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
目的
核融合研Jan4(05)
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
量子
制御
化学反応量子制御
Contents
核融合研Jan4(05)
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
Key principles of quantum control
核融合研Jan4(05)
Quantum control: manipulating quantum interferences
|1 > + | 2 >
1
| 2 >=
[ | A > - | B >]
2
| 1 >=
|A>
1
[ | A > + | B >]
2
h
|1 > - | 2 >
shaped laser pulse
|B>
1
| Φ ± >=
[ | 1 > ± | 2 >] =| A > or | B >
2
Laser-field manipulation of constructive and destructive
interferences of the evolving molecular wave function.
Optimal control experiment (OCE)
核融合研Jan4(05)
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
Optimal control experiment (OCE)
核融合研Jan4(05)
closed-loop experiment
adjusting control knobs
learning algorithm
pulse
shaper
molecular samples
pulse Enew (t )
Wilson (97)
photochemistry of dyes
Gerber (98)
measured coordination complexes
results
Bucksbaum (99)
pulse propagation in liquid
Motzkus (02)
biological system
crystalline polymer
antenna complex LH2 of
rhodopseudomonas acidophila
Other examples of OCE’s
核融合研Jan4(05)
Isolated
systems
complex systems
charge-transfer coordination complex, ...
strong-field dynamics
dissociation & rearrangement of chemical bonds
selective generation of high harmonics
Condensed
systems
pulse propagation
self-phase modulation
application to biological systems
branching ratio between intramolecular and
intermolecular energy transfer processes
Why optimal control theory ?
核融合研Jan4(05)
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
My current research subjects
核融合研Jan4(05)
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
Contents
核融合研Jan4(05)
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
核融合研Jan4(05)
Part I
Introduction of optimal control theory
Optimal control method in wave function formalism
Schrödinger’s equation
ih

|  (t )  [ H 0   E (t ) ] |  (t ) 
t
 : electric dipole moment operator
E (t )
: electric field (semiclassical approximation)
optimal control method
(1) Introducing a target operator W 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
 (t ) that constrains the system to obey
the equation of motion.
核融合研Jan4(05)
objective functional
核融合研Jan4(05)
unconstrained objective functional
J   (t f ) W  (t f ) 
(1) expectation value
tf
1
  dt
[ E (t )]2
hA
0
 i t f
 Re   d t   (t )
 h 0
(2) penalty term


ih  H t
t


 (t )  

(3) constraint due to the Schrödinger equation
 (t ) 
Lagrange multiplier
coupled pulse design equations
核融合研Jan4(05)
optimal control pulse
E(t )  2 A Im   (t )   (t ) 
A parameter that weighs the significance of penalty
the Schrödinger equation
ih

 (t )  [ H 0   E (t ) ]  (t ) 
t
initial condition
 (0)   0 
the equation for Lagrange multiplier

ih
 (t )  [ H 0   E (t ) ]  (t ) 
t
final condition  (t f )  W  (t f ) 
density matrix formalism
核融合研Jan4(05)
Quantum Liouville equation
ih

 (t )  [ H t ,  (t ) ]  ih   (t )
t
H t  H 0   E (t )
 (t ) density matrix
  (t )
relaxation term
unconstrained objective functional
t
1 f
2
J  tr{W  (t f )} 
d
t
[
E
(
t
)]

hA0
tf
  d t tr{(t )  ih
0

 LtT
t
  (t )}
pulse design
equations
density matrix formalism
Double (Liouville)-space notation
核融合研Jan4(05)
Definition of inner product between operators A and B
 A B  tr ( A† B)
ih

 (t )  Lt
t
 (t )  ih   (t ) 
unconstrained objective functional
J  W
tf
 (t f )  
  d t  (t )
0

tf
1
d t [ E (t )]2

hA0

ih  LtT
t
  (t ) 
(t )  Lagrange multiplier density
Ohtsuki, Zhu & Rabitz, J. Chem. Phys. (99)
density matrix formalism
核融合研Jan4(05)
Step 1
 (0)
ih
 (t )  L (0) (t )  M  (0) (t ) E (0) (t )
t
Step 2
ih
ih
 (1)
i
 (t )  L†(1) (t )  M (1) (t ) A Tr{(1) (t )  (0) (t )}
t
2
 (1)
i
 (t )  L  (1) (t )  M  (1) (t ) A Tr{(1) (t )  (1) (t )}
t
2
Step 3
 (2)
i
 (t )  L†(2) (t )  M (2) (t ) A Tr{(2) (t )  (1) (t )}
t
2
 (2)
i
(2)
(2)
ih
 (t )  L  (t )  M  (t ) A Tr{(2) (t )  (2) (t )}
t
2
ih
multiple target
核融合研Jan4(05)
Minimization of the objective functional
J    X k (t f )   xk
2
: J1
k

l

J1
J2
tf

d t  l (t )  Rl (t )  rl (t )
2
: J2
0
tf
1
2
d
t
[
E
(
t
)
]

hA0
: J3
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
Ohtsuki et al., J. Chem. Phys. (01)
multiple target
核融合研Jan4(05)
J1    X k (t f )   xk
2
k
  
k
X k†
|  (t f )   xk  1|  (t f ) 
W   | Wk  Wk |
k
2
( | Wk | X k†   |1  xk )
Quadruple-Space Representation
inner product between double-space operators X  and Y
†
 X  | Y  tr ( X 
Y )
J1  W |  (t f ) 
multiple target
核融合研Jan4(05)
Y (t )   | Yl (t )   l (t )  Yl (t ) | ( | Yl (t ) | Rl   |1  rl (t ) )
l
J2 
tf
 d t  Y (t ) |  (t ) 
0
Objective functional in
quadruple-space representation
tf
J  W |  (t f )    d t  Y (t ) |  (t ) 
0
t
1 f
2

d
t
[
E
(
t
)
]

hA0
with a constraint of satisfying

ih
|  (t )  Lt |  (t ) 
t
most general formalism
核融合研Jan4(05)
t

| u (t )  [    E (t ) ] | u(t )    d  (t   ) | u ( ) 
t
0
tf
t
1 f
J I  2 Re  X | u (t f )  2 Re  d t  Y (t ) | u (t )  
d t [ E (t ) ]2

hA0
0
tf
J II
t
1 f
 u (t f ) | X | u (t f )    d t  u (t ) | Y (t ) | u (t )  
d t [ E (t ) ]2

hA0
0
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).
核融合研Jan4(05)
Part II
Relaxation effects on quantum control:
Manipulating dissociation wave packets of
I2- in water
(省略)
Nishiyama et al., J. Chem. Phys. (04)
核融合研Jan4(05)
Part III
Application to Molecular Quantum Computer
Y. Ohtsuki, Chem. Phys. Lett. in press. (2005)
Qubit and quantum parallel processing
核融合研Jan4(05)
“classical” bit and “quantum” bit
|0
|    | 0    |1 
binary number
3  1 21  1 20
| 3 2 |11 
| 3 3 | 011 
|1 
| 3 4 | 0011 
integer
computation
the number of bit basis
We freely use a string of qubits and an integer with a subscript.
Qubit and quantum parallel processing
核融合研Jan4(05)
3ビットの例
{000 001 010 011 100 101 110 111}
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
210  1024 ビットマシン(Itanium2 16CPU相当)
213  8192 ビットマシン(Itanium2 128CPU相当)
Qubit and quantum parallel processing
核融合研Jan4(05)
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
1994 Shor 因数分解アルゴリズム
1996 Grover 高速検索アルゴリズム
Qubit and quantum parallel processing
核融合研Jan4(05)
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...
Qubit and quantum parallel processing
核融合研Jan4(05)
Example of fundamental gate
H | 0 
Hadamard transform
1
2
(| 0   |1 )
U f | x n | y | x n | f ( x)  y 
protocol
parallel processing
| 1 n1 | 0 n | 0 
binary
sum
1  0  1
1  1  0 


output/
input/
control bit target bit
|  2  n 1  H n  I |  1  n 1

1
2
(| 0   |1 )  L 
|  3 n1  U f |  2 n1 
1
2
n2
1
2
n
1 2 1
(| 0   |1 ) | 0 
2n 1
 | x  n | f ( x) 
x 0
2
n2
 | x n | 0 
x 0
idea of quantum computation
核融合研Jan4(05)
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
Purpose
核融合研Jan4(05)
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.
vibrational states as qubits
核融合研Jan4(05)
example of the mapping in 2-qubit case
I2
B state
v  16  0 0 
v  17  01 
v  18  1 0 
v  19  11 
gate pulse
mapping
X state
intramolecular
states
qubits
vibrational states as qubits
核融合研Jan4(05)
gate operations acting upon the mapped qubits
Qubits are realized by
a set of independent spins.
 qubit






qubit
O






qubit 
1-qubit operator acts on
each subspace
Qubits are realized by
multilevel systems.

L

M O

L



M


1-qubit operator acts on
the whole space
vibrational states as qubits
1 qubit
2 qubits
核融合研Jan4(05)
a b 
A1(1)  

c d 
 A (1)
A1(2)   1
 0(1)
matrix representation of
1-qubit gates
 a 1(1)
0(1) 
(2)
 A2  
(1) 
 c 1(1)
A1 
3 qubits
 A (2)
A1(3)   1
 0(2)
n qubits
 A ( n 1)
A1( n)   1
 0( n 1)
0(2)  (3)  A (2)
 A2   2
 0(2)
A1(2) 
0( n 1) 

( n 1) 
A1

...
b 1(1) 

(1) 
d1 
(2)

a
1
0(2) 
(3)
 A3  
(2)


c
1
A (2)

2 
A(nn)
 a 1( n1)

 c 1( n1)
b 1(2) 

(2) 
d1 
b 1( n1) 

( n 1) 
d1

The Deutsch problem
核融合研Jan4(05)
Consider certain global properties of functions on n-bit binary numbers:
a set of binary numbers
function
Xn  (0, 1)n  {| 0 n , |1 n , L ,| 2n  1 n ,}
f : Xn  {0, 1}
balanced: the number of times the function returns 0 is equal to
the number of times the function returns 1.
constant: f : Xn  {0} or {1}
The Deutsch problem
Given a function that is either balanced or constant,
determine which type it is.
Deutsch-Jozsa algorithm (2-qubit case)
A classical algorithm requires that f be evaluated for
2n-1+1
核融合研Jan4(05)
values.
A quantum algorithm requires only one evaluation of f .
(1) Step 1: Initial preparation
|1 2 | 0 2 | 00 
(2) Step 2: Applying the Hadamard transformation to (1)
1
1
| 2 2  H 2 |1 2  (| 0   |1 )  (| 0   |1 )  n 2
2
2
2n 1
 | x n2
x 0
(3) Step 3: function evaluation via the f-controlled gate, U f
U f | x n  (1) f ( x) | x n
| 3 2  U f | 2 2
1
 [(1) f (0) | 0  2 (1) f (1) |1  2 (1) f (2) | 2  2 (1) f (3) | 3  2 ]
2
Deutsch-Jozsa algorithm
核融合研Jan4(05)
(4) Step 4: Applying the Hadamard transformation to (3)
n
n
1 2 1 2 1
|  4 2  H 2 |  3 2 
(1) f ( x )  x y | y  n 2


2n 2 x 0 y 0
binary inner product: x  y  x0 y0  x1 y1 | x 2 | x1 x0 , etc.
(5) Step 5: observation
| 0 0 |  4 2 |2 
balanced: 0
1
| (1) f (0)  (1) f (1)  (1) f (2)  (1) f (3) |2
16
constant: 1
1
1
H | x0  1 2 [ | 0  (1) x0 |1 ]  1 2  (1) x0 y0 | y0 
2
2 y0 0,1
1
1
H 2 | x1x0  1 2 [ | 0  (1) x0 |1 ]  2 2 
(1) x0 y0  x1 y1 | y1 y0 
2
2 y0 , y1 0,1
Equations of motion
quantum Liouville equation
ih
核融合研Jan4(05)
(Liouville-space notation)

 (t )  [ H   E (t ),  (t ) ]  ih(t )  (t )  [ L  ME (t )]|  (t ) 
t
system Hamiltonian H
electric dipole interaction
laser fields E (t )
dipole moment operator 
time-dependent relaxation operator (t )
Liouville-space time evolution operator
|  (t )  G (t ,0) |  (t  0)  G (t ,0) |  0 
ih

G(t ,0)  [ L  ME (t )]G(t ,0)
t
Optimal-gate-pulse design
核融合研Jan4(05)
An optimal-gate-pulse maximizes the following
objective functional for an arbitrary initial state:
t
1 f
0
0
2
J  W  | G (t f ,0) |   
d
t
|
E
(
t
)|

hA 0
gate operator
penalty due to fluence
Specifying the independent matrix elements by {j},
we introduce a modified objective functional:
J 
j
f j  W  0j | G (t f
weight factor
,0) |  0j
tf
1
2
 
d
t
|
E
(
t
)|

hA 0
Pulse design equations
核融合研Jan4(05)
Optimal laser pulse
i
E (t )  A  f j  W  0j | G (t f , t ) MG (t ,0) |  0j 
2 j
Introducing the auxiliary density {|  j (t )  }

ih
|  j (t )  [ L†  ME (t )] |  j (t ) 
t
with a final condition |  j (t f ) | W  0j 
non-Markovian effects: Y. Ohtsuki, J. Chem. Phys.(03)
Markovian dissipation: Y. Ohtsuki et al., Chem. Phys.(03)
Hadamard transform in 1-qubit case
核融合研Jan4(05)
1 qubit obtained by the mapping
I2
B state
v  16 
0
v  17  1 
gate pulse
X state
computation
・24 vibrational states
・Runge-Kutta-Fehlberg method
Hadamard transform in 1-qubit case
核融合研Jan4(05)
9
E(t) (10 V/m)
1.0
High fidelity:99.8%
0.5
0.0
A modulated structure suggests
the importance of phase control.
-0.5
-1.0
0.5
Medium fidelity:97.8%
9
E(t) (10 V/m)
1.0
0.0
-0.5
-1.0
0
500
time (fs)
1000
Hadamard transform in 1-qubit case
核融合研Jan4(05)
H1
fidelity (%)
100
H5
95
H10
The gate pulse designed with high
accuracy can achieve a correct
transform even after ten successive
operations.
100
fidelity (%)
80
The accuracy is reduced rapidly
with the increase in the number of
gate operations.
H1
60
40
20
When using a pulse that lacks
0
extreme precision, it will be safe
-1
0
1
  1 coefficient of the |0> state   1 to avoid using the pulse several
times.
|    | 0   1   2 |1 
H5
H10
Simulating the Deutsch-Jozsa algorithm
核融合研Jan4(05)
2 qubits obtained by mapping
I2
B state
v  16  0 0 
v  17  01 
v  18  1 0 
v  19  11 
numerical procedure
gate pulse
X state
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.
Simulating the Deutsch-Jozsa algorithm
核融合研Jan4(05)
examples of oracle pulses
Hadamard transform pulse
5
0
constant
100% population
in |00>
balanced
no population
in |00>
otherwise
25% population
in each states
5
8
E(t) (10 V/m)
-5
0
-5
5
0
-5
0
500
time (fs)
1000
Simulating the Deutsch-Jozsa algorithm
核融合研Jan4(05)
constant
balanced
balanced
balanced
0.3
population
population
1.0
0.5
0.2
0.1
0.0
0.0
|00>
|01>
|10>
|11>
computational basis
balanced & constant
|00>
|01>
|10>
computational basis
other cases
Universal computation can be realized by
combining optimally designed gate pulses.
|11>
CNOT pulse
time evolution of populations of computational basis
核融合研Jan4(05)
9
E(t) (10 V/m)
2
greater than 99%
0
-2
population
1.0
The CNOT gate can be realized
by using only one pulse even in
a multilevel system.
0.5
0.0
0
500
time (fs)
1000
summary
核融合研Jan4(05)
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.
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