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Quantum Computing via Local Control Einat Frishman Shlomo Sklarz David Tannor

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“ 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

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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)

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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

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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)

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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

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The Model: Producing Unitary Transformations on the Vibrational Ground Electronic States of Na 2 Register Mediating states X1g+X1g+ A1u+A1u+ E(t) H=H 0 +H int, H int = ( ) E* EE

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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 )

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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). SS PP ! SS PP

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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

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

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Femto-second pulse shaping

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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.]

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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

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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!

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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:

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Sørensen-Mølmer Scheme |n+1 |n |n-1 |ee |ge |eg |gg Field internalexternal

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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:

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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

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