introduction to quantum computing

11

Introduction to Quantum Introduction to Quantum ComputingComputing

Lecture 1Lecture 1

22

OUTLINEOUTLINE Why Quantum Computing?Why Quantum Computing?

What is Quantum Computing?What is Quantum Computing?

HistoryHistory

Quantum WeirdnessQuantum Weirdness

Quantum Properties Quantum Properties

Quantum ComputationQuantum Computation

33

Why Quantum Computing?

44

Transistor DensityTransistor Density

1970 1975 1980 1985 1990 1995 2000 2005 2010103

104

105

106

107

108

109Transistors per chip

Year

80786PentiumPro

Pentium80486

8038680286

8086

80804004

?

55

Transistor SizeTransistor Size

1985 1990 1995 2000 2010 2015 202010-1

100

101

102

103

104Electrons per device

2005Year

(16M)

(Transistors per chip)

(4M)

(256M)(1G)

(4G)

(16G)

(64M)

??1 electron/transistor

66

Why Quantum Computing?Why Quantum Computing?

By 2020 we will hit natural limits on the size of By 2020 we will hit natural limits on the size of transistorstransistors Max out on the number of transistors per chipMax out on the number of transistors per chip Reach the minimum size for transistors Reach the minimum size for transistors Reach the limit of speed for devicesReach the limit of speed for devices

Eventually, all computing will be done using Eventually, all computing will be done using some sort of alternative structuresome sort of alternative structure DNADNA Cellular AutomatonCellular Automaton QuantumQuantum

77

What is Quantum Computing?

88

IntroductionIntroduction

The common characteristic of any digital The common characteristic of any digital computer is that it stores bits computer is that it stores bits Bits represent the state of some physical systemBits represent the state of some physical system Electronic computers use voltage levels to Electronic computers use voltage levels to

represent bits represent bits

Quantum systems possess properties that Quantum systems possess properties that allow the encoding of bits as physical statesallow the encoding of bits as physical states Direction of spin of an electronDirection of spin of an electron The direction of polarization of a photonThe direction of polarization of a photon The energy level of an excited atomThe energy level of an excited atom

99

Spin StatesSpin States

An electron is always in one of two spin An electron is always in one of two spin statesstates ““spin up” – the spin is parallel to the particle spin up” – the spin is parallel to the particle

axisaxis ““spin down” – the spin is antiparallel to the spin down” – the spin is antiparallel to the

particle axisparticle axis

Notation:Notation: Spin up:

Spin down:

1010

qubitqubit

A A qubitqubit is a bit represented by a is a bit represented by a quantum systemquantum system

By convention:By convention: A qubit state 0 is the spin up stateA qubit state 0 is the spin up state A qubit state 1 is the spin down stateA qubit state 1 is the spin down state

01

1111

DefinitionsDefinitions

A qubit is governed by the laws of A qubit is governed by the laws of quantum physicsquantum physics While a quantum system can be in one While a quantum system can be in one

of a discrete set of states, it can also be of a discrete set of states, it can also be in a blend of states called a in a blend of states called a superpositionsuperposition

That is a qubit can be in:That is a qubit can be in:

01

10c0 + c1|c0|2+|c1|2 = 1

1212

MeasurementMeasurement

If a qubit is realized by the spin of If a qubit is realized by the spin of an electron, it is possible to measure an electron, it is possible to measure the qubit value by passing the the qubit value by passing the electron through a magnetic fieldelectron through a magnetic field If the qubit encodes a |0> then it will If the qubit encodes a |0> then it will

be deflected upwardbe deflected upward If the qubit encodes a |1> then it will If the qubit encodes a |1> then it will

be deflected downwardbe deflected downward

1313

Superposition MeasurementSuperposition Measurement

If the qubit is in a superposition state If the qubit is in a superposition state it cannot be determine if it will it cannot be determine if it will deflect up or downdeflect up or down

However, the probability of each However, the probability of each possible deflection can be foundpossible deflection can be found

10c0 + c1

Probability of 0 c02

Probability of 1 c12

1414

Quantum Computing History

1515

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

began to look into the possibility of reversible began to look into the possibility of reversible computation to avoid power loss.computation to avoid power loss. Since quantum mechanics is reversible, a possible link Since quantum mechanics is reversible, a possible link

between computing and quantum devices was between computing and quantum devices was suggestedsuggested

Some early work on quantum computation Some early work on quantum computation occurred in the 80’soccurred in the 80’s Benioff 1980,1982 explored a connection between Benioff 1980,1982 explored a connection between

quantum systems and a Turing machinequantum systems and a Turing machine Feynman 1982, 1986 suggested that quantum systems Feynman 1982, 1986 suggested that quantum systems

could simulate reversible digital circuitscould simulate reversible digital circuits Deutsch 1985 defined a quantum level XOR mechanismDeutsch 1985 defined a quantum level XOR mechanism

1616

Existing Quantum ComputersExisting Quantum Computers

liquid NMR quantum computers with liquid NMR quantum computers with 2 – 12 qubit registers.2 – 12 qubit registers.

Ion Trap method have achieved a Ion Trap method have achieved a single CONTROLLED NOT and 4 qubit single CONTROLLED NOT and 4 qubit entangled statesentangled states

linear optics, linear optics, Superconductive Device… Superconductive Device…

1717

Quantum Weirdness

1818

Weird Measurement Weird Measurement

One of the unusual features of One of the unusual features of Quantum Mechanics is the interaction Quantum Mechanics is the interaction between an event and its between an event and its measurementmeasurement Measurement changes the state of a Measurement changes the state of a

quantum systemquantum system Measurement of the superposition state Measurement of the superposition state

of a qubit forces it into one of the qubit of a qubit forces it into one of the qubit states in an unpredictable mannerstates in an unpredictable manner

1919

Comparison IComparison I Compare qubits to classical bitsCompare qubits to classical bits

A bit always has adefinite value

True False, a qubit need not have adefinite value until the momentafter it is observed

A bit can only be 0 or 1 True False, a qubit can be in asuperposition of 0 and 1simultaneously

A bit can be copied withoutaffecting its value

True False, a qubit in an unknownstate cannot be copied withoutdisrupting its state

Assumption Classical Quantum

A bit can be read withoutaffecting its value

True False, reading a qubit that isinitially in a superposition willchange the value of the qubit

2020

Comparison IIComparison II

Reading one bit has no effecton another unread bit

True False, if the qubit being read isentangled with another qubitreading one will affect the other

Assumption Classical Quantum

2121

Quantum Phenomena

2222

Quantum PhenomenaQuantum Phenomena

There are five quantum phenomena There are five quantum phenomena that make quantum computing weirdthat make quantum computing weird SuperpositionSuperposition InterferenceInterference EntanglementEntanglement Non-determinismNon-determinism Non-clonabilityNon-clonability

2323

SuperpositionSuperposition

The Principal of Superposition states if The Principal of Superposition states if a quantum system can be measured to a quantum system can be measured to be in one of a number of states then it be in one of a number of states then it can also exist in a blend of all its states can also exist in a blend of all its states simultaneouslysimultaneously

RESULT: An n-bit qubit register can be RESULT: An n-bit qubit register can be in all 2in all 2nn states at once states at once Massively parallel operationsMassively parallel operations

2424

InterferenceInterference We see interference patterns when We see interference patterns when

light shines through multiple slitslight shines through multiple slits

This is a quantumThis is a quantumphenomena which isphenomena which isalso present in quantumalso present in quantumcomputerscomputers A quantum computerA quantum computer

can operate on severalcan operate on severalinputs at once, the results inputs at once, the results interfere with each otherinterfere with each otherproducing a collectiveproducing a collectiveresultresult

2525

EntanglementEntanglement If two or more qubits are made to If two or more qubits are made to

interact, they can emerge from the interact, they can emerge from the interaction in a joint quantum state which interaction in a joint quantum state which is different from any combination of the is different from any combination of the individual quantum statesindividual quantum states

RESULT: If two entangled qubits are RESULT: If two entangled qubits are separated by any distance and one of separated by any distance and one of them is measured then the other, at the them is measured then the other, at the same instant, enters a predictable statesame instant, enters a predictable state

2626

Non-DeterminismNon-Determinism

Quantum non-determinism refers to Quantum non-determinism refers to the condition of unpredictabilitythe condition of unpredictability

If a quantum system is in a If a quantum system is in a superposition state and then superposition state and then measured, the measured state can measured, the measured state can not be predicted.not be predicted.

2727

Non-ClonabilityNon-Clonability

It is impossible to copy an unknown It is impossible to copy an unknown quantum state exactlyquantum state exactly

If you asked a friend to prepare a qubit If you asked a friend to prepare a qubit in a superposition state without telling in a superposition state without telling you which superposition state, then you you which superposition state, then you could not make a perfect copy of the could not make a perfect copy of the qubitqubit Useful in quantum cryptologyUseful in quantum cryptology

2828

Quantum Computation

2929


Changes to a quantum state can be described using Changes to a quantum state can be described using the language of quantum computationthe language of quantum computation

Single Qubit GatesSingle Qubit Gates Classical Not GateClassical Not Gate - Truth table - Truth table

Quantum Not Gate - Truth tableQuantum Not Gate - Truth table

0 1 and 1 0

0 1 and 1 0

3030


Superposition of states? Superposition of states?

Not without further knowledge of the properties of Not without further knowledge of the properties of quantum gatesquantum gates

The quantum NOT gate acts LINEARLY…The quantum NOT gate acts LINEARLY…

Linear behaviour is a general property of quantum Linear behaviour is a general property of quantum mechanicsmechanics

Non-linear behaviour can lead to apparent paradoxesNon-linear behaviour can lead to apparent paradoxes

- Time Travel- Time Travel

- Faster than light communication- Faster than light communication

- Violates the 2- Violates the 2ndnd Law of Thermodynamics Law of Thermodynamics

0 1 1 0

3131


NOT gate representationNOT gate representation

for anyfor any

we get… we get…

to summarize… to summarize…

0 1X

1 0

0 1

0 1X or 0 1

1 0

0 1 1 0

3232


Are there any constraints on what matrices may be used Are there any constraints on what matrices may be used as quantum gates? Of course!as quantum gates? Of course!

We require the normalization conditionWe require the normalization condition

for for

and the result after the gate has and the result after the gate has actedacted

The appropriate condition for this (of course) is The appropriate condition for this (of course) is that the matrix representing the gate is that the matrix representing the gate is UNITARYUNITARY

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

2 21 0 1

' ' 0 ' 1

† =U U I †where is the adjoint of U U

3333


Two more important gates…Two more important gates… Z gateZ gate

Hadamard GateHadamard Gate

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

1 0Z

0 -1

1 11

1 -12H

leaves 0 unchanged

flips the sign of 1 to - 1

turns 0 into 0 1 2

turns 1 into 0 1 2

2H I

3434


Hadamard Gate: A most useful gate indeed!Hadamard Gate: A most useful gate indeed!

1 0 =1, =0 0 0 1

21

1 0, 1 1 0 12

H H

H H

for

for

1if and 0 1

21

2

0 1 1 01 1 1

1 0 0 12 2 2

H X Z then

H X Z

3535


Review: Important single-qubit Review: Important single-qubit gatesgates

0 1

0 1

0 1

X

Z

H

0 1

0 1

0 1 1 1

2 2

3636

Quantum ComputationQuantum Computation Arbitrary Single Qubit Quantum GateArbitrary Single Qubit Quantum Gate

- complete set from properties of a much smaller - complete set from properties of a much smaller setset

2 2

2 2

cos sin0 02 2

sin cos0 02 2

i i

i

i i

e eU e

e e

Global Phase Factor

Rotation about z

Rotation

Scaling Constant

, , and are all real valued

3737

Quantum ComputationQuantum Computation Classical Universal Gates (example)Classical Universal Gates (example)

- The NAND gate is a classical Universal Gate. Why?- The NAND gate is a classical Universal Gate. Why?

Universal Quantum GatesUniversal Quantum Gates

- An arbitrary quantum Computation on n qubits can be - An arbitrary quantum Computation on n qubits can be generated by a finite set of gates that are UNIVERSAL generated by a finite set of gates that are UNIVERSAL for quantum computationfor quantum computation

* Need to introduce some multiple quibit quantum * Need to introduce some multiple quibit quantum gatesgates

NOT gate using NAND

AND gate using NAND

OR gate using NAND

3838

Multiple Qubit GatesMultiple Qubit Gates Controlled-NOT (CNOT) GateControlled-NOT (CNOT) Gate

- - two input qubits: control and targettwo input qubits: control and target

- In General- In General

A

B

A

B A

00 00 or 01 01

10 11 or 11 10

if control is 0 target left alone

else control is 1 target qubit is flipped

, ,A B A B A

3939

CNOT quantum gateCNOT quantum gate

A

B

A

B A

0 0

0 0

0 0

0

0 1

1 0

0

0

1

0 1CNU

0

0 1

0 10 00

1 0 1

1 001

1 1 01

1 0

1

1 0

0 0

0

0

0 0 1

1

B

A BA

B AA BA

B A BA B

A BBA

A B ABA

A B

B

if then we get

if then we get

Any multiple qubit logic gate may be composed from Any multiple qubit logic gate may be composed from

CNOT and Single Qubit GatesCNOT and Single Qubit Gates

4040

Other Computational BasesOther Computational Bases MeasurementsMeasurements

- In terms of basis states- In terms of basis states

- Generally any basis state can represent an - Generally any basis state can represent an arbitrary qubit statearbitrary qubit state

- If orthonormal then we can perform a - If orthonormal then we can perform a measurement in keeping with probability measurement in keeping with probability interpretationinterpretation

0 12 2 2 2

0 1 0 1,

2 2

a b

4141

Quantum CircuitsQuantum Circuits

Elements of a Quantum CircuitElements of a Quantum Circuit- each line in a circuit represents a "wire" - each line in a circuit represents a "wire"

* passage of time * passage of time

* photon moving from one location to another* photon moving from one location to another

- assume the state input is a computational basis - assume the state input is a computational basis statestate

- input is usually the state consisting of all s- input is usually the state consisting of all s

- no loops allowed ie: acyclic- no loops allowed ie: acyclic

- No FANIN(not reversible therefore not Unitary) - No FANIN(not reversible therefore not Unitary)

- FANOUT (can't copy a qubit)- FANOUT (can't copy a qubit)

0

4242


Quantum Qubit Swap CircuitQuantum Qubit Swap Circuit

, ,

, ,

, ,

a b a a b

a a b a b b a b

b a b b b a

a

b

,b a,b a b,a a b

x

x

4343


Controlled-U GateControlled-U Gate- A Controlled-U Gate has one control qubit and n - A Controlled-U Gate has one control qubit and n target qubits target qubits

- where U is any unitary matrix acting on n qubits- where U is any unitary matrix acting on n qubits

U

4444


Measurement OperationMeasurement Operation- Converts a single qubit state into a probabilistic - Converts a single qubit state into a probabilistic classical bit Mclassical bit M

M

4545


Can we make a Qubit Copying Can we make a Qubit Copying Circuit?Circuit?- Copying a classical bit can be done with the - Copying a classical bit can be done with the

Classical CNOT gateClassical CNOT gate

x

0

x

x

x x

y x yscratch-padinitialized to zero

bit to becopied

originalbit

copiedbit

4646


Can we make a Qubit Copying Can we make a Qubit Copying Circuit?Circuit?- How about copying a qubit in an unknown state - How about copying a qubit in an unknown state using a controlled-CNOT gate?using a controlled-CNOT gate?

scratch-padinitialized to zero

bit to becopied

Output State

0 1a b

0 1a b

0

00 11a b00 10a b

4747


Can we make a Qubit Copying Circuit?Can we make a Qubit Copying Circuit?- Does ?- Does ?

- Unless this does not copy the quantum - Unless this does not copy the quantum state input state input

- It is impossible to make a copy of the unknown - It is impossible to make a copy of the unknown quantum statequantum state

- NO CLONING THEOREM -- NO CLONING THEOREM -

00 11a b

2 20 1 0 1 00 01 10 11a b a b a ab ab b

0ab

2 200 01 10 11 00 11a ab ab b a b

4848


Bell States, EPR States, EPR PairsBell States, EPR States, EPR Pairs

x H

xyy

0 10 00 10 00 11

22 2

0

InIn OutOut00

01

10

11

0000 11 2

0101 10 2

0000 11 2

0000 11 2

4949

Quantum AlgorithmsQuantum Algorithms

, ,x y x y f x

Data Register

Target Register

Initial State Final State

00 10,

2x y

0 1

2

0

x x

y y f x

0,0 0 1,0 1 0, 0 1, 1

2 2

f f f f

Uf

5050

Eureka!!!! Both values of the Eureka!!!! Both values of the function show up in the final state function show up in the final state

solution.solution.

This can be generalized to functions on This can be generalized to functions on

arbitrary number of bits using the… arbitrary number of bits using the…

HADAMARD TRANSFORM HADAMARD TRANSFORM or or

WALSH-HADAMARD TRANSFORM WALSH-HADAMARD TRANSFORM


0, 0 1, 1

2

f f

5151

Quantum AlgorithmsQuantum Algorithms Deutsch's Algorithm CircuitDeutsch's Algorithm Circuit

- Combines quantum parallelism and interference- Combines quantum parallelism and interference

0 1

2

x x

y y f x

0 H

1 H

0 1

2

0

1

2

3

HUf

5252

Quantum AlgorithmsQuantum Algorithms Deutsch's Algorithm CalculationsDeutsch's Algorithm Calculations

- Combines quantum parallelism and interference- Combines quantum parallelism and interference

0 01

0 1

0 1 0 1

2 2

1 2

0 1 0 1 if 0 1

2 2

0 1 0 1 if 0 1

2 2

f f

f f

5353

Quantum AlgorithmsQuantum Algorithms Deutsch's Algorithm ConclusionDeutsch's Algorithm Conclusion

realizing is 0 if and 1 otherwise… realizing is 0 if and 1 otherwise…

2 3

0 10 if 0 1

2

0 11 if 0 1

2

f f

f f

3

0 10 1

2f f

0 1f f 0 1f f

measuring the 1st qubit gives

0 1f f

5454

Quantum AlgorithmsQuantum Algorithms Deutsch's Algorithm ResultsDeutsch's Algorithm Results

- The quantum circuit has given us the ability to - The quantum circuit has given us the ability to determine a GLOBAL PROPERTY of namelydetermine a GLOBAL PROPERTY of namely

using only ONE evaluation ofusing only ONE evaluation of

- A classical computer would require at least two - A classical computer would require at least two evaluations! evaluations!

- Difference between quantum parallelism and - Difference between quantum parallelism and classical randomized algorithmsclassical randomized algorithms

* * One might think the state corresponds to probabilistic classical computer that evaluates with probability 1/2 or 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

0 1f f f x

f x

0 0 1 1f f 0f

1f

5555

Deutsch-Jozsa AlgorithmDeutsch-Jozsa Algorithm- A simple case of a more general algorithm- A simple case of a more general algorithm

- Application is called Deutsch's Problem- Application is called Deutsch's Problem

- Classically Alice can only send one value of x each time- Classically Alice can only send one value of x each time

- Best classical algorithm requires up to queries- Best classical algorithm requires up to queries


Alice Bob

x is a number from 0 to 2n-1

Constant for all values of

Balanced: 1 for 1/ 2 the values of or 0 otherwise

xf x

x

x

2 / 2 0 1n 's and one Balanced

2 / 2 1n

n bits each time

5656

Deutsch-Jozsa AlgorithmDeutsch-Jozsa Algorithm- If Bob and Alice were able to exchange qubits instead of classical bits and if Bob calculated - If Bob and Alice were able to exchange qubits instead of classical bits and if Bob calculated f(x)f(x) using a unitary transform using a unitary transform UUff then Alice could determine the function in one query. 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- 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- Prepares query and answer register in a superposition state

- Bob evaluates f(x) and puts result into answer register- 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- Alice interferes the states in the superposition using a hadamard transform on the query register


1

2

3

0

5757

Quantum AlgorithmsQuantum Algorithms Deutsch-Jozsa Algorithm CircuitDeutsch-Jozsa Algorithm Circuit

x x

y y f x

0

1 H

0

1

2

3

nnH

Uf

nH

0 1

0,1

0 10 1

22n

n

nx

x

1 2

0,1

1 0 1

22n

f x

nx

x

Bob's function evaluation is stored in the amplitude

5858

Hadamard transform: helps to calculate effect on a state By checking the cases x=0 and x=1 separately for a single qubit…

thus

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


Deutsch-Jozsa Algorithm - detourDeutsch-Jozsa Algorithm - detourx

12

xz

z

zH x

1 1

1

... 11 ,...,

,...,,..., 1

2

n n

n

x z x z nnn z z n

z zH x x

12

x zn

z n

zH x

x z

5959

Quantum AlgorithmsQuantum Algorithms Deutsch-Jozsa Algorithm CircuitDeutsch-Jozsa Algorithm Circuit

- amplitude for is…- amplitude for is…

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

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

is unit length then all other amplitudes must be zero.is unit length then all other amplitudes must be zero.

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

2 3

1 0 1

2 2

x z f x

nz x

z

query register

0n

1

2

f x

nx

0n

3

6060

Quantum AlgorithmsQuantum Algorithms Deutsch-Jozsa Algorithm CircuitDeutsch-Jozsa Algorithm CircuitCase 2: If f is balanced then the positive and negative contributions to the amplitude for cancel, leaving an amplitude of 0Case 2: If f is balanced then the positive and negative contributions to the amplitude for cancel, leaving an amplitude of 0

- A measurement must yield a result other than 0 on at least one qubit - A measurement must yield a result other than 0 on at least one qubit

Summary: Summary:

- If Alice measures all zeros then the function is constant- If Alice measures all zeros then the function is constant

- Otherwise the function is balanced.- Otherwise the function is balanced.

- Deutsch's problem on a quantum computer can be solved in one evaluation. - Deutsch's problem on a quantum computer can be solved in one evaluation.

0n

6161

Quantum AlgorithmsQuantum Algorithms Other Quantum AlgorithmsOther Quantum Algorithms

- Generally there are three classes- Generally there are three classes

* Discrete * Discrete Fourier Transform Fourier Transform AlgorithmsAlgorithms

~Deutsch-Jozsa Algorithm~Deutsch-Jozsa Algorithm

~Shor's Algorithm for Factoring~Shor's Algorithm for Factoring

~Shor's Discrete Logarithm Algorithm~Shor's Discrete Logarithm Algorithm

* Quantum Search Algorithms* Quantum Search Algorithms

* Quantum Simulation Algorithms* Quantum Simulation Algorithms

~Quantum Computer is used to ~Quantum Computer is used to simulate quantum systemssimulate quantum systems

6262

Experimental Quantum Experimental Quantum Information ProcessingInformation Processing

The Stern-Gerlach ExperimentThe Stern-Gerlach Experiment Optical TechniquesOptical Techniques Nuclear Magnetic ResonanceNuclear Magnetic Resonance Quantum DotsQuantum Dots Traps: Ion Traps & Neutral Atom Traps: Ion Traps & Neutral Atom

TrapsTraps

6363

NMR Quantum ComputingNMR Quantum Computing

Lecture 2Lecture 2

6464

Nuclear Magnetic Nuclear Magnetic Resonance Resonance

Quantum ComputersQuantum ComputersQubit representation: spin of an atomic nucleusUnitary evolution: using magnetic field pulses

applied to spins in a strong magnetic field.

Chemical bonds between atoms couple the spinsState preparation: using a strong magnetic field to polarize the spinsReadout: using magnetic-moment induced free

induction decay signals

6565

Nuclear Magnetic Resonance Q.C.Nuclear Magnetic Resonance Q.C.Physical ApparatusPhysical Apparatus

Tesla 8.11B

(uniform to 1 part in 109)

Liquid samplePNFC 31151912 , , ,

Regard as an ensemble of n-bit quantumcomputers

Computer

amplifier

amplifier-pre

source-RF

coil-RF Typical ExperimentWait a few minutesfor the sample to cometo thermal equilibrium2. Send RF pulses to manipulate nuclear

spinsinto desired state.3. Switch off the ampsand switch on the pre-amplifier to measure the free-induction

decay

6666

BB00 11

Nuclear Spins as qubitsNuclear Spins as qubits

High field magnetHigh field magnet

RF WaveRF Wave

sample sample test tubetest tube

SpectrometerSpectrometer

ADC for data acquisitionADC for data acquisitionRF synthesizer and amplifierRF synthesizer and amplifier

Gradient controlGradient control

wave guideswave guides I SJIS

2-3 Dibromothiophene

9.6 T

RF wave

Nuclear Magnetic Resonance Nuclear Magnetic Resonance Q.C.Q.C.

Physical ApparatusPhysical Apparatus

6767

InternalInternal Hamiltonian Hamiltonian

The evolution of a spin system is The evolution of a spin system is generated by Hamiltoniansgenerated by Hamiltonians Internal Hamiltonian:Internal Hamiltonian:

HHintint==IIIIzz++SSSSzz+2+2JJISISIIzzSSzz

spin-spin couplingspin-spin coupling

interaction with B fieldinteraction with B field

I SJIS


9.6 T

6868

External HamiltonianExternal Hamiltonian Experimentally Controlled Hamiltonian:Experimentally Controlled Hamiltonian:

Total Hamiltonian:Total Hamiltonian:

HHextext(t)(t) ==RFxRFx(t)·(I(t)·(Ixx+S+Sxx)+)+RFyRFy(t)·(I(t)·(Iyy+S+Syy))

HHtotaltotal(t)(t)

controlled viacontrolled via

HHextext(t)(t)

I SJIS


9.6 T

RF wave

spins couple to RF fieldspins couple to RF field

HHtotal total (t)(t) = H= Hintint + H + Hextext(t)(t)

6969

2x 32

zx

TomographyTomographyNot all elements of the density matrix are observable on an

NMR spectra.

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.

7070

Quantum Games:Quantum Games:theoretical and experimental theoretical and experimental

resultsresults

One Example of NMR One Example of NMR QC:QC:

7171

OutlineOutline Introduction of quantum gamesIntroduction of quantum games

Classical game: Prisoner’s DilemmaClassical game: Prisoner’s Dilemma Maximal entangled quantum gameMaximal entangled quantum game

Some of our resultsSome of our results Theoretical extensions with non-Theoretical extensions with non-

maximal entanglement, more players, maximal entanglement, more players, larger strategy space, and so on.larger strategy space, and so on.

Experimental realization of quantum Experimental realization of quantum gamegame

Future Plan and discussionFuture Plan and discussion

7272

Prisoner's dilemmaPrisoner's dilemma Game theoryGame theory --an important branch of applied mathematics. It is --an important branch of applied mathematics. It is

the theory of decision-making and conflict between the theory of decision-making and conflict between different agents. Since the seminal book of Von Neumann different agents. Since the seminal book of Von Neumann and Morgenstern, modern game theory has found and Morgenstern, modern game theory has found applications ranging from economics through to biology.applications ranging from economics through to biology.

It concludes: Players, Strategy space, Payoff functionIt concludes: Players, Strategy space, Payoff function

Classifications: Classifications: Time (static & Dynamic). Time (static & Dynamic).

Information (complete &incomplete)Information (complete &incomplete)

Prisoner’s DilemmaPrisoner’s Dilemma --a famous game in game theory.--a famous game in game theory.

7373

Nash Equilibrium: mutual defect (D,D)Nash Equilibrium: mutual defect (D,D) Nash Equilibrium implies that no player can increase his payoff by Nash Equilibrium implies that no player can increase his payoff by

unilaterally changing his strategy.unilaterally changing his strategy. Pareto optimal: mutual cooperation (C,C)Pareto optimal: mutual cooperation (C,C) A pair of strategies is called pareto optimal if it is not possible to 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 increase one player’s payoff without lessening the payoff of the other player.other player.

Prisoner’s Dilemma: Nash Equilibrium strategy Prisoner’s Dilemma: Nash Equilibrium strategy profile is not equivalent to Pareto optimalprofile is not equivalent to Pareto optimal

• 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.

BobBobCCAliceAlice

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Maximal entangled quantum Maximal entangled quantum gamegame

Quantum game theoryQuantum game theory Recently, new effect involving quantum information has Recently, new effect involving quantum information has

been discovered theoritically in the area of game theory by been discovered theoritically in the area of game theory by some pioneers.some pioneers.

1.1. L.Goldenberg, L.Vaidman, S.Wiesner, Phys.Rev.Lett. 82, 3356 L.Goldenberg, L.Vaidman, S.Wiesner, Phys.Rev.Lett. 82, 3356 (1999).(1999).

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

3.3. J.Eisert, M.Wilkens, M.Lewenstein, Phys.Rev.Lett. 83, 3077 J.Eisert, M.Wilkens, M.Lewenstein, Phys.Rev.Lett. 83, 3077 (1999).(1999).

Maximal entangled quantum gameMaximal entangled quantum game Eisert et al. showed that the classical problem of Eisert et al. showed that the classical problem of

Prisoner's DilemmaPrisoner's Dilemma is a subset of the quantum game by is a subset of the quantum game by using a using a physical model of the quantum gamephysical model of the quantum game, and there is no , and there is no longer a dilemma when employ a maximally entangled game.longer a dilemma when employ a maximally entangled game.

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Interesting resultsInteresting resultsFor For a separable gamea separable game with , with ,

there exists a pair of quantum strategies (D,D) is a there exists a pair of quantum strategies (D,D) is a Nash Equilibrium and yields payoff (1,1) which is not Nash Equilibrium and yields payoff (1,1) which is not Pareto optimal. Indeed, this quantum game behaves Pareto optimal. Indeed, this quantum game behaves “classically”.“classically”. For For a maximally entangled quantum gamea maximally entangled quantum game with with , ,

(Q,Q) is the Nash equilibrium of the game and has the (Q,Q) is the Nash equilibrium of the game and has the property to be property to be Pareto optimalPareto optimal . .

So Prisoner’s Dilemma is removed So Prisoner’s Dilemma is removed if quantum strategies are allowed for.if quantum strategies are allowed for.

2

0

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Correlation between quantum Correlation between quantum game and quantum entanglementgame and quantum entanglement

Yet it is legitimate for us to ask:Yet it is legitimate for us to ask:

----Whether a quantum game will Whether a quantum game will still outperform its classical version if still outperform its classical version if it is not maximally entangled? and it is not maximally entangled? and how a quantum game depends on how a quantum game depends on the entanglement of the game's the entanglement of the game's state?state?

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A physical model of quantum A physical model of quantum gamegame

J. Eisert have proposed a physical model of this J. Eisert have proposed a physical model of this game and the elegant quantum network is game and the elegant quantum network is illustrated as:illustrated as:

UUAA and U and UBB are the strategy moves available to are the strategy moves available to the players:the players:

Unitary operatorUnitary operator

2cos2sin2sin2cos

,

i

i

e

eU

2/exp DDiJ

01

100,

^

UD

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Two player’s initial state is Two player’s initial state is

The entanglement of the game's initial state can The entanglement of the game's initial state can be denoted asbe denoted as

therefore, can be denoted as a measure for therefore, can be denoted as a measure for the entanglement.the entanglement.

The final state isThe final state is

Then the expected payoff for Alice and Bob areThen the expected payoff for Alice and Bob are

00JUUJ BAf

112sin002cos00 iJi

2cosln

2cos

2sinln

2sin 2222

110100

111000

53

53

PPP

PPP

B

2

fij ijP

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Theoretical ResultsTheoretical Results Nash Equilibrium:Nash Equilibrium:

There exist two threshold:There exist two threshold:

Expected payoff as game’s entanglement Expected payoff as game’s entanglement varies varies

51sin1 Arcth 52sin2 Arcth

i

iQD

QQ

DQ

QD

DD

UU

th

thth

thth

th

BA 0

0ˆ,01

10ˆ,

2

0

,ˆˆ,ˆ,ˆˆ,ˆˆ

ˆˆ

2

21

21

1

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Other Theoretical ResultsOther 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 Experimental realization Physics Review Letter 88Physics Review Letter 88, 137902, 137902 （（ 20022002 ））

Technologies for quantum information processing(QIP) Technologies for quantum information processing(QIP)

--There are a number of proposed device technologies for QIP. There are a number of proposed device technologies for QIP. --Of them, NMR have given the many successful results experimentally --Of them, NMR have given the many successful results experimentally

for QIP, such as quantum teleportation, quantum error correction, for QIP, such as quantum teleportation, quantum error correction, quantum simulation, quantum algorithm etc. quantum simulation, quantum algorithm etc.

We add game theory to the list: Quantum We add game theory to the list: Quantum games was experimental realized on games was experimental realized on nuclear magnetic resonance quantum nuclear magnetic resonance quantum computer.computer.

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Two-qubit: Nuclear Coupled SpinsTwo-qubit: Nuclear Coupled Spins QubitsQubits Partially deuterated cytosine Partially deuterated cytosine

molecule contains two protons, molecule contains two protons, in a magnetic field, each spin in a magnetic field, each spin state of proton could be used as state of proton could be used as a qubit. a qubit.

Distinguish each qubitDistinguish each qubit Different Larmor frequencies Different Larmor frequencies

(the chemical shift) enable us to (the chemical shift) enable us to address each qubit individually.address each qubit individually.

Quantum logic gates Quantum logic gates Radio Frequency (RF) fields and Radio Frequency (RF) fields and

spin--spin couplings between the spin--spin couplings between the nuclei are used to implement nuclei are used to implement quantum logic gates. quantum logic gates.

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Quantum network and gatesQuantum network and gates Quantum networkQuantum network

Entangled gate: Entangled gate:

The strategy moves UThe strategy moves UAA and U and UBB are are

Each gate can be realized by NMR technique.Each gate can be realized by NMR technique.

i

iQD

n

n

n

n

QQ

DQ

QD

DD

UU BA 0

0ˆ,01

10ˆ,

187

76

76

50

,ˆˆ,ˆ,ˆˆ,ˆˆ

ˆˆ

}18,...1,0{,36

,2/expˆ nn

DDiJ

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Experiments for quantum gameExperiments for quantum game

Experimentally, we performed nineteen separate Experimentally, we performed nineteen separate sets of experiments which was distinguished by:sets of experiments which was distinguished by:

In each set, the full process of the game was In each set, the full process of the game was executed.executed.

1.1. Create an effective pure state Create an effective pure state

2.2. Prepare the initial entangled state by applying gate JPrepare the initial entangled state by applying gate J

3.3. Players Alice and Bob executed their strategic moves UPlayers Alice and Bob executed their strategic moves UA A and Uand UB B

4.4. Apply the unentangled gate JApply the unentangled gate J++

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

}18,...1,0{,36

,2/expˆ nn

DDiJ

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NMR SpectrometerNMR Spectrometer

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The player Alice's payoffs as a function of the The player Alice's payoffs as a function of the parameter . parameter .

It is easy to see that (n=0) corresponds to It is easy to see that (n=0) corresponds to Eisert et al.'s separable game and (n=18) Eisert et al.'s separable game and (n=18) corresponds to their maximally entangled quantum corresponds to their maximally entangled quantum game.game.

02

Experimental resultsExperimental results

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Good agreement between theory and experiment. Good agreement between theory and experiment. Experimental Error: Experimental Error: --an estimated error is less than 0.08, the errors are --an estimated error is less than 0.08, the errors are

primarily due to inhomogeneity of magnetic field, imperfect primarily due to inhomogeneity of magnetic field, imperfect RF selective pulses, and the variability over time of the RF selective pulses, and the variability over time of the mesurement process.mesurement process.

Decoherence:Decoherence: --each experiment took less than 300 milliseconds, --each experiment took less than 300 milliseconds,

which was well within the the decoherence time (3 which was well within the the decoherence time (3 seconds). seconds).

This experiment was referred by :This experiment was referred by : Physics News update (APS), Physics web (IOP),Physics News update (APS), Physics web (IOP), New Scientist, Science Update (Nature).New Scientist, Science Update (Nature). Physics worldPhysics world

Experimental resultsExperimental results

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2002.42002.4 《《 NatureNature 》》 Science Science UpdateUpdate

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2001.9:2001.9: APS- Physics News UpdateAPS- Physics News Update

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2002.12002.1 －－ New ScientistsNew Scientists

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ThanksThanks

introduction to quantum computing

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