Introduction to Quantum Computing

**Introduction** **to** **Quantum**

**Computing**

Lecture 1

1

OUTLINE

• Why **Quantum** **Computing**

• What is **Quantum** **Computing**

• His**to**ry

• **Quantum** Weirdness

• **Quantum** Properties

• **Quantum** Computation

2

Why **Quantum** **Computing**

3

Transis**to**rs per chip

10 9

10 8

Transis**to**r 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

Transis**to**r Size

Electrons per device

10 4

(4M)

10 3

10 2

(16M)

(64M)

(256M)

(1G)

(4G)

(Transis**to**rs per chip)

(16G)

10 1

10 0

10 -1

1 electron/transis**to**r

1985 1990 1995 2000 2005 2010 2015 2020

Year

5

Why **Quantum** **Computing**

• By 2020 we will hit natural limits on the size

of transis**to**rs

• Max out on the number of transis**to**rs per chip

• Reach the minimum size for transis**to**rs

• Reach the limit of speed for devices

• Eventually, all computing will be done using

some sort of alternative structure

• DNA

• Cellular Au**to**ma**to**n

• **Quantum**

6

What is **Quantum** **Computing**

7

**Introduction**

• The common characteristic of any digital

computer is that it s**to**res 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 pho**to**n

• The energy level of an excited a**to**m

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** His**to**ry

14

His**to**ry

• In the 1970’s s Fredkin, Toffoli, Bennett and others

began **to** look in**to** 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 in**to** 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 cryp**to**logy

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 in**to** 0 1 2

turns 1 in**to** 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

Fac**to**r

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

* pho**to**n 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 in**to** 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 in**to** 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

s**to**red 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 - de**to**ur

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 Fac**to**ring

~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 A**to**m Traps

62

NMR **Quantum** **Computing**

Lecture 2

63

Nuclear Magnetic Resonance

**Quantum** Computers

Qubit representation: spin of an a**to**mic nucleus

Unitary evolution: using magnetic field pulses

applied **to** spins in a strong magnetic

field.

Chemical bonds between a**to**ms 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

in**to** 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 Hamil**to**nian

• The evolution of a spin system is

generated by Hamil**to**nians

• Internal Hamil**to**nian:

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 Hamil**to**nian

• Experimentally Controlled Hamil**to**nian:

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 Hamil**to**nian:

H **to**tal (t) = H int + H ext (t)

H **to**tal (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.

• Pare**to** optimal: mutual cooperation (C,C)

• A pair of strategies is called pare**to** 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** Pare**to** 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 Pare**to**

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 Pare**to** 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:

⎛

iφ

⎞

( ) = ⎜

e cos θ sin θ

U θ , φ

2 2 ⎟

• Unitary opera**to**r

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:

Uˆ

A

⊗Uˆ

B

Theoretical Results

⎧ Dˆ

⊗ Dˆ , 0 ≤ γ ≤ γ

th1

⎪

Dˆ

⊗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

⎛

Dˆ

0

= ⎜

⎝−1

th

=

2

Arc

−1⎞

⎟,

0 ⎠

⎛ i

Qˆ

= ⎜

⎝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 cy**to**sine

molecule contains two pro**to**ns, in a

magnetic field, each spin state of

pro**to**n 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

Uˆ

A

⊗Uˆ

B

⎧ Dˆ

⊗ Dˆ ,

⎪

Dˆ

⊗Qˆ,

= ⎨

⎪Q

⊗ Dˆ ,

⎪

⎩ Qˆ

⊗Qˆ,

0 ≤ n ≤ 5

6 ≤ n ≤ 7

,

6 ≤ n ≤ 7

7 ≤ n ≤18

⎛ − ⎞ ⎛ i ⎞

Dˆ

0 1

0

= ⎜ ⎟,

Qˆ

= ⎜ ⎟

⎝−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 **to**ok 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