Introduction to Quantum
Computing
Lecture 1
1
OUTLINE

Why Quantum Computing?

What is Quantum Computing?

History

Quantum Weirdness

Quantum Properties

Quantum Computation
2
Why Quantum Computing?
3
Transistors per chip
Transistor Density
109
?
108
Pentium 80786
Pro
107
80486
106
Pentium
80386
80286
105
8086
8080
104
4004
103
1970
1975
1980
1985
1990
Year
1995
2000
2005
2010
4
Transistor Size
Electrons per device
104
(4M)
(16M)
103
(64M)
(256M)
(Transistors per chip)
(1G)
102
(4G)
(16G)
101
?
100
1 electron/transistor
10-1
1985
1990
1995
2000
Year
2005
2010
2015
2020
5
Why Quantum Computing?

By 2020 we will hit natural limits on the size
of transistors




Max out on the number of transistors per chip
Reach the minimum size for transistors
Reach the limit of speed for devices
Eventually, all computing will be done using
some sort of alternative structure



DNA
Cellular Automaton
Quantum
6
What is Quantum Computing?
7
Introduction

The common characteristic of any digital
computer is that it stores bits



Bits represent the state of some physical system
Electronic computers use voltage levels to represent
bits
Quantum systems possess properties that allow
the encoding of bits as physical states



Direction of spin of an electron
The direction of polarization of a photon
The energy level of an excited atom
8
Spin States

An electron is always in one of two spin states



“spin up” – the spin is parallel to the particle axis
“spin down” – the spin is antiparallel to the particle
axis
Notation:
Spin up:
Spin down:
9
qubit


A qubit is a bit represented by a
quantum system
By convention:


A qubit state 0 is the spin up state
A qubit state 1 is the spin down state
0
1
10
Definitions

A qubit is governed by the laws of
quantum physics


While a quantum system can be in one of
a discrete set of states, it can also be in a
blend of states called a superposition
That is a qubit can be in:
0
1
c0 0 + c1 1
|c0|2+|c1|2 = 1
11
Measurement

If a qubit is realized by the spin of an
electron, it is possible to measure the
qubit value by passing the electron
through a magnetic field


If the qubit encodes a |0> then it will be
deflected upward
If the qubit encodes a |1> then it will be
deflected downward
12
Superposition Measurement


If the qubit is in a superposition state it
cannot be determine if it will deflect up or
down
However, the probability of each possible
deflection can be found
Probability of 0
c0
2
Probability of 1
c1
2
c0 0 + c1 1
13
Quantum Computing History
14
History

In the 1970’s Fredkin, Toffoli, Bennett and others
began to look into the possibility of reversible
computation to avoid power loss.


Since quantum mechanics is reversible, a possible link
between computing and quantum devices was suggested
Some early work on quantum computation occurred
in the 80’s



Benioff 1980,1982 explored a connection between quantum
systems and a Turing machine
Feynman 1982, 1986 suggested that quantum systems could
simulate reversible digital circuits
Deutsch 1985 defined a quantum level XOR mechanism
15
Existing Quantum Computers




liquid NMR quantum computers with 2 –
12 qubit registers.
Ion Trap method have achieved a single
CONTROLLED NOT and 4 qubit entangled
states
linear optics,
Superconductive Device…
16
Quantum Weirdness
17
Weird Measurement

One of the unusual features of
Quantum Mechanics is the interaction
between an event and its
measurement


Measurement changes the state of a
quantum system
Measurement of the superposition state
of a qubit forces it into one of the qubit
states in an unpredictable manner
18
Comparison I

Compare qubits to classical bits
Assumption
Classical
Quantum
A bit always has a
definite value
True
False, a qubit need not have a
definite value until the moment
after it is observed
A bit can only be 0 or 1
True
False, a qubit can be in a
superposition of 0 and 1
simultaneously
A bit can be copied without
affecting its value
True
False, a qubit in an unknown
state cannot be copied without
disrupting its state
A bit can be read without
affecting its value
True
False, reading a qubit that is
initially in a superposition will
change the value of the qubit
19
Comparison II
Assumption
Reading one bit has no effect
on another unread bit
Classical
True
Quantum
False, if the qubit being read is
entangled with another qubit
reading one will affect the other
20
Quantum Phenomena
21
Quantum Phenomena

There are five quantum phenomena
that make quantum computing weird





Superposition
Interference
Entanglement
Non-determinism
Non-clonability
22
Superposition


The Principal of Superposition states if a
quantum system can be measured to be in
one of a number of states then it can also
exist in a blend of all its states
simultaneously
RESULT: An n-bit qubit register can be in all
2n states at once

Massively parallel operations
23
Interference


We see interference patterns when light
shines through multiple slits
This is a quantum
phenomena which is
also present in quantum
computers

A quantum computer
can operate on several
inputs at once, the results
interfere with each other
producing a collective
result
24
Entanglement


If two or more qubits are made to interact,
they can emerge from the interaction in a joint
quantum state which is different from any
combination of the individual quantum states
RESULT: If two entangled qubits are separated
by any distance and one of them is measured
then the other, at the same instant, enters a
predictable state
25
Non-Determinism


Quantum non-determinism refers to the
condition of unpredictability
If a quantum system is in a superposition
state and then measured, the measured
state can not be predicted.
26
Non-Clonability


It is impossible to copy an unknown
quantum state exactly
If you asked a friend to prepare a qubit in a
superposition state without telling you
which superposition state, then you could
not make a perfect copy of the qubit

Useful in quantum cryptology
27
Quantum Computation
28
Quantum Computation
Changes to a quantum state can be described using the
language of quantum computation

Single Qubit Gates
Classical Not Gate - Truth table
0  1 and 1  0
Quantum Not Gate - Truth table
0  1 and 1  0
29
Quantum Computation
Superposition of states?
Not without further knowledge of the properties of
quantum gates
The quantum NOT gate acts LINEARLY…
 0   1  1   0
Linear behaviour is a general property of quantum
mechanics
Non-linear behaviour can lead to apparent paradoxes
- Time Travel
- Faster than light communication
- Violates the 2nd Law of Thermodynamics
30
Quantum Computation
NOT gate representation
0 1 
X

1 0 
for any
 
 0  1  
 
we get…
  0 1     
X   
   or  0   1



   1 0     
to summarize…
 0   1  1   0
31
Quantum Computation
Are there any constraints on what matrices may be used as
quantum gates? Of course!
We require the normalization condition
    1 for    0   1
2
2
and the result  '   ' 0   ' 1 after the gate has
acted
The appropriate condition for this (of course) is
that the matrix representing the gate is
UNITARY
U †U =I
where U † is the adjoint of U
That's it!!! Anything else is a valid quantum gate.
32
Quantum Computation
Two more important gates…
 Z gate
1 0 
Z

0
-1



leaves 0 unchanged
flips the sign of 1 to - 1
Hadamard Gate
1 1 1 
H
1 -1
2

0 1
into  0  1 
turns 0 into
2
turns 1
2
Note: Applying H twice to a state does nothing to it.
H I
2
33
Quantum Computation
Hadamard Gate: A most useful gate indeed!
1
if H 
 X  Z  and    0   1 then
2
1
H 
X  Z  
2
1   0 1    1 0     1         1    




     








2  1 0     0 1    
2        
2    
for H 0
for H 1
1
 =1, =0 H 0 
0 1
2
1
  0,   1 H 1 
0 1
2

34
Quantum Computation

Review: Important single-qubit gates
 0  1
X
 0  1
 0  1
Z
 0  1
 0  1
H

0 1
2

1 1
2
35

Quantum Computation
Arbitrary Single Qubit Quantum Gate
- complete set from properties of a much smaller set

 i 2
i  e
U e 
 0
Global
Phase
Factor


 cos
0 
2

 

i
2   sin
e 

2
Rotation
about z
 

 sin 
2
e
2 

 
cos
 0
2 
Rotation
i

0
 
i
e 2 
Scaling
Constant
 ,  ,  and  are all real valued
36
Quantum Computation

Classical Universal Gates (example)
- The NAND gate is a classical Universal Gate. Why?
NOT gate using NAND

AND gate using NAND
OR gate using NAND
Universal Quantum Gates
- An arbitrary quantum Computation on n qubits can be
generated by a finite set of gates that are UNIVERSAL
for quantum computation
* Need to introduce some multiple quibit quantum gates
37
Multiple Qubit Gates
Controlled-NOT (CNOT) Gate

-
two input qubits: control and target
A
A
B
B A
if control is 0 target left alone 00  00 or 01  01
else control is 1 target qubit is flipped 10  11 or 11  10
- In General
A, B  A, B  A
38
CNOT quantum gate
A
A
U CN
B
  B0    A0 B0 
 A0    

B
A
B

1

   0 1
A B 
  B    A1 B0 
 A1  0   

A
B

B
  1    1 1 
B A
1
0

0

0
0
1
0
0
0
0
0
1
0
0 
1

0
 B0 
B 
A0  1
if A  0 then
we get  1 
0
A1  0
 
0
0
0
A0  0
if A  1 then
we get  
 B0 
A1  1
 
 B1 
Any multiple qubit logic gate may be composed from
CNOT and Single Qubit Gates
39

Other Computational Bases
Measurements
- In terms of 
0


  0   1 
1
2
  
2
,
0



1
basis states
2
 
2


2
 
 
2

- Generally any basis state can represent an arbitrary
qubit state
  a   b
- If orthonormal then we can perform a measurement in
keeping with probability interpretation
40
Quantum Circuits

Elements of a Quantum Circuit
- each line in a circuit represents a "wire"
-
* passage of time
* photon moving from one location to another
assume the state input is a computational basis state
input is usually the state consisting of all 0 s
no loops allowed ie: acyclic
No FANIN(not reversible therefore not Unitary)
FANOUT (can't copy a qubit)
41
Quantum Circuits

Quantum Qubit Swap Circuit
a, b  a, a  b
 a   a  b  , a  b  b, a  b
 b,  a  b   b  b , a
a
a, a  b
b, a  b
b, a
b
x
x
42
Quantum Circuits

Controlled-U Gate
- A Controlled-U Gate has one control qubit and n target
qubits
- where U is any unitary matrix acting on n qubits
U
43
Quantum Circuits

Measurement Operation
- Converts a single qubit state into a probabilistic
classical bit M

M
44
Quantum Circuits

Can we make a Qubit Copying Circuit?
- Copying a classical bit can be done with the
Classical CNOT gate
bit to be
copied
scratch-pad
initialized to zero
original
bit
x
x
x
x
0
y
x y
x
copied
bit
45
Quantum Circuits

Can we make a Qubit Copying Circuit?
- How about copying a qubit in an unknown state using
a controlled-CNOT gate?  a 0  b 1
bit to be
copied
Output State
a 0 b 1
a 00  b 11
a 00  b 10
0
scratch-pad
initialized to zero
46
Quantum Circuits

Can we make a Qubit Copying Circuit?
- Does    a 00  b 11 ?
   a 0  b 1
 a 0
 b 1   a 2 00  ab 01  ab 10  b 2 11
- Unless ab  0this does not copy the quantum state
input 2
a 00  ab 01  ab 10  b2 11  a 00  b 11
- It is impossible to make a copy of the unknown
quantum state
- NO CLONING THEOREM 47
Quantum Circuits

Bell States, EPR States, EPR Pairs
In
x
H
00
 xy
y
01
10
11
Out
 00  11 
 01  10 
 00  11 
 00  11 
2   00
2   01
2   00
2   00
0 1
0 
00  10
00  11


2 
2
2

0 

48
Quantum Algorithms
x, y  x, y  f  x 
Initial State
0 1
x
2
Data Register
0
x, y 
 
y
Uf
Final State
x

Target Register
y  f  x
00  10
2
0,0  f  0  1,0  f 1
2

0, f  0   1, f 1
2
49
Quantum Algorithms
Eureka!!!! Both values of the function
show up in the final state solution.
 
0, f  0  1, f 1
2
This can be generalized to functions on
arbitrary number of bits using the…
HADAMARD TRANSFORM
or
WALSH-HADAMARD TRANSFORM
50

Quantum Algorithms
Deutsch's Algorithm Circuit
- Combines quantum parallelism and interference
0 1
H
0
2
x
0 1
H
1
2
y
Uf
x
H
y  f  x




0
1
2
3
51

Quantum Algorithms
Deutsch's Algorithm Calculations
- Combines quantum parallelism and interference
 0  01
 0  1  0  1 
 0  1  


2 
2 

  0  1  0  1
 

2 
2
 
1   2  
  0  1  0  1

 
2 
2
 

 if f  0   f 1


 if f  0   f 1

52

Quantum Algorithms
Deutsch's Algorithm Conclusion
2  3

 0



 1

0


0


1
 if f  0   f 1
2 
1
 if f  0   f 1
2 
realizing f  0  f 1 is 0 if f  0  f 1 and 1 otherwise…
3
0 1
  f  0   f 1 

2 

measuring the 1st qubit gives f  0  f 1
53

Quantum Algorithms
Deutsch's Algorithm Results
- The quantum circuit has given us the ability to
determine a GLOBAL PROPERTY of f  x  namely f  0  f 1
using only ONE evaluation of f  x 
- A classical computer would require at least two
evaluations!
- Difference between quantum parallelism and classical
randomized algorithms
* One might think the state 0 f  0   1 f 1 corresponds to
probabilistic classical computer that evaluates f  0 with probability 1/2
or f 1with probability ½. These are classically mutually exclusive.
* Quantum mechanically these two alternatives can INTERFERE to
yield some global property of the function f and by using a Hadamard gate
can recombine the different alternatives
54
Quantum Algorithms

Deutsch-Jozsa Algorithm
- A simple case of a more general algorithm
- Application is called Deutsch's Problem
n bits each time
x is a number
from 0 to 2n-1
x
Alice
Bob
Constant for all values of x

f  x 
Balanced: 1 for 1/ 2 the values of x or 0 otherwise
- Classically Alice can only send one value of x each
time
- Best classical algorithm requires up to 2n / 2  1 queries
2n / 2 0's and one 1  Balanced
55
Quantum Algorithms

0
1
2
3
Deutsch-Jozsa Algorithm
- If Bob and Alice were able to exchange qubits instead
of classical bits and if Bob calculated f(x) using a unitary
transform Uf then Alice could determine the function in
one query.
- Alice has an n qubit register and a single qubit register
which she gives to Bob
- Prepares query and answer register in a superposition
state
- Bob evaluates f(x) and puts result into answer register
- Alice interferes the states in the superposition using a
hadamard transform on the query register
56

Quantum Algorithms
Deutsch-Jozsa Algorithm Circuit
0
1
n

H n
x
H
y
0  0
Uf

y  f  x
1
0
n
1  1 
1   2 

H n
x
x0,1
 1
x0,1
n

f  x
n


2
3
x 0 1


n
2
2 

x 0 1


n
2
2


Bob's function
evaluation is
stored in the
amplitude
57
Quantum Algorithms

Deutsch-Jozsa Algorithm - detour
Hadamard transform: helps to calculate effect on a state x
By checking the cases x=0 and x=1 separately for a single qubit…
H x   z  1
thus
H
n
z
2
x1 ,..., xn   z ,..., z  1
1
H
where
xz
n
x1z1 ... xn zn
2n
n
x   z  1
x z
z1 ,..., zn
z
2
n
x  z is the bitwise inner product of x and z, modulo 2
58

Quantum Algorithms
Deutsch-Jozsa Algorithm Circuit
 2   3  
z
 1
x z  f  x 
2n
x
z 0 1


2 

query register
- amplitude for 0
n
is…

x
 1
f  x
2n
n
Case 1: If f is constant the amplitude for 0
is +1 or -1
depending on the constant value f(x) takes. Since  3
is unit length then all other amplitudes must be zero.
- An observation will yield 0s for all qubits in the register
59

Quantum Algorithms
Deutsch-Jozsa Algorithm Circuit
Case 2: If f is balanced then the positive and negative
n
contributions to the amplitude for 0 cancel, leaving an
amplitude of 0
- A measurement must yield a result other than 0 on at
least one qubit
Summary:
- If Alice measures all zeros then the function is constant
- Otherwise the function is balanced.
- Deutsch's problem on a quantum computer can be
solved in one evaluation.
60

Quantum Algorithms
Other Quantum Algorithms
- Generally there are three classes
* Discrete Fourier Transform Algorithms
~Deutsch-Jozsa Algorithm
~Shor's Algorithm for Factoring
~Shor's Discrete Logarithm Algorithm
* Quantum Search Algorithms
* Quantum Simulation Algorithms
~Quantum Computer is used to
simulate quantum systems
61
Experimental Quantum Information
Processing





The Stern-Gerlach Experiment
Optical Techniques
Nuclear Magnetic Resonance
Quantum Dots
Traps: Ion Traps & Neutral Atom Traps
62
NMR Quantum Computing
Lecture 2
63
Nuclear Magnetic Resonance
Quantum Computers
Qubit representation: spin of an atomic nucleus
Unitary evolution: using magnetic field pulses
applied to spins in a strong magnetic
field.
Chemical bonds
between atoms couple the spins
State preparation: using a strong magnetic field to
polarize the spins
Readout: using magnetic-moment induced free
induction decay signals
64
Nuclear Magnetic Resonance Q.C.
Physical Apparatus
pre- amplifier
RF - source
RF - coil
amplifier

B  11.8 T esla Typical Experiment
(uniform to
1 part in 109)
Liquid sample
12
C, 19 F , 15 N , 31P
Computer
Regard as an ensemble
of n-bit quantum
computers
Wait a few minutes
for the sample to come
to thermal equilibrium
2. Send RF pulses to
manipulate nuclear spins
into desired state.
3. Switch off the amps
and switch on the preamplifier to measure
the free-induction decay
65
Nuclear Magnetic Resonance Q.C.
Physical Apparatus
Spectrometer
Nuclear Spins as qubits
ADC for data acquisition
RF synthesizer and amplifier
Gradient control
wave guides
sample
test tube
0
1B
I
JIS
S
9.6 T
RF Wave
RF wave
High field magnet
2-3 Dibromothiophene
66
Internal Hamiltonian

The evolution of a spin system is
generated by Hamiltonians

Internal Hamiltonian:
Hint=wIIz+wSSz+2p JISIzSz
I
JIS
S
9.6 T
interaction with B field
spin-spin coupling
2-3 Dibromothiophene
67
External Hamiltonian

Experimentally Controlled Hamiltonian:
Hext(t) =wRFx(t)·(Ix+Sx)+wRFy(t)·(Iy+Sy)
spins couple to RF field

Total Hamiltonian:
I
JIS
S
9.6 T
Htotal (t) = Hint + Hext(t)
Htotal(t)
controlled via
Hext(t)
RF wave
2-3 Dibromothiophene
68
Tomography
Not all elements of the density matrix are observable on an
NMR spectra.
 x2
 x2 z3
To observe the other elements of the density matrix
requires repeating the experiment 7 times with
readout pulses appended to the pulse program.
This is done without changing any other parameters
of the pulse program.
69
One Example of NMR QC:
Quantum Games:
theoretical and experimental results
70
Outline

Introduction of quantum games



Some of our results



Classical game: Prisoner’s Dilemma
Maximal entangled quantum game
Theoretical extensions with non-maximal
entanglement, more players, larger
strategy space, and so on.
Experimental realization of quantum game
Future Plan and discussion
71
Prisoner's dilemma

Game theory
--an important branch of applied mathematics. It is the
theory of decision-making and conflict between different agents.
Since the seminal book of Von Neumann and Morgenstern,
modern game theory has found applications ranging from
economics through to biology.

It concludes: Players, Strategy space, Payoff function
Classifications: Time (static & Dynamic).
Information (complete &incomplete)

Prisoner’s Dilemma
--a famous game in game theory.
72
• Table: Payoff matrix for the Prisoner's Dilemma. The first entry in the
parenthesis denotes the payoff of Alice and the second to Bob's.
Alice





C
D
C
(3,3)
(5,0)
Bob
D
(0,5)
(1,1)
Nash Equilibrium: mutual defect (D,D)
Nash Equilibrium implies that no player can increase his payoff by
unilaterally changing his strategy.
Pareto optimal: mutual cooperation (C,C)
A pair of strategies is called pareto optimal if it is not possible to
increase one player’s payoff without lessening the payoff of the
other player.
Prisoner’s Dilemma: Nash Equilibrium strategy
profile is not equivalent to Pareto optimal
73
Maximal entangled quantum game

1.
2.
3.

Quantum game theory
Recently, new effect involving quantum information has
been discovered theoritically in the area of game theory by
some pioneers.
L.Goldenberg, L.Vaidman, S.Wiesner, Phys.Rev.Lett. 82, 3356
(1999).
D.A.Meyer, Phys.Rev.Lett. 82, 1052 (1999).
J.Eisert, M.Wilkens, M.Lewenstein, Phys.Rev.Lett. 83, 3077
(1999).
Maximal entangled quantum game
Eisert et al. showed that the classical problem of
Prisoner's Dilemma is a subset of the quantum game by
using a physical model of the quantum game, and there is no
longer a dilemma when employ a maximally entangled game.
74
Interesting results
For a separable game with   0 ,
there exists a pair of quantum strategies (D,D) is a
Nash Equilibrium and yields payoff (1,1) which is not Pareto
optimal. Indeed, this quantum game behaves “classically”.
For a maximally entangled quantum game   p
with
,
2
(Q,Q) is the Nash equilibrium of the game and has the
property to be Pareto optimal .
So Prisoner’s Dilemma is removed
if quantum strategies are allowed for.
75
Correlation between quantum game
and quantum entanglement

Yet it is legitimate for us to ask:
--Whether a quantum game will still
outperform its classical version if it is not
maximally entangled? and how a quantum
game depends on the entanglement of the
game's state?
76
A physical model of quantum game



J. Eisert have proposed a physical model of this game
and the elegant quantum network is illustrated as:
UA and UB are the strategy moves available to the
 ei cos

players:
sin 


2
2
U  ,   
i


 
Unitary operator
J  expiD  D/ 2

 sin
2
e
cos
2
 0 1

D  U p ,0  
 1 0 
^
77


Two player’s initial state is
 i  J 00  cos 2 00  i sin 2 11
The entanglement of the game's initial state can be
denoted as
2 
2 
2 
2 
 sin


2
ln sin
2
 cos
2
ln cos
2
therefore,  can be denoted as a measure for the
entanglement.
The final state is
 f  J  U A  U B J 00
Then the expected payoff for Alice and Bob are
   3P00  5P10  P11
2
 B  3P00  5P01  P11
Pij  ij  f
78
Theoretical Results

Nash Equilibrium:
 Dˆ  Dˆ , 0     th1
ˆ ˆ
 0  1 ˆ  i 0 
 D  Q,  th1     th2
ˆ
ˆ
ˆ
, Q  

U A U B  
, D  
ˆ
 1 0 
0  i
 Q  D,  th1     th2
 Qˆ  Qˆ ,  th2    p 2


There exist two threshold:

 th1  Arc sin 1 5



 th2  Arc sin 2 5

Expected payoff as game’s entanglement varies
79
Other Theoretical Results
•
Differnet sets of strategies.
J. Du et al., Physics Letter A, 289 (2001) 9
•
Multi players more than 2-player.
J. Du et al., Physics Letter A, 302 (2002) 229
•
Phase-transition-like behavior of quantum games
J. Du et al., Journal of Physics A: Mathematical and General 36,
6551-6562 (2003) .
•
One Review
J. Du et al., Fluctuation and Noise Letters Vol 2, Iss 4, R189-R203.
•
Quantum games in econophysics
H. Li, J. Du and S. Massar, Physics Letter A, 306 (2002) 73
J. Du et al., Physics Review E 68, 016124 (2003)
80
Experimental realization
Physics Review Letter 88, 137902（2002）

Technologies for quantum information processing(QIP)
-There are a number of proposed device technologies for QIP.
--Of them, NMR have given the many successful results
experimentally for QIP, such as quantum teleportation, quantum
error correction, quantum simulation, quantum algorithm etc.

We add game theory to the list: Quantum games
was experimental realized on nuclear magnetic
resonance quantum computer.
81
Two-qubit: Nuclear Coupled Spins

Qubits
Partially deuterated cytosine
molecule contains two protons, in a
magnetic field, each spin state of
proton could be used as a qubit.

Distinguish each qubit
Different Larmor frequencies (the
chemical shift) enable us to address
each qubit individually.

Quantum logic gates
Radio Frequency (RF) fields and
spin--spin couplings between the
nuclei are used to implement
quantum logic gates.
82
Quantum network and gates



Quantum network
Entangled gate:
Jˆ  exp iD  D / 2, 

np
, n  {0,1,...18}
36
The strategy moves UA and UB are
 Dˆ  Dˆ , 0  n  5
ˆ ˆ
 0  1 ˆ  i 0 
 D  Q, 6  n  7
, Q  

Uˆ A  Uˆ B  
, Dˆ  
ˆ
6

n

7

1
0
0

i




 Q  D,
 Qˆ  Qˆ , 7  n  18


Each gate can be realized by NMR technique.
83
Experiments for quantum game

Experimentally, we performed nineteen separate sets
of experiments which was distinguished by:
Jˆ  exp iD  D / 2,

1.
2.
3.
4.
5.
 
np
, n  {0,1,...18}
36
In each set, the full process of the game was
executed.
Create an effective pure state
Prepare the initial entangled state by applying gate J
Players Alice and Bob executed their strategic moves UA and UB
Apply the unentangled gate J+
Measure the final state and calculate the expected payoff.
84
NMR Spectrometer
85
Experimental results


The player Alice's payoffs as a function of the
parameter  .
It is easy to see that   0 (n=0) corresponds to
Eisert et al.'s separable game and   p 2 (n=18)
corresponds to their maximally entangled quantum
game.
86
Experimental results


Good agreement between theory and experiment.
Experimental Error:
--an estimated error is less than 0.08, the errors are primarily
due to inhomogeneity of magnetic field, imperfect RF selective
pulses, and the variability over time of the mesurement process.

Decoherence:
--each experiment took less than 300 milliseconds, which was
well within the the decoherence time (3 seconds).

This experiment was referred by :



Physics News update (APS),
New Scientist,
Physics world
Physics web (IOP),
Science Update (Nature).
87
2002.4《 Nature 》Science
Update
88
2001.9: APS- Physics News Update
89
2002.1－ New Scientists
90
Thanks
91