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.
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• 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
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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.
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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.
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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
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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 operator
J
{ i D ⊗ 2}
⎜
⎝
− sin θ
2
e
−iφ
cos θ ⎟
2 ⎠
⎛ 0
⎜
⎝−1
1⎞
0⎠
= exp γ D/
D=
Uπ ( ,0) = ⎟
^
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• 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
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• 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⎠
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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)
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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.
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• 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.
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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
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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.
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• 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
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2002.4《 Nature 》Science
Update
88

2001.9: APS- Physics News Update
89

2002.1- New Scientists
90

Thanks
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