# Introduction to Quantum Computing Introduction to Quantum Computing

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

10 9

10 8

Transistor Density

10 7

Pentium

Pro

80786

10 6

10 5

8086

80286

80386

80486

Pentium

10 4

4004

8080

10 3

1970 1975 1980 1985 1990 1995 2000 2005 2010

Year

4

Transistor Size

Electrons per device

10 4

(4M)

10 3

10 2

(16M)

(64M)

(256M)

(1G)

(4G)

(Transistors per chip)

(16G)

10 1

10 0

10 -1

1 electron/transistor

1985 1990 1995 2000 2005 2010 2015 2020

Year

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

c 0 0 + c 1

1

|c 0 | 2 +|c 1 | 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

2

Probability of 0 c 0

2

Probability of 1 c 1

c 0 + c 1

0

1

13

Quantum Computing History

14

History

• In the 1970’s 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

A bit can be read without

affecting its value

True

True

False, a qubit in an unknown

state cannot be copied without

disrupting its state

False, reading a qubit that is

initially in a superposition will

change the value of the qubit

19

Comparison II

Assumption Classical Quantum

Reading one bit has no effect

on another unread bit

True

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 n

qubit register can be in all

2 n 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…

α 0 + β 1 → α 1 + β 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 2 nd Law of Thermodynamics

30

Quantum Computation

NOT gate representation

X

we get…

⎡0 1⎤

≡ ⎢

1 0 ⎥

⎣ ⎦

for any

⎡α

α 0 + β 1 ≡ ⎢

β ⎥

⎣ ⎦

⎡α⎤ ⎡0 1⎤⎡α⎤ ⎡β⎤

X ⎢ or β 0 α 1

β

⎥ = ⎢ = +

1 0

⎥⎢

β

⎥ ⎢

α

⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

to summarize…

α 0 + β 1 → α 1 + β 0

31

Quantum Computation

Are there any constraints on what matrices may be used as

quantum gates Of course!

We require the normalization condition

α

2 2

+ β = 1 ψ = α 0 + β 1

for

and the result ψ ' = α' 0 + β'1

after the gate has

acted

The appropriate condition for this (of course) is

that the matrix representing the gate is

UNITARY

UU=

I

where U is the adjoint of

That's it!!! Anything else is a valid quantum gate.

U

32

Quantum Computation

Two more important gates…

• Z gate

Z

⎡1 0⎤

≡ ⎢

0 -1 ⎥

⎣ ⎦

leaves 0 unchanged

flips the sign of 1 to - 1

• Hadamard Gate

H

1 ⎡1 1⎤

≡ ⎢

2 1 -1

⎣ ⎦

( + )

( − )

turns 0 into 0 1 2

turns 1 into 0 1 2

Note: Applying H twice to a state does nothing to it.

H

2

=

I

33

Quantum Computation

Hadamard Gate: A most useful gate indeed!

if H =

1

( X + Z)

and ψ = α 0 + β 1 then

2

H ψ =

1

( X ψ + Z ψ )

2

=

1 ⎛⎡0 1⎤⎡α ⎤ ⎡1 0 ⎤⎡α⎤⎞ 1 ⎛⎡β⎤ ⎡α ⎤⎞ 1 ⎡α + β⎤

⎜⎢ 2 1 0

⎥⎢

β

⎥+ ⎢ ⎟= ⎜ + ⎟=

0 −1

⎥⎢

β

⎥ ⎢

2 α

⎥ ⎢

−β ⎥ ⎢

2 α −β

⎝⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦⎠ ⎝⎣ ⎦ ⎣ ⎦⎠

⎣ ⎦

for

for

H

H

1

0 α=1, β=0 H 0 = ( 0 + 1 )

2

1

1 α = 0, β = 1 H 1 = 0 − 1

2

( )

34

Quantum Computation

• Review: Important single-qubit

gates

α 0 + β 1

α 0 + β 1

α 0 + β 1

X

Z

H

β 0 + α 1

α 0 − β 1

0 + 1 1 − 1

α + β

2 2

35

Quantum Computation

• Arbitrary Single Qubit Quantum Gate

- complete set from properties of a much smaller set

U

β

δ

γ γ

−i

⎤⎡

2

cos sin ⎡ −i

2

i

e 0 ⎢

2 2⎥

α

e 0

= e

⎢ ⎥ ⎢ ⎥

⎢ β ⎥⎢ ⎥

δ

i γ γ ⎢ ⎥

i

⎢ 2 ⎢sin

cos ⎥

2

⎣ 0 e ⎥⎦ ⎢ 0 e ⎥

⎢⎣

2 2 ⎥⎦⎣ ⎦

Global

Phase

Factor

Rotation

about z

Rotation

Scaling

Constant

α, βγ , and δare

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

AB , → AB , ⊕A

38

CNOT quantum gate

⎡1

0 0 0⎤

A

A

0 1 0 0

U CN

= ⎢ ⎥

⎢ 0 0 0 1 ⎥

B

B ⊕ A

⎢ ⎥

⎣0

0 1 0⎦

⎡B0

A0 = 1

B

1

if A = 0 then we get ⎢ ⎥

⎡ ⎡B0⎤⎤ ⎡AB

A

0 0⎤

1

= 0 ⎢ 0 ⎥

⎢A0

⎢ B

⎥⎥

⎢ ⎢ ⎥

1 AB

⎣ ⎦

0

0 1

⎣ ⎦

A B →

⎢ ⎥

= ⎢ ⎥

⎢ ⎡B

⎤⎥ ⎢AB

⎥ ⎡ 0 ⎤

0

1 0

⎢A1

B

⎥⎥ AB

1

⎢ 1 1⎥

A0

= 0

0

⎢ ⎥ ⎣ ⎦

⎣ ⎣ ⎦⎦ if A = 1 then we get ⎢ ⎥

A1 = 1 ⎢B

0

⎢⎣

B1

⎥⎦

Any multiple qubit

qubit logic gate may be composed from

CNOT and Single Qubit Gates

39

Other Computational Bases

• Measurements

( 0 + 1 ) ( 0 − 1 )

- In terms of + = , − =

basis states

2 2

+ + − + − − α + β α −β

ψ = α 0 + β 1 = α + β = + + −

2 2 2 2

- Generally any basis state can represent an arbitrary

qubit state

ψ = α a + β 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

ab , → aa , ⊕b

( )

( )

→ a⊕ a⊕b , a⊕ b = b,

a⊕b

→ b, a⊕b ⊕ b = b,

a

a

b

aa ,

⊕ b

ba ,

⊕ b

ba ,

x

x

42

• Controlled-U U Gate

Quantum Circuits

- A Controlled-U 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

ψ

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

x

x

x

x

original

bit

0

y x⊕

y

x

scratch-pad

initialized to zero

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

ψ = a 0 + b 1

bit to be

copied

a

0 + b 1

Output State

a

00 + b 10

0

a

00 + b 11

scratch-pad

initialized to zero

46

Quantum Circuits

• Can we make a Qubit Copying Circuit

- Does ψ ψ = a 00 + b 11

( )( )

2 2

ψ ψ = a 0 + b 1 a 0 + b 1 = a 00 + ab 01 + ab 10 + b 11

- Unless ab = 0this does not copy the quantum state

input

2 2

a 00 + ab 01 + ab 10 + b 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

x

y

H

Out

00

β xy 01

10

11

+ ⎫

2 ⎬ → →

→ ⎪ ⎭

0 1

0 → 00 + 10 00 + 11

0

In

Out

( 00 + 11 ) 2 ≡ β00

( 01 + 10 ) 2 ≡ β01

( 00 + 11 ) 2 ≡ β00

( 00 + 11 ) 2 ≡ β00

2 2

48

Quantum Algorithms

Initial State

( )

xy , → xy , ⊕ f x

Final State

Data Register

0 + 1

2

0

x

U f

x

y y⊕

f ( x)

ψ

Target Register

00 + 10

xy , =

2

ψ

0,0 ⊕ f 0 + 1,0 ⊕ f 1 0, f 0 + 1, f 1

= =

2 2

( ) ( ) ( ) ( )

49

Quantum Algorithms

Eureka!!!! Both values of the function

show up in the final state solution.

ψ

=

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 HADAMARD TRANSFORM

50

Quantum Algorithms

• Deutsch's Algorithm Circuit

- Combines quantum parallelism and interference

0 H

0 + 1

2

x

x

H

1 H

0 − 1

2

U f

y y⊕

f ( x)

ψ 0

ψ 1

ψ 2

ψ 3

51

Quantum Algorithms

• Deutsch's Algorithm Calculations

- Combines quantum parallelism and interference

ψ

0

= 01

ψ

ψ

⎡ 0 + 1 ⎤⎡ 0 − 1 ⎤

→ ψ =⎢ ⎥⎢ ⎥

⎣ 2 ⎦⎣ 2 ⎦

0 1

⎧ ⎡ 0 + 1 ⎤⎡ 0 − 1 ⎤

⎪± ⎢ ⎥⎢ ⎥ if f 0 = f 1

⎪ ⎣ 2 ⎦⎣ 2 ⎦

→ ψ =⎨

⎪ ⎡ 0 − 1 ⎤⎡ 0 − 1 ⎤

⎪ ± ⎢ ⎥⎢ ⎥ if f 0 ≠ f 1

⎩ ⎣ 2 ⎦⎣ 2 ⎦

1 2

( ) ( )

( ) ( )

52

Quantum Algorithms

• Deutsch's Algorithm Conclusion

ψ

⎧ ⎡ 0 − 1 ⎤

⎪± 0 ⎢ ⎥ if f 0 = f 1

⎪ ⎣ 2 ⎦

→ ψ =⎨

⎪ ⎡ 0 − 1 ⎤

⎪ ± 1 ⎢ ⎥ if f 0 ≠ f 1

⎩ ⎣ 2 ⎦

2 3

( ) ( )

( ) ( )

realizing f ( 0) ⊕ f ( 1)

is 0 if f ( 0) = f ( 1)

and 1 otherwise…

ψ

3

f

0 f 1

⎡ 0 − 1 ⎤

=± ( ) ⊕ ( ) ⎢ ⎥

2

measuring the 1 st 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

x is a number

from 0 to 2 n -1

Alice

x

n bits each time

Bob

f

( x)

⎧ Constant for all values of 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 2 n /2+

1queries

n

2 /2 0 's and one 1⇒

Balanced

55

ψ 0

ψ 1

ψ 2

ψ 3

Quantum Algorithms

• 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 U f 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

n

⊗n

H

x

x

⊗n

H

U f

1 H

y y⊕

f ( x)

ψ 0

ψ 1

ψ 2

ψ 3

ψ

⊗n

x ⎡ 0 + 1 ⎤

= 0 1 → ψ = ∑ ⎢ ⎥

n

n

x∈

2 ⎣ 2 ⎦

0 1

{ 0,1}

ψ

1 2

{ 0,1}

f

( ) ( x )

−1 x ⎡ 0 − 1 ⎤

→ ψ = ∑

⎢ ⎥

n

n

x∈

2 ⎣ 2 ⎦

Bob's function

evaluation is

stored in the

amplitude

57

Hadamard transform: helps to calculate effect on a state x

By checking the cases x=0 and x=1 separately for a single qubit…

thus

Quantum Algorithms

• Deutsch-Jozsa

Algorithm - detour

H x =

( −1)

where is the bitwise inner product of x and z, modulo 2

z

xz

z

2

( ) 1 1

⊗n

H x ,..., x = ∑ −1

x•

z

xz+ ... + x z 1 n

n n

n z1

zn

n

1 ,...,

H

⊗n

x

= ∑ −

z

1

( )

x•

z

z

2

n

z

,...,

2

z

58

Quantum Algorithms

• Deutsch-Jozsa

Algorithm Circuit

ψ

2 3

- amplitude for is…

( )

−1 z ⎡ 0 − 1 ⎤

→ ψ = ∑∑

n ⎢ ⎥

z x 2 ⎣ 2 ⎦

0 n

Case 1: If f is constant the amplitude for is +1 or -1

depending on the constant value f(x) ) takes. Since

ψ

is unit length then all other amplitudes must be zero.

- An observation will yield 0s for all qubits in the register

( )

x• z+

f x

query register

f

( ) ( x )

x

−1

2

n

0 n

⊗ 3

59

Quantum Algorithms

• Deutsch-Jozsa

Algorithm Circuit

Case 2: If f is balanced then the positive and negative

contributions to the amplitude for 0 ⊗n

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

Computer

Liquid sample

12

C,

RF - coil

19

F,

15

N,

31

P

=11.8 Tesla

B (uniform to

1 part in 10 9 )

Regard as an ensemble

of n-bit quantum

computers

amplifier

Typical Experiment

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.

Spectrometer

Physical Apparatus

Nuclear Spins as qubits

ADC for data acquisition

RF synthesizer and amplifier

Gradient control

0

1

B

wave guides

sample

test tube

9.6 T

I

J IS

S

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:

H int =ω I I z +ω S S z +2π J IS I z S z

interaction with B field

9.6 T

I

J IS

S

spin-spin coupling

2-3 Dibromothiophene

67

External Hamiltonian

• Experimentally Controlled Hamiltonian:

H ext (t) =ω RFx (t)·(I

(I x +S x )+

spins couple to RF field

)+ω RFy (t)

(t)·(I(I y +S y )

• Total Hamiltonian:

H total (t) = H int + H ext (t)

H total (t

(t)

controlled via

H ext (t)

9.6 T

I J IS S

RF wave

2-3 Dibromothiophene

68

Tomography

Not all elements of the density matrix are observable on an

NMR spectra.

2

σ

x

σ

2 3

x

σ

z

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

• Classical game: Prisoner’s s Dilemma

• Maximal entangled quantum game

• Some of our results

• Theoretical extensions with non-maximal

entanglement, more players, larger

strategy space, and so on.

• Experimental realization of quantum game

• Future Plan and discussion

71

• Game theory

Prisoner's dilemma

--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 s Dilemma

--a 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 s payoff without lessening the payoff of the

other player.

• Prisoner’s s Dilemma: Nash Equilibrium strategy

profile is not equivalent to Pareto optimal

73

Maximal entangled quantum game

Quantum game theory

Recently, new effect involving quantum information on has

been discovered theoritically in the area of game theory by

some pioneers.

1. L.Goldenberg, L.Vaidman, S.Wiesner, Phys.Rev.Lett. . 82, 3356

(1999).

2. D.A.Meyer, Phys.Rev.Lett. . 82, 1052 (1999).

3. 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 γ =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

π

γ =

2

with ,

(Q,Q) is the Nash equilibrium of the game and has the

property to be Pareto optimal .

So Prisoner’s 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:

• U A and U B are the strategy moves available to the

players:

( ) = ⎜

e cos θ sin θ

U θ , φ

2 2 ⎟

• Unitary operator

J

{ i D ⊗ 2}

− sin θ

2

e

−iφ

cos θ ⎟

2 ⎠

⎛ 0

⎝−1

1⎞

0⎠

= exp γ D/

D=

Uπ ( ,0) = ⎟

^

77

• Two player’s s initial state is

( ) ( )11

ψi = J 00 = cos γ 2 00 + isin

γ 2

• The entanglement of the game's initial state can be

denoted as

γ γ γ

γ

− sin

therefore, can be denoted as a measure for the

entanglement.

• The final state is

ψ = J

+ U ⊗U

J

f

2

ln sin

− cos

2

( ) 00

• Then the expected payoff for Alice and Bob are

A

B

2

ln cos

2 2

2

2 γ

ξ

Α

= 3P00

+ 5P10

+ P11

2

ξ

B

= 3P00

+ 5P01

+ P

Pij

= ij ψ

f

11

2

78

• Nash Equilibrium:

A

⊗Uˆ

B

Theoretical Results

⎧ Dˆ

⊗ Dˆ , 0 ≤ γ ≤ γ

th1

⊗Qˆ,

γ

th1

≤ γ ≤ γ

th2

= ⎨

,

⎪Q

⊗ Dˆ , γ

th1

≤ γ ≤ γ

th2

⎩ Qˆ

⊗Qˆ,

γ

th2

≤ γ ≤ π 2

• There exist two threshold:

γ sin ( 1 5 ) γ sin ( 2 5 )

th 1

= Arc

0

= ⎜

⎝−1

th

=

2

Arc

−1⎞

⎟,

0 ⎠

⎛ i

= ⎜

⎝0

• Expected payoff as game’s s entanglement varies

0 ⎞

− i⎠

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

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

• Qubits

Two-qubit

qubit: : Nuclear Coupled Spins

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:

ˆ nπ

J = exp { iγD

⊗ D / 2} , γ = , n = {0,1,... 18}

• The strategy moves U A and U B are

A

⊗Uˆ

B

⎧ Dˆ

⊗ Dˆ ,

⊗Qˆ,

= ⎨

⎪Q

⊗ Dˆ ,

⎩ Qˆ

⊗Qˆ,

0 ≤ n ≤ 5

6 ≤ n ≤ 7

,

6 ≤ n ≤ 7

7 ≤ n ≤18

⎛ − ⎞ ⎛ i ⎞

0 1

0

= ⎜ ⎟,

= ⎜ ⎟

⎝−1

0 ⎠ ⎝0

− i⎠

• Each gate can be realized by NMR technique.

36

83

Experiments for quantum game

• Experimentally, we performed nineteen separate sets

of experiments which was distinguished by:

ˆ nπ

J = exp iγD

⊗ D / 2 , γ = , n = {0,1,... 18

{ } }

• In each set, the full process of the game was

executed.

1. Create an effective pure state

2. Prepare the initial entangled state by applying gate J

3. Players Alice and Bob executed their strategic moves U A and U B

4. Apply the unentangled gate J +

5. Measure the final state and calculate the expected payoff.

36

84

NMR Spectrometer

85

Experimental results

• The player Alice's payoffs as a function of the

parameter γ .

• It is easy to see that γ = 0 (n=0) corresponds to

Eisert et al.'s separable game and γ = π 2 (n=18)

corresponds to their maximally entangled quantum

game.

86

• 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), Physics web (IOP),

• New Scientist, Science Update (Nature).(

• Physics world

Experimental results

87

2002.4《 Nature 》Science

Update

88

2001.9: APS- Physics News Update

89

2002.1- New Scientists

90

Thanks

91

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