Introduction to Quantum Computers Goren Gordon The Gordon Residence July 2006.

Introduction to Quantum Computers

Goren Gordon

The Gordon ResidenceJuly 2006

Outline• Introduction to quantum physics

– Superposition

• Software– Deutsch-Jozsa algorithm – Grover search algorithm– “No Shor for you, come back one year!!!”

• Hardware• Why aren’t there any QC around?• The weird stuff

– Cluster state quantum computers

Classical (regular) computer

0 or 1

Introduction to Quantum Physics

Classical bit (binary):

00

1

f(00)


0 or 1



01

2

f(01)


0 or 1



10

3

f(10)


0 or 1



11

4

f(11)

One computation per input number

2N computation for N bits

Computational complexity: how many computations as a function of number of bits

Classical (regular) computerIntroduction to Quantum Physics

01 f(01)

(Classical) Parallel computing

10 f(10)

11 f(11)

00 f(00)

FasterSame number of computationsSame computational complexity

Slow simple computationVERY large parallelism

http://images.google.com/imgres?imgurl=http://www.cogneuro.ox.ac.uk/images/brain.gif&imgrefurl=http://www.cogneuro.ox.ac.uk/&h=203&w=266&sz=22&hl=en&start=15&tbnid=t5hELx8UWEO34M:&tbnh=86&tbnw=113&prev=/images%3Fq%3Dbrain%26svnum%3D10%26hl%3Den%26lr%3D

The Dream…Introduction to Quantum Physics

00 and 01 and10 and 11

1

f(00) and f(01) andf(10) and f(11)

One computation for ALL possible numbers

How can this happen?

Quantum Superposition

0 and 1A quantum bit (qubit): |0> + |1>

You can process all the numbers at the same time !!!


In the quantum world you can have:

|00> +|01>+|10>+|11>

1

|f(00)>+|f(01)>+|f(10)>+|f(11)>

Quantum SuperpositionWhat does it mean to have a superposition?

A qubit: a|0> + b|1>|0> |1>

If I open the boxes, (measurement)Probability a2 to be in box |0>Probability b2 to be in box |1>

Closed boxes.Contain one particle.


a2+b2=1 Ring a bell? cos2+sin2 =1

Qubit: cos|0> +sin|1>

An axiom of QM:Born’s Rule

a and b are numbers

Quantum MeasurementIntroduction to Quantum Physics

Example:Polarization of light

|0>

|1>

|0>

|1>|0>+|1>

|0> 0

|1> 90

|0> + |1> 45 superposition

Rotation by 45|0> OR |1>0 90

45 OR 135

Measure: 50% |0>, 50% |1>

Classical

|0> + |1> 45

|1>90

Measure: 100% |1>

Quantum

Rotation by 45

|0>+|1>(+|0>+|1>)/2 + (-|0>+|1>)/2 = |1>

Cancel out

Qubit: cos|0> +sin|1>

+

Quantum MeasurementDistinguishability: two states are distinguishable ifI can tell with 100% which state I have.


|0>, |1> are distinguishable

|0>, |0>+|1> are not distinguishable: if I measure and get |0>, there is 50% that the state was |0>+|1>.


|0>

|1>

|0>+|1>, |0>-|1> are distinguishable

|0>

|1>|0>+|1>

|0>-|1>

The sign matters

Quantum Logic GateIntroduction to Quantum Physics

|0>

|1>|0>+|1>

|0>-|1>

Single qubit logic gate: Rotation

Hadamard |0> |0> +|1>|1> |0> - |1>

Not |0> |1>|1> |0>


|0>

|1>

Computation with Quantum Superposition

A qubit: a|0> + b|1>


qubit Logicgate

output

|0>NOT

|1>

|1>NOT

|0>

a|0>+b|1>NOT

a|1>+b|0> Two operations of the Gate in one step!!!


Many qubits: |>=a0|000> + a1|001>+…+a8|111>


|> ComplexComputation

|>=a0|f(000)> + a1|f(001)>+…+a8|f(111)>

Many operations of the function in one step!!!

Classical computation:

x f(x)


Many qubits: |>=a0|000> + a1|001>+…+a8|111>


|>=a0|f(000)> + a1|f(001)>+…+a8|f(111)>Result of computation:

Problem:We need to measure the result!!!

Each measurement gives only one result of the calculation!!!

With probability a02 we will get f(000),

With probability a12 we will get f(001),

…With probability a8

2 we will get f(111)

Does not save time, or number of calculations!!!

Deutsch-Jozsa algorithm Quantum Algorithms

Function: f(x)Either constant: f(x) = +1 always

f(x) = -1 always

Or balanced: f(x) = +1 half of the time -1 half of the time

x=0-8 (3 bits)

f(000) = +1f(001) = -1f(010) = +1f(011) = +1f(100) = -1f(101) = -1f(110) = +1f(111) = -1

Balanced

f(000) = +1f(001) = +1f(010) = +1f(011) = +1f(100) = +1f(101) = +1f(110) = +1f(111) = +1

Constant


Function: f(x)Either constant, or balanced.

x=0-8 (3 bits)

How many calculations of f(x) do I need to do to know if it is constant or balanced?

On a classical computer we need:At worst 5 calculations (more than half)

f(000) = +1f(001) = +1f(010) = +1f(011) = +1f(100) = -1

Balanced Computational complexity:2N-1+1


Function: f|x>=±|x>Either constant, or balanced.

|>=|000>+|001>+|010>+|011>+|100>+|101>+|110>+|111>

On a quantum computer:Only one calculation:

Constant:f|>=±(|000>+|001>+|010>+|011>+|100>+|101>+|110>+|111>)

Balanced:f|>=|000>-|001>+|010>+|011>-|100>-|101>+|110>-|111>


Constant:fc|>=±(|000>+|001>+|010>+|011>+|100>+|101>+|110>+|111>)

Balanced:fb|>=|000>-|001>+|010>+|011>-|100>-|101>+|110>-|111>

|0>+|1>, |0>-|1> are distinguishable

|0>

|1>|0>+|1>

|0>-|1>

The sign matters

Reminder:


Constant:fc|>=±(|000>+|001>+|010>+|011>+|100>+|101>+|110>+|111>)

Balanced:fb|>=|000>-|001>+|010>+|011>-|100>-|101>+|110>-|111>

These two states are distinguishable

One can make a measurement to distinguish between the two states.

Only one calculation of the function is needed to know if it is constant or balanced!!!

Grover’s Search AlgorithmQuantum Algorithms

Find a specific number out of N numbers.Example: Searching in a database

1

3

2

5

6

8

7

4

Is it the right number?

NO

YES!!!


Find a specific number out of N numbers.Example: Searching in a database

Given: f(x) = -1 for a specific (unknown) x+1 for all other x

Goal: Find x

Classical computer:Worst case: Go over all x until you find.

Quantum computer:Use superposition to shorten the search


Stage 1: Prepare |> = |000>+|001>+…+|111> (superposition of all states)

Stage 2: Do:2.a. Apply f| > calculate f2.b. Apply 2|><|-I do another simple calculation

Stage 3: measure

Example:|>=a|0>+b|1>2.b. (2a-1)|0> + (2b-1)|1>


Example: 2 qubits, f(|01>)=-1

Stage 1: ½|00>+½|01>+½|10>+½|11>

Stage 2.a. ½|00>-½|01>+½|10>+½|11>

Stage 2.b. |01>

Stage 3. Measure |01>

(2x½-1)=0, (2x(-½)-1)=-1

We applied f(x) only once !!!

Normalization: (½)2+(½)2+(½)2+(½)2=1

f(|01>)=-1


Stage 1: Prepare |> = |000>+|001>+…+|111> (superposition of all states)

Stage 2: Do N times:2.a. Apply f| >2.b. Apply 2|><|-I

Stage 3: measure|x>

|>|x>

Final result:Instead of N times in the classical computerYou need N times in the quantum computer

DesiredUnknownstate

Orthogonal to desired state

Very small error

One word on Shor

• The algorithm that started everything

• Proves that a quantum computers can break the RSA code in polynomial times

• Uses Fourier Transforms (and other mathematical stuff)

• Too complicated to show it here

Quantum Algorithms

Software: Conclusions

• The quantum computer uses superposition

• The quantum algorithms are only useful for global, or collective results (Deustch-Jozsa)

• There are many (many) new quantum algorithms, which are exponentially faster than classical computers

• There isn’t any quantum computer, yet

Quantum Algorithms

Building a Quantum Computer

Problems

1. Distinguishable qubits single particles1. Preparation reproducibility

2. Readout deterministic

2. Single qubit gates control, short

3. Two-qubits gates interaction

4. (noise issues) always

Quantum Hardware

Building a Quantum Computer

1. Optics qubits = polarization

2. Atoms qubits = electron energy levels

3. Molecules qubits = nuclear spins

4. QDots qubits = electron charge

Quantum Hardware

OpticsQuantum Hardware

Distinguishable qubits: Polarization of single photons

Preparation: single photon sourcesUsually, two photon sources

© Stanford University

Laser

Strong filter

Single photons

Laser

Specialmaterial

Entangled photon pairD

Single photons

Polarization of light|0>

|1>

http://www.qcaustralia.org/crp_sl.htm



Distinguishable qubits: Polarization of single photons

Readout: single photon detectors

© LC Technologies

There are two ways a detector can fail:1. It counts too few photons (loss);2. it counts too many photons (dark counts).


|1>




Single photon gates: polarization rotation

Two-photon gates: THE PROBLEM

Photons do not interact. Solutions:1. Non-liner materials – low efficiency2. Non-deterministic schemes – probabilistic,

requires auxiliary resourcesNon-linear

|1>

|1>

|1>

|0>

XOR gate


|1>





|1>

© Stanford University © LC Technologies

The whole setup

Single photonsource

Singlephotons

ComputationSingle photondetection



Optics

Pros:

• There are demonstrations of quantum computations with optics

• Low noise – good!!!

Cons:

• Requires too many resources

• Not scalable (yet)

Quantum HardwarePolarization of light

|0>

|1>



Neutral AtomsQuantum Hardware

THE PROBLEM:Working with single atoms

Optical TweezerScattering force:

Gradient Force:

http://www.iap.uni-bonn.de/ag_meschede/english/index_eng.html



THE PROBLEM:Working with single atoms

Source of atom beam

Single atom

Magneto Optical Trap (MOT)

lasers

magnets




Optical conveyer belt

Single atoms

Moving and controlling single atoms












Distinguishable qubits: electrons energy levels

|0>

|1>

Energy levels

electron

|0>

|1>|0>+|1>




Distinguishable qubits: electrons energy levels

|0>

|1>

Energy levels

Resonant LASER:A laser with a specific frequency thatMatches the energy levels

laser

Creates transition between the two levels

|0>/4 pulse

|0>+|1>

|0>/2 pulse

|1>




Distinguishable qubits: electrons energy levels |0>

|1>

Energy levels

Preparation:All electrons decay to |0>

Readout:Usually fluorescence

|0>

|1>laser

|0>

|1>

laser

lightNothing happens

Non-resonant

Resonant




|0>

|1>

Energy levels

laser

Single qubit gates: laser pulses

|0>

|1>

Two qubit gate:

Instead of LASER:Interaction between two atomsAnd EM field in the cavity

|00>

|10>|01>

|11>laser

SWAP gate: |01>|10>




|0>

|1>

Energy levels

The whole setup

lase

r lase

r

Single atom source

computation readout



|0>

|1>

Energy levels

Pros:

• Single atom manipulation

• Scalability to many atoms

Cons:

• Two-atom gate not accomplished yet

• A lot of noise


MoleculesQuantum Hardware

Liquid state NMR (nuclear magnetic resonance):Room temperature liquid of molecules

Each one of the atoms of the molecule can be a qubit

Distinguishable qubits: nuclear spins

http://qso.lanl.gov/qc/

Several words on SpinsQuantum Hardware

Spin: self magnetic field

Can have two values:Up / Down

Energy level

Energy level

One of the spins have less energyIn a magnetic field



Spin: self magnetic field

RF field

Strong magnetic fieldRF (radio frequency) field

Spins can become qubits

|0> |1>



Liquid state NMR (nuclear magnetic resonance):A room temperature liquid of molecules

Each one of the atoms of the molecule can be a qubit

Distinguishable qubits: nuclear spins

|0> |1>



Preparation: THE PROBLEMAt room temperature, the nuclear spins are a mess

We want We have

Some solutions: cool the liquid (new solid state NMR)

|0> |1>



Readout: Spectroscopy|0> |1>

|0> |1>

frequency

intensity 91.8



Readout: spectroscopy|0> |1>



Single and two-qubit gates: RF fields

RF field

|0> |1>


MoleculesQuantum Hardware |0> |1>

The whole setup

RF field

Preparation Computation Readout


MoleculesQuantum Hardware |0> |1>

Pros:

• Easy gates and readout

• Easy access to single qubits

Cons:

• Not scalable: never more than 12 qubits

• Preparation problematic


Quantum DotsQuantum Hardware

Fabricated nanostructure trapping single electrons

Single electron

Distinguishable qubits: electron charge

http://www.qcaustralia.org/crp_asd.htm


Preparation: Putting the electrons in the right place

Readout: reading the voltage of the circuit

Single and two-qubits gates: applying the right voltages



Pros:

• Scalable

• Easy manufacture

Cons:

• Hard to create two-qubit gate

• 3 qubits computation not yet demonstrated


Why aren’t there any QC around?

• Noise \ loss \ decoherence• The quantum information is lost due to

interaction with environment:– Fluctuation in magnetic fields– Collision with hot particles

• Systems:– Photons’ polarization fluctuates with time– Electrons decay to lower levels \ lose phase– Nuclear spins fluctuates with time– Electrons’ spins fluctuates


• The cohernece time = how long until 1% of the information is lost

• Quantum computation possible only when

Coherence time >> Computation time

• In all systems, this is a problem


• Scalability– if N qubits requires X resources, do 2N

qubits require 2X?

• All systems are not yet scalable

• The status today:1. NMR: 12 qubits

2. Ions: 8 qubits

3. Photons, Qdots, atoms: 1-3 qubits

The Weird StuffCluster state quantum computer

• All qubits are entangled

• Measure one qubit

• Change another according to result of measurement

• Continue until final result is left

measure Conditional gateresult

Final result ofComputation

http://arxiv.org/abs/quant-ph/0504097

Summary

• Quantum computers can do magic

• We are only in the beginning

• Software more advanced than hardware

• Competition between different setups

• Real quantum computers in 10-100 years

Thank you!!!

Topics for next lecture

1. Entanglement and non-locality

2. Shor & Co. algorithms

3. Specific implementation of QC

4. Quantum Games

Introduction to Quantum Computers Goren Gordon The Gordon Residence July 2006.

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