大槻幸義 東北大・院理 & CREST-JST

1. 超短レーザーパルスによる分子量子演算の可能性超短レーザーパルスによる分子量子演算の可能性 大槻幸義 東北大・院理 & CREST-JST

2. 量子 制御 目的 分子分光法を使った，量子計算の（原理）実証実験法の開発 量子情報資源としての分子基盤技術の確立 目的：量子計算アルゴリズムは報告されている 　　　　→　分子への実装シミュレーション 量子コンピュータ 化学反応量子制御

3. Contents Part I Introduction of optimal control theory Part II Relaxation effects on quantum control: Manipulating dissociation wave packets of I2- in water Part III Application to molecular quantum computer

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

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

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

7. Other examples of OCE’s complex systems Isolated systems charge-transfer coordination complex, ... strong-field dynamics dissociation & rearrangement of chemical bonds selective generation of high harmonics pulse propagation Condensed systems self-phase modulation application to biological systems branching ratio between intramolecular and intermolecular energy transfer processes

8. Why optimal control theory ? It is natural to employ optimal control procedures for clarifying the mechanisms of OCE results. New Rules numerical simulations & model analyses Optimal Control Experiment Optimal Control Simulation: step 1 Design of optimal pulses step 2 Decoding of designed pulses

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

10. Contents Part I Introduction of optimal control theory Part II Relaxation effects on quantum control: Manipulating dissociation wave packets of I2- in water Part III Application to molecular quantum computer

11. Part I Introduction of optimal control theory

12. : electric dipole moment operator : electric field (semiclassical approximation) Optimal control method in wave function formalism Schrödinger’s equation optimal control method • 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.

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

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

15. density matrix relaxation term density matrix formalism Quantum Liouville equation unconstrained objective functional pulse design equations

16. Double (Liouville)-space notation Lagrange multiplier density density matrix formalism Definition of inner product between operators A and B unconstrained objective functional Ohtsuki, Zhu & Rabitz, J. Chem. Phys. (99)

17. density matrix formalism Step 1 Step 2 Step 3

18. The objective state is specified by a set of target operators with a set of expectation values The constraints imposed on the time behavior of certain observables on the interval multiple target Minimization of the objective functional Ohtsuki et al., J. Chem. Phys. (01)

19. Quadruple-Space Representation inner product between double-space operators and multiple target

20. multiple target Objective functional in quadruple-space representation with a constraint of satisfying

21. most general formalism Y. Ohtsuki, and H. Rabitz, CRM Proceedings and Lectures, 33, 163 (2003). Y. Ohtsuki,J. Chem. Phys. 119, 661 (2003). Y. Ohtsuki, G. Turinici, and H. Rabitz, J. Chem. Phys.120, 5509 (2004).

22. Part II Relaxation effects on quantum control: Manipulating dissociation wave packets of I2- in water （省略） Nishiyama et al., J. Chem. Phys. (04)

23. Part III Application to Molecular Quantum Computer Y. Ohtsuki, Chem. Phys. Lett. in press. (2005)

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

25. Qubit and quantum parallel processing ３ビットの例 従来のコンピュータは，いずれか１つの値を表現 量子コンピュータは，８通りの値を同時に表現できる 入力 プロセッサ（CPU） 同時に８つの計算を処理できる 出力 ８つの計算結果を同時に出力 量子ビット　１０ 　　　　　　　 １３ ビットマシン（Itanium2 16CPU相当） ビットマシン（Itanium2 128CPU相当）

26. Qubit and quantum parallel processing 量子力学の重ね合わせ 量子系をシミュレーションするのに必要なコンピュータ資源は 自由度の増加とともに指数関数的に増大， Feynman（’82）　量子力学に従い動作するコンピュータの提案 1994Shor　因数分解アルゴリズム 1996Grover　高速検索アルゴリズム

27. Qubit and quantum parallel processing NP問題と量子コンピュータ P（Polynomial）　：従来の決定論的なコンピュータで解ける NP完全問題（Non-deterministic Polynomial）： 非決定論的コンピュータ（仮想マシン）を用いれば 多項式時間で解ける ・巡回セールスマン問題 ・充足問題，．．． 量子コンピュータはNP問題を解くことができるか？←未解決 因数分解はNP問題ではないらしい．．．

28. Qubit and quantum parallel processing Example of fundamental gate Hadamard transform binary sum protocol output/ target bit input/ control bit parallel processing

29. idea of quantum computation quantum algorithm: constructive interference at a position corresponding to a solution. All the operations are reversible (unitary). (Wave packet corruption due to measurements is often utilized.) A solution is obtained with high probability schematic illustration of quantum computation

30. Purpose Numerically study molecular quantum computation using qubits that are implemented in the vibrational states of I2 Optimally designed pulses are shown to act as universal gates through a case study of the simulation of the Deutsch-Jozsa algorithm.

31. vibrational states as qubits example of the mapping in 2-qubit case I２ B state gate pulse mapping intramolecular states qubits X state

32. vibrational states as qubits gate operations acting upon the mapped qubits Qubits are realized by a set of independent spins. Qubits are realized by multilevel systems. 1-qubit operator acts on each subspace 1-qubit operator acts on the whole space

33. vibrational states as qubits matrix representation of 1-qubit gates 1 qubit 2 qubits 3 qubits n qubits ．．．

34. constant: The Deutsch problem Given a function that is either balanced or constant, determine which type it is. The Deutsch problem Consider certain global properties of functions on n-bit binary numbers: a set of binary numbers function balanced: the number of times the function returns 0 is equal to the number of times the function returns 1.

35. (1) Step 1: Initial preparation (2) Step 2: Applying the Hadamard transformation to (1) (3) Step 3: function evaluation via the f-controlled gate, Deutsch-Jozsa algorithm (2-qubit case) A classical algorithm requires that f be evaluated for 2n-1+1 values. A quantum algorithm requires only one evaluation of f .

36. Deutsch-Jozsa algorithm (4) Step 4: Applying the Hadamard transformation to (3) binary inner product: (5) Step 5: observation constant: 1 balanced: 0

37. Equations of motion quantum Liouville equation (Liouville-space notation) system Hamiltonian electric dipole interaction laser fields dipole moment operator time-dependent relaxation operator Liouville-space time evolution operator

38. An optimal-gate-pulse maximizes the following objective functional for an arbitrary initial state: penalty due to fluence gate operator Specifying the independent matrix elements by {j}, we introduce a modified objective functional: weight factor Optimal-gate-pulse design

39. Pulse design equations Optimal laser pulse Introducing the auxiliary density with a final condition non-Markovian effects: Y. Ohtsuki, J. Chem. Phys.(03) Markovian dissipation: Y. Ohtsuki et al., Chem. Phys.(03)

40. Hadamard transform in 1-qubit case 1 qubit obtained by the mapping I２ B state gate pulse computation 　・24 vibrational states 　・Runge-Kutta-Fehlberg method X state

41. Hadamard transform in 1-qubit case High fidelity：99.8% A modulated structure suggests the importance of phase control. Medium fidelity：97.8%

42. Hadamard transform in 1-qubit case The gate pulse designed with high accuracy can achieve a correct transform even after ten successive operations. The accuracy is reduced rapidly with the increase in the number of gate operations. When using a pulse that lacks extreme precision, it will be safe to avoid using the pulse several times.

43. Simulating the Deutsch-Jozsa algorithm 2 qubits obtained by mapping I２ B state numerical procedure gate pulse Applying the Hadamard transform pulse to the qubits in which either the constant state or one of the balanced states is encoded by an oracle pulse. X state

44. Simulating the Deutsch-Jozsa algorithm examples of oracle pulses Hadamard transform pulse constant 100% population in |00> balanced no population in |00> otherwise 25% population in each states

45. Simulating the Deutsch-Jozsa algorithm other cases balanced & constant Universal computation can be realized by combining optimally designed gate pulses.

46. CNOT pulse time evolution of populations of computational basis greater than 99% The CNOT gate can be realized by using only one pulse even in a multilevel system.

47. summary We have investigated the possibility of molecular quantum computation using the vibrational states of I2, in which we have introduced qubits by mapping the computational basis onto the vibrational states in the B state. The matrix representation of a 1-qubit operator is derived in an n-qubit case. We have numerically shown that the optimally designed gate pulses act as universal gates through a case study of the Deutsch-Jozsa algorithm in a 2-qubit case.