Quantum Computing via Local Control Einat Frishman Shlomo Sklarz David Tannor

“ Rule ” (logical operation) U(t) same for all input “ Rule ” (logical operation) U(t) same for all input The Schr ö dinger Equation: The Schr ö dinger Equation: We can formally Solve: We can formally Solve: Unitary propagator U(t) creates Unitary propagator U(t) creates mapping between (0) and (t): mapping between (0) and (t): Quantum Circuits = Unitary Transformations ↔ T H T T input output 

The Unitary Control Problem U(t) is determined by the laser field E(·) : U(t)=U([E],t) U(t) is determined by the laser field E(·) : U(t)=U([E],t) Given a desired U(T)=O can we find a field E(·) that produces it? Given a desired U(T)=O can we find a field E(·) that produces it? Inverse problem  Control problem Inverse problem  Control problem [1] C.M. Tesch and R. de Vivie-Riedle, PRL 89, (2002) [2] J.P. Palao and R. Kosloff, PRL 89, (2002) External laser Field E(t)

Control of a State vs. Control of a Transformation What is usually done in quantum control: What is usually done in quantum control: - Control of a State: find E(t) such that - Control of a State: find E(t) such that  f   i. Controls the evolution of one state What we have here – a harder problem ! What we have here – a harder problem ! - Control of a Transformation: find E(t) such that - Control of a Transformation: find E(t) such that  f   U  i ,  f   U  i ,      f (n)  U  i  n .  f (n)  U  i  n . Controls simultaneously the evolution of all possible states and phases

System=Register+Mediating states System=Register+Mediating states Two alternative realizations: Two alternative realizations: Direct sum space Direct product space Objective: Produce Target Unitary Transformation on register without intermediate population of auxiliary mediating states Objective: Produce Target Unitary Transformation on register without intermediate population of auxiliary mediating states Quantum Register and Mediating States Mediating states Register states E(t)

Projection onto Register Separable Unitary transformation on space: Separable Unitary transformation on space: Define P a projection operator onto the quantum register sub-manifold: U R =PUP Define P a projection operator onto the quantum register sub-manifold: U R =PUP Register states Mediating states U URUR Entire Hilbert Space

The Model: Producing Unitary Transformations on the Vibrational Ground Electronic States of Na 2 Register Mediating states X1g+X1g+ A1u+A1u+ E(t) H=H 0 +H int, H int = ( )  E* EE

Definition of Constrained Unitary Control Problem System equation of motion: System equation of motion: Control: laser field Control: laser field E(t) Objective: target unitary transformation O R Maximize Objective: target unitary transformation O R Maximize J=|Tr(O R † U R (T))| 2 Constraint: No depopulation of register Conserve Constraint: No depopulation of register Conserve C=Tr(U R † U R )

Motivation: Stimulated Raman Adiabatic Passage (STIRAP) Bergmann et al. (1990). Bergmann, Theuer and Shore, Rev Mod. Phys. 70, 1003 (1998). V. Malinovsky and D. J. Tannor, Phys. Rev. A 56, 4929 (1997). SS PP ! SS PP

At each point in time: Enforce constraint C Enforce constraint C dC/dt=Imag(g E(t))=0  E(t)=a g *  direction Monotonic increase in Objective J Monotonic increase in Objective J dJ/dt=Real(f E(t))=a Real(f g * )>0  a=Real(f g * )  Sign and magnitude Local Optimization Method Re Im g*g* f E(t) g

Creating a Hadamard Gate in a Three-Level  -System |1 |2 |3 E(t) Register states Mediating states

Femto-second pulse shaping

Register Mediating states Fourier Transform on a Quantum Register: with (7+3) level sub-manifold of Na 2 ; w=e 2  i/6 [24 p.s.]

Direct-Sum vs. Direct-Product Space (separable transformations) URUMURUM Direct Sum U=U R  U M Direct product U=U R  U M U R 11 U M …U R 1n U M   U R n1 U M …U R nn U M

Ion-Trap Quantum Gates E(t) Atoms in linear trap Internal states External Center of mass modes |e |g |n+1  |n  |n-1  |ee |ge |eg |gg |n+1  |n  |n-1      [1] J.I. Cirac and P. Zoller, PRL 74, 4091 (1995) øø [2] A. S ø rensen and K. M ø lmer, PRL 82, 1971 (1999) [3] T. Calarco, U. Dorner, P.S. Julienne, C.J. Williams and P. Zoller, PRA 70, , (2004) Problem: Entanglement of the Quantum register with the external modes!

Liouville-Space Formulation Projection P onto register must trace out the environment producing, in general, mixed states on the register. Projection P onto register must trace out the environment producing, in general, mixed states on the register. Liouville space description is required! Liouville space description is required! Space: H→L,Space: H→L, Density Matrix:  →|  |  R  |  E Density Matrix:  →|  |  R  |  E  Inner product: Tr(  †  ) →  | Inner product: Tr(  †  ) →  |  Super Operators:  [H,  ] →H |   U  U † U | Super Operators:  [H,  ] →H |   U  U † U |  Evolution Equation:Evolution Equation:

Sørensen-Mølmer Scheme |n+1  |n  |n-1  |ee |ge |eg |gg     Field internalexternal

Local Control (Initial) Results for a two-qubit entangling gate We assumed each pulse is near-resonant with one of the sidebands We assumed each pulse is near-resonant with one of the sidebands We fixed the total summed intensity We fixed the total summed intensity Results close to the S ø rensen-M ø lmer scheme Results close to the S ø rensen-M ø lmer scheme Fields (amp,phase) and evolution of propagator:

Summary Control of unitary propagators implies simultaneously controlling all possible states in system Control of unitary propagators implies simultaneously controlling all possible states in system We devised a Local Control method to eliminate undesired population leakage We devised a Local Control method to eliminate undesired population leakage We considered two general state-space structures: We considered two general state-space structures: Direct Sum  E.g.:* Hadamard on a  system, * SU(6)-FT on Na 2Direct Sum  E.g.:* Hadamard on a  system, * SU(6)-FT on Na 2 Direct Product  E.g.:* S ø rensen-M ø lmer Scheme to directly produce arbitrary2-qubit gatesDirect Product  E.g.:* S ø rensen-M ø lmer Scheme to directly produce arbitrary2-qubit gates