1. 分子・レーザー相互作用を使っ た 量子情報処理 大槻幸義 ( 東北大院理. & JST-CREST)

2. Key principles of quantum control Quantum control: manipulating quantum interferences Laser-field manipulation of constructive and destructive interferences of the evolving molecular wave function. shaped laser pulse

3. Optimal control experiment (OCE) Learning Control Measurements can destroy a molecular wave function. lack of the knowledge about molecular Hamiltonian (existence of experimental noises) need minimal or even no knowledge of the Hamiltonian statistical solution search development of laser shaping techniques

4. Optimal control experiment (OCE) closed-loop experiment pulse shaper molecular samples learning algorithm measured results adjusting control knobs pulse Wilson (97) photochemistry of dyes Gerber (98) coordination complexes Bucksbaum (99) pulse propagation in liquid Motzkus (02) biological system crystalline polymer antenna complex LH2 of rhodopseudomonas acidophila

5. My current research subjects Applications Fundamentals Development of solution algorithms Relaxation effects on quantum control Laser-induced surface dynamics Isotope separation Ultrafast processes including non-adiabatic transitions Electronic dynamics in surface and nano-structures Molecular quantum computer Challenges Suppression of decoherence

6. Part I Y. Ohtsuki, Chem. Phys. Lett. 404, 126 (2005). Y. Ohtsuki, to be submitted. Quantum optimal control simulation of Molecular quantum computation

7. Qubit and quantum parallel processing “classical” bit and “quantum” bit binary number computation basis the number of bit integer We freely use a string of qubits and an integer with a subscript.

8. Simulating Grover’s algorithm subroutine (oracle) How many times do we have to call the subroutine to find the unknown number, a ? Answer X state B state gate pulse I２I２ I I 2 qubits obtained by mapping purpose simulating Grover’s search algorithm by combining optimally designed pulses

9. Grover iteration ．．． ① preparing the initial state ② applying the subroutine (oracle) to ③ applying the inversion operator to ④ repeating ②＆③ times The inversion operator folds back the amplitudes about mean.

10. the angle between and is Grover iteration (geometric visualization) oracle (subroutine) reflection in the line through the origin along inversion about mean After k-th Grover iteration, reflection in the line through the origin along

11. Optimal control method in wave function formalism optimal control method (1)Introducing a target operator to specify a physical objective. (2) Adding a penalty term due to pulse fluence in order to reduce pulse energy. (3) Introducing a Lagrange multiplier density that constrains the system to obey the equation of motion. Schrödinger’s equation : electric field (semiclassical approximation) : electric dipole moment operator

12. objective functional unconstrained objective functional Lagrange multiplier (1) expectation value (2) penalty term (3) constraint due to the Schrödinger equation

13. coupled pulse design equations optimal control pulse parameter that weighs the significance of penalty the equation for Lagrange multiplier final condition the Schrödinger equation initial condition

14. generalized solution algorithms Y. Ohtsuki, and H. Rabitz, CRM Proceedings and Lectures, 33, 163 (2003). Y. Ohtsuki, K. Nakagami, W. Zhu and H. Rabitz, Chem. Phys. 287, 197 (2003). Y. Ohtsuki, J. Chem. Phys. 119, 661 (2003). Y. Ohtsuki, G. Turinici, and H. Rabitz, J. Chem. Phys. 120, 5509 (2004). Y. Ohtsuki et al. to be submitted. Type I (linear with respect to the state vector) Type II (bilinear with respect to the state vector)

15. Gate pulse design for an arbitrary initial state: purpose: designing a gate operator, G for arbitrary states: minimization

16. Gate pulse design Pulse design equations These have the same form as those for the “state-to-state” control in the density matrix formalism

17. Summary of Part I We have developed a new optimal control simulation algorithm within the density matrix formalism in order to design gate pulses (not shown here). We have simulated the Grover algorithm and clarified the importance of the non-Markovian relaxation effects on the computational accuracy. The “normalized” target populations are shown to be robust against relaxation effects.

18. Part II Quantum optimal control simulation of Implementation of quantum gate operations in molecules with weak laser fields Y. Teranishi, Y. Ohtsuki, and K. Ohmori et al., J. Chem. Phys. in press. collaboration with Prof. Ohmori’s group (IMS, Japan)

19. Basic Idea Key experimental techniques A logical state is defined as a superposition of molecular eigenstates. 1. To precisely prepare the initial logical input state (pulse shaping). 2. To precisely measure relative phases of the wave packet (“high-resolution” quantum interferogram). gate operation = free propagation of the wave packet The computational results are encoded in the wave packet.

20. Basic Idea gate operator diagonal matrix logical state gate matrix (known) unitary matrix that diagonalizes G matrix Find a set of vibrational states and the gate time T so that they diagonalize the gate matrix.

21. 3-qubit quantum Fourier transform (QFT) example of transformation matrix gate operation time

22. In a real molecule B state input preparation pulse gate computational results B state reference (readout) pulse quantum interferogram interferogram can be read by a ns laser input preparation & gate operationreadout preparing the logical input by a shaping pulse ・ Anharmonic effects on the fidelity

23. experimental setup (theoretical proposal) pulse shaper laser sample ① ② ① input preparation pulse ② reference (readout) pulse ② ① gate time (free propagation) phase shifter detector time delay measure population distribution

24. 3-qubit Fourier transform (QFT) input Fidelity 0.99173 0.99824 0.99819 0.98468 0.99173 0.98468 0.99819 0.99824 average0.99321 Probability amplitude after 3-qubit Fourier transform （ real part & imaginary part ）

25. overlap=0.999993 (periodic function) OUTPUT 3-qubit Fourier transform (QFT) QFT period T =2 overlap=0.999991 INPUT period ω =4 logical state

26. Summary of Part II We have developed a molecular quantum computation scheme, in which a logical state is expressed as a superposition state. 1. A logical input state is prepared by a designed fs laser pulse. 2. The free propagation of the wave packet for a gate operation time performs the computations through the change in the relative phases of the molecular basis. 3. The computational results are retrieved by quantum interferometry. The numerical results showed the initial preparation, gate operation, and readout steps are performed with high fidelities (~99%).