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Quantum Computing via Local Control

Quantum Computing via Local Control. Einat Frishman Shlomo Sklarz David Tannor. Quantum Circuits = Unitary Transformations. output. input. Y. T. T. H. T. “ Rule ” (logical operation) U(t) same for all input The Schr ö dinger Equation: We can formally Solve:

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Quantum Computing via Local Control

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  1. Quantum Computingvia Local Control Einat Frishman Shlomo Sklarz David Tannor

  2. Quantum Circuits = Unitary Transformations output input Y T T H T • “Rule” (logical operation) U(t) same for all input • The Schrödinger Equation: • We can formally Solve: • Unitary propagator U(t) creates mapping between y(0) and y(t): ↔

  3. The Unitary Control Problem External laser Field 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? • Inverse problem  Control problem [1] C.M. Tesch and R. de Vivie-Riedle, PRL 89, 157901 (2002) [2] J.P. Palao and R. Kosloff, PRL 89, 188301 (2002)

  4. Control of a State vs. Control of a Transformation • What is usually done in quantum control: - Control of a State:find E(t) such that Yf  Yi . Controls the evolution of one state • What we have here – a harder problem ! - Control of a Transformation: find E(t) such that Yf(1)= U Yi(1) , Yf(2)= U Yi(2) ,   Yf(n)= U Yi(n) . Controls simultaneously the evolution of allpossible states and phases

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

  6. U UR Entire Hilbert Space Register states Mediating states Projection onto Register • Separable Unitary transformation on space: • Define Pa projection operator onto the quantum register sub-manifold: UR=PUP

  7. A1Su+ E(t) Mediating states X1Sg+ Register The Model: Producing Unitary Transformations on the Vibrational Ground Electronic States of Na2 H=H0+Hint , Hint= ( ) mE mE*

  8. Definition of ConstrainedUnitary Control Problem • System equation of motion: • Control: laser field E(t) • Objective: target unitary transformation ORMaximize J=|Tr(OR†UR(T))|2 • Constraint: No depopulation of register Conserve C=Tr(UR†UR)

  9. 2 WS WP 3 1 Motivation: Stimulated Raman Adiabatic Passage (STIRAP) WS WP ! 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).

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

  11. |3 E(t) |2 |1 Mediating states Register states Creating a Hadamard Gate in a Three-Level L-System

  12. Femto-second pulse shaping

  13. Mediating states Register Fourier Transform on a Quantum Register: with (7+3) levelsub-manifold of Na2; w=e2pi/6 [24 p.s.]

  14. UR11UM …UR1nUM    URn1UM…URnnUM Direct-Sum vs. Direct-Product Space(separable transformations) UR UM • Direct Sum U=URUM • Direct product U=URUM

  15. Atoms in linear trap Internal states |ee |e |g E(t) w2 w1 |n+1|n|n-1 |eg |ge |n+1|n|n-1 External Center of mass modes w1 w2 |gg Ion-Trap Quantum Gates Problem: Entanglement of the Quantum register with the external modes! [1] J.I. Cirac and P. Zoller, PRL 74, 4091 (1995) [2] A. Sørensen and K. Mølmer, PRL 82, 1971 (1999) [3] T. Calarco, U. Dorner, P.S. Julienne, C.J. Williams and P. Zoller, PRA 70, 012306, (2004)

  16. Liouville-Space Formulation • Projection P onto register must trace out the environment producing, in general, mixed states on the register. • Liouville space description is required! • Space:H → L, • Density Matrix:r → |r =|rR|rE, • Inner product: Tr(r†s) → r|s • Super Operators:[H,r] → H |r, UrU† U |r • Evolution Equation:

  17. |ee w2 w1 |eg |ge w1 w2 |gg Sørensen-Mølmer Scheme Field internalexternal |n+1|n|n-1

  18. Local Control (Initial) Resultsfor a two-qubit entangling gate Fields (amp,phase) and evolution of propagator: • We assumed each pulse is near-resonant with one of the sidebands • We fixed the total summed intensity • Results close to the Sørensen-Mølmer scheme

  19. Summary • Control of unitary propagators implies simultaneously controlling all possible states in system • We devised a Local Control method to eliminate undesired population leakage • We considered two general state-space structures: • Direct Sum E.g.:* Hadamard on a L system, * SU(6)-FT on Na2 • Direct ProductE.g.:* Sørensen-Mølmer Scheme to directly produce arbitrary2-qubit gates

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