = e (T -t) - 1 + · 1 (t) · 1 ^ 1 1

Konnov & Krotov, Automation Remote Control 60, 10 (1999)

construction of kl (t) ^

can be done locally or globally Sklarz/Tannor's discussion local (but results coincide with global derivation due to choice of JT ) Krotov: constructive proof for global conditions derivation for global conditions leads to much simpler solution for fourth-degree tensor ^ (t) = e (T -t) - 1 + · 1 (t) · 1 ^ 1 1

Krotov's proof

main idea: assure that nothing goes wrong for very large & very small and very large

If:

1

The right-hand side of the equation of motion, f (t, , ), is bounded. Specifically, for large values of the state vector, , the right-hand side of the equations of motion does not grow faster than quadratically with respect to for all t and possible fields . The Jacobian of the right-hand side of the equations of motion, J, is bounded for any time t, field and state vector . The functionals JT () and g ( , , t) are twice differentiable and bounded. In particular, for large values of the state vector , the functionals JT and g do not grow faster than quadratically with respect to .

2

3

then

(t) = e (T -t) - 1 + · 1 (t) · 1 ^ 1 1

quantum control

state vectors (i) , (i+1) inherently bounded boundedness conditions already guaranteed if f (t, , ), J, JT () and g ( , , t) regular change in states compact subset of R2NM

quantum control

state vectors (i) , (i+1) inherently bounded boundedness conditions already guaranteed if f (t, , ), J, JT () and g ( , , t) regular change in states compact subset of R2NM

fulfilling G () 0

G = 1 (T ) · + (T ) · · + JT (T )(i) + - JT (T )(i) 2

for = 0: G 0 = 0 for = 0:

fulfilling G () 0

G = 1 (T ) · + (T ) · · + JT (T )(i) + - JT (T )(i) 2

for = 0: G 0 = 0 for = 0: G = · (T ) · + JT (T )(i) + -JT (T )(i) 1 (T ) + 2 ·

T · + JT (T )(i) + - JT (T )(i) A = sup

·

fulfilling G () 0

G = 1 (T ) · + (T ) · · + JT (T )(i) + - JT (T )(i) 2

for = 0: G 0 = 0 for = 0: G = · (T ) · + JT (T )(i) + -JT (T )(i) 1 (T ) + 2 ·

T · + JT (T )(i) + - JT (T )(i) A = sup

·

(T ) < -2A

fulfilling R() 0

R (t), t = 1 (t) · (t) + (t) · (t) · (t) 2 + (t) + (t)(t) · f (t), t - g (t), t

for = 0: R 0, t = 0 t for = 0:

1 (t) - | (t) · B| + C > 0 2

fulfilling R() 0

R (t), t = 1 (t) · (t) + (t) · (t) · (t) 2 + (t) + (t)(t) · f (t), t - g (t), t

for = 0: R 0, t = 0 t for = 0: R (t), t = (t) · (t) (t) · f 1 (t) · (t) + (t) · f - g (t) + (t) + 2 (t) · (t) (t) · (t) (t) · f (t) · (t) (t) · (t) + (t) · f - g (t) · (t)

B C

= =

sup

(t)R2NM ;t[0,T ]

inf

(t)R2NM ;t[0,T ]

fulfilling R() 0

R (t), t = 1 (t) · (t) + (t) · (t) · (t) 2 + (t) + (t)(t) · f (t), t - g (t), t

for = 0: R 0, t = 0 t for = 0: B C = = sup

(t)R2NM ;t[0,T ]

(t) · f (t) · (t) (t) · (t) + (t) · f - g (t) · (t)

inf

(t)R2NM ;t[0,T ]

1 (t) - | (t) · B| + C > 0 2

maximizing L wrt

(T ) < -2A 1 (t) - | (t) · B| + C > 0 2 one solution (t) = e B(T -t)

¯

¯ ¯ C C ¯ -A - ¯ ¯ B B

¯ ¯ ¯ with B = 2B + , C = min (-, 2C - ) and A = max (, 2A + )

or more generally (t) = e (T -t) - 1 +

how to get A, B, C ?

A, B, C are Taylor expansions of certain quantities starting at the first or second order estimate the remainder (Lagrange's form)

W () =

||n-1

1 ( W ) (i) · + R(i) ,n !

how to get A, B, C ?

A, B, C are Taylor expansions of certain quantities starting at the first or second order estimate the remainder (Lagrange's form)

W () =

||n-1

1 ( W ) (i) · + R(i) ,n ! 1 W M (i) , || = n ! n W (i) +

R(i) ,n

W Mn (i) = sup ,||=n

,

how to get A, B, C ?

A, B, C are Taylor expansions of certain quantities starting at the first or second order estimate the remainder (Lagrange's form)

W () =

||n-1

1 ( W ) (i) · + R(i) ,n ! 1 W M (i) , || = n ! n W (i) +

R(i) ,n

W Mn (i) = sup ,||=n

,

estimate that is independent of (i)

W Mn = sup Mn (i) (i) X

estimate of A

A = sup

JT ,2 ·

.

estimate JT ,2 by its Lagrange remainder: 1 J 1 A M2 T = sup JT 2 2 ,||=2

estimate of A

A = sup

JT ,2 ·

.

estimate JT ,2 by its Lagrange remainder: 1 J 1 A M2 T = sup JT 2 2 ,||=2 for functionals JT that are linear or quadratic in A=0

estimate of B

B X sup

+

sup

(t);t[0,T ]

(t),

(i)

,t

mat

estimate of B

B X sup

+

sup

(t);t[0,T ]

(t),

(i)

,t

mat

for Hamiltonians that do not depend on the state B = = sup

;t[0,T ]

(t) ·

(i)

(i)

, t · (t)

(t) · (t) ,t

mat

sup

t[0,T ]

for unitary evolution: B = 0 for non-unitary evolution: max. eigenvalue

estimate of C

-C sup

(t);t[0,T ]

(t) · 1 · (t) (t) · (t)

+

sup

(t);t[0,T ]

g2 (t) · (t)

-C sup (M1 · (t) ) + t[0,T ]

1 sup g 2 ,||=2

.

estimate of C

-C sup

(t);t[0,T ]

(t) · 1 · (t) (t) · (t)

+

sup

(t);t[0,T ]

g2 (t) · (t)

-C sup (M1 · (t) ) + t[0,T ]

1 sup g 2 ,||=2

.

^ for state-independent H and g depending on only up to linear order : 1 = 0 and g2 = 0 C =0

estimate of C

-C sup

(t);t[0,T ]

(t) · 1 · (t) (t) · (t)

+

sup

(t);t[0,T ]

g2 (t) · (t)

-C sup (M1 · (t) ) + t[0,T ]

1 sup g 2 ,||=2

.

^ for state-independent H and g depending on only up to linear order : 1 = 0 and g2 = 0 C =0 for certain (!) functionals and EoMs: A = 0 & B = 0 & C = 0 (t) = 0 and the second order contribution to vanishes: Palao-Kosloff version of Krotov method (Krotov-PK) (still ensuring monotonic convergence globally)

Krotov's step (ii) second order construction of

equation for the field

remember R (i+1) (i+1) , ,t = 0 R = R (i+1) , (i+1) , t - R (i+1) , (i) , t 0

equations of motion in basis set expansion: 2M × 2M matrix H k,R H k,I -H k,I H k,R !

=

k

first order condition yields:

1 2a S(t)

N 2M

(i+1)

(t) =

ref (t) +

km

k=1 m,n=1 N

(i)

k (i+1) mn kn km k mn kn

2M

+(t)

k=1 m,n=1

we now have an algorithm that is monotonically convergent for arbitary targets/constraints

remark: combine Krotov with BFGS

1

Quasi-Newton algorithms are approximate solutions to an extremization problem using information from the second-order Taylor expansion of the function BFGS is a quasi-Newton algorithm using a rank-two update formula involving only gradient information to approximate the (inverse) Hessian for convex functions it is globally and monotonically convergent if supplemented by a line search fulfilling the Wolfe conditions L-BFGS uses only information from the gradients and state vectors of previous steps to solve the memory problem in storing the approximate inverse Hessian under certain additional assumptions convergence remains monotonic

2

3

remark: combine Krotov with BFGS

1

Quasi-Newton algorithms are approximate solutions to an extremization problem using information from the second-order Taylor expansion of the function BFGS is a quasi-Newton algorithm using a rank-two update formula involving only gradient information to approximate the (inverse) Hessian for convex functions it is globally and monotonically convergent if supplemented by a line search fulfilling the Wolfe conditions L-BFGS uses only information from the gradients and state vectors of previous steps to solve the memory problem in storing the approximate inverse Hessian under certain additional assumptions convergence remains monotonic

2

3

remark: combine Krotov with BFGS

1

Quasi-Newton algorithms are approximate solutions to an extremization problem using information from the second-order Taylor expansion of the function BFGS is a quasi-Newton algorithm using a rank-two update formula involving only gradient information to approximate the (inverse) Hessian for convex functions it is globally and monotonically convergent if supplemented by a line search fulfilling the Wolfe conditions L-BFGS uses only information from the gradients and state vectors of previous steps to solve the memory problem in storing the approximate inverse Hessian under certain additional assumptions convergence remains monotonic

2

3

remark: combine Krotov with BFGS

remember: 1 2a S(t)

N 2M (i)

(t)Krotov =

km

k=1 m,n=1 N

k (i+1) mn kn km k mn kn

2M

+(t)

k=1 m,n=1

remark: combine Krotov with BFGS

remember: 1 2a S(t)

N 2M (i)

(t)Krotov =

km

k=1 m,n=1 N

k (i+1) mn kn km k mn kn

2M

+(t)

k=1 m,n=1

^ compare gradient ascent w/ BFGS (for linear dependence of H on !):

-1 N 2M

^ (t)BFGS = B

(t)

k=1 m,n=1

km

(i)

k (i) mn kn

^ with B(t) approximated Hessian

remark: combine Krotov with BFGS

remember: 1 2a S(t)

N 2M (i)

(t)Krotov =

km

k=1 m,n=1 N

k (i+1) mn kn km k mn kn

2M

+(t)

k=1 m,n=1

^ compare gradient ascent w/ BFGS (for linear dependence of H on !):

-1 N 2M

^ (t)BFGS = B

(t)

k=1 m,n=1

km

(i)

k (i) mn kn

^ with B(t) approximated Hessian

choose a S(t) according to L-BFGS this does not affect the monotonicity that is ensured by Krotov's construction

where are we ? outline !

1

introduction: optimal control

terminology & basic concepts functionals: how to convey the physics to the algorithm

2

a functional to optimize the entangling power of two-qubit gates

local equivalence classes the new functional 2nd order Krotov algorithm

3

application to a Rydberg gate in cold atoms where can we go from here?

4

Rydberg qubits

one-atom level scheme

optical tweezers

Gaetan et al., Nat. Phys. 5, 115 (2009)

Rydberg qubits

one-atom level scheme one-atom Hamiltonian

0 ^ (1) ^r r H = |0 0| T^ + Vtrap (^) 1 ^r +|1 1| T^ + Vtrap (^) r i ^r +|i i| T^ + Vtrap (^) r

optical tweezers

+B (t) (|0 i| + |i 0|) µ(^) r r ^r +|r r | T^ + Vtrap (^) r +R (|i r | + |r i|) µ(^) r

Gaetan et al., Nat. Phys. 5, 115 (2009)

Rydberg qubits

one-atom level scheme two-atom level scheme

u = u(|r1 - r2 |)

two-atom Hamiltonian

^ ^ (1) 1 H = H1 1 4,2 1 ^2 1r

(1) +1 4,1 1 ^1 H2 1 1r ^

^ +Hint

(1,2)

^ (1,2) = |rr rr | Hint

u0 |^1 - ^2 |3 r r

Jaksch et al., PRL 85, 2208 (2000)

Rydberg qubits

one-atom level scheme two-atom level scheme

u = u(|r1 - r2 |)

two-atom Hamiltonian

^ ^ (1) 1 H = H1 1 4,2 1 ^2 1r

(1) +1 4,1 1 ^1 H2 1 1r ^

^ +Hint

(1,2)

^ (1,2) = |rr rr | Hint

u0 |^1 - ^2 |3 r r

Jaksch et al., PRL 85, 2208 (2000)

controlled Rydberg phasegate

gate time T=20 ns ^ functional to obtain O

1 0.5 0 -0.5 -1 -1 1 0.5 0 -0.5 -1 -1 1

^ functional to obtain [O]

1 0.5 0 -0.5 -1 -1 1 0.5 0 -0.5 -1 -1 1

|00>

|01>

0.5 0 -0.5 -1 1

|00>

|01>

0.5 0 -0.5 -1 1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

|10>

|11>

0.5 0 -0.5 -1

|10>

|11>

0.5 0 -0.5 -1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

F = 0.993

F = 0.996

controlled Rydberg phasegate

gate time T=20 ns ^ functional to obtain O

1 0.8

^ functional to obtain [O]

1 0.8

pulse envelope

0.6 0.4 0.2 0 0

pulse envelope

0.6 0.4 0.2 0 0

5

10 time [ ns ]

15

20

5

10 time [ ns ]

15

20

F = 0.993

F = 0.996

controlled Rydberg phasegate

approaching the quantum speed limit

1×10

0

1×10

-1

T = 15 ns

error

1×10

-2

T = 20 ns

1×10

-3

T = 30 ns

1×10 0

-4

1000

2000 3000 iterations

4000

5000

controlled Rydberg phasegate

to be continued analyse error sources check role of pulse parametrization test with Hamiltonian allowing for ^ non-diagonal O

summary

optimal control is an extremely versatile tool but

summary

optimal control is an extremely versatile tool but you need to know how to ask questions! we derived a new class of optimization functionals suitable for quantum information purposes based on geometric classification of entangling operations (Cartan decomposition & representation in Weyl chamber) requires optimization algorithm ensuring monotonic convergence ­ 2nd order Krotov method first results for a Rydberg gate encouraging full power of approach still needs to be explored

summary

optimal control is an extremely versatile tool but you need to know how to ask questions! we derived a new class of optimization functionals suitable for quantum information purposes based on geometric classification of entangling operations (Cartan decomposition & representation in Weyl chamber) requires optimization algorithm ensuring monotonic convergence ­ 2nd order Krotov method first results for a Rydberg gate encouraging full power of approach still needs to be explored

summary

optimal control is an extremely versatile tool but you need to know how to ask questions! we derived a new class of optimization functionals suitable for quantum information purposes based on geometric classification of entangling operations (Cartan decomposition & representation in Weyl chamber) requires optimization algorithm ensuring monotonic convergence ­ 2nd order Krotov method first results for a Rydberg gate encouraging full power of approach still needs to be explored

summary

optimal control is an extremely versatile tool but you need to know how to ask questions! we derived a new class of optimization functionals suitable for quantum information purposes based on geometric classification of entangling operations (Cartan decomposition & representation in Weyl chamber) requires optimization algorithm ensuring monotonic convergence ­ 2nd order Krotov method first results for a Rydberg gate encouraging full power of approach still needs to be explored

summary

optimal control is an extremely versatile tool but you need to know how to ask questions! we derived a new class of optimization functionals suitable for quantum information purposes based on geometric classification of entangling operations (Cartan decomposition & representation in Weyl chamber) requires optimization algorithm ensuring monotonic convergence ­ 2nd order Krotov method first results for a Rydberg gate encouraging full power of approach still needs to be explored

summary

optimal control is an extremely versatile tool but you need to know how to ask questions! we derived a new class of optimization functionals suitable for quantum information purposes based on geometric classification of entangling operations (Cartan decomposition & representation in Weyl chamber) requires optimization algorithm ensuring monotonic convergence ­ 2nd order Krotov method first results for a Rydberg gate encouraging full power of approach still needs to be explored

where can we go from here?

where can we go from here?

1

optimize for the complete Weyl chamber, i.e. for an arbitrary perfect entangler

problem: no simple inversion of g1 , g2 , g3 c1 , c2 , c3 solution: define ellipsoid in g -space containing almost all of the Weyl chamber

2

optimize for a specified trajectory in the Weyl chamber ...

3

thank you !

Information

[2ex]Telling optimal control how to maximize the entangling power of two-qubit gates

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