Lecture 33: Quantum Computing 2

8/14/2019 Lecture 33: Quantum Computing 2

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Artificial IntelligenceDr. Richard Spillman

PLU

Fall 2003

Lecture 33: Quantum Computing 2


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Class Topics

Intro to AI Lisp

Search Prolog

ExpertSystems

GeneticAlgorithms

NLP Learning

Future

Expert

Systems

Lisp

NLP

Prolog

Intro to AI

Learning

Search

Genetic

Algorithms

Future


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Last Class

Why Quantum Computing?

What is Quantum Computing?

History

Quantum Weirdness

Quantum Properties

Quantum Devices


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Review The Need

The size of components will drop down to the

one atom per device level by 2020


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Review - Superposition

The Principal of Superposition states if aquantum 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 qubit register can be in all

2n states at once Massively parallel operations


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Outline

Quantum Logic Gates II

Quantum Dots

Quantum Error Correction


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Quantum LogicQuantum Logic

Gates IIGates II


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Controlled NOT

One of the first quantum logic gatesproposed was the Controlled-NOT gate

which implements an XOR

It has two inputs and two outputs (required for

reversibility)

c t c t

0 0 0 0

0 1 0 1

1 0 1 1

1 1 1 0

c

t

c

t

The target, t, is inverted when

the control, c, is 1


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Toffoli Gate

Example of a reversible AND sometimescalled controlled-controlled-NOT gate

It has three inputs and three outputs

The target input is XORed with the AND of thetwo control inputs

C1 c2 t c1 c2 t

0 0 0 0 0 0

0 0 1 0 0 1

0 1 0 0 1 0

0 1 1 0 1 1

1 0 0 1 0 0

1 0 1 1 0 1

1 1 0 1 1 1

1 1 1 1 1 0

c2

t

c2

t

c1 c1


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Quantum Gate Operation

Suppose the control input is in asuperposition state, what happens to the

target, does it get flipped or not?

The answer is that it does both

In fact, c and t become entangled

ct

ct

10 +

0

1100 +

Entangled states that is

a superposition of states in

which c and t are either both

spin up or spin down


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Quantum DotsQuantum Dots


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Quantum Dots

Quantum dots are small metal or semi-conductor

boxes that hold well defined number of electrons

The number of electrons in a box may be adjusted

by changing the dots electrostatic environment Dots have been made which vary from 30 nm to 1 micron

They hold from 0 to 100 electrons

ee

Quantum dot

w/electronQuantum dot

wo/electron


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Quantum Dot Wireless Logic

Lent and Porod of Notre Dame proposed awireless two-sate quantum dot device

called a cell

Each cell consists of 5 quantum dots and two

electrons

ee

ee

State 1

ee

ee

State 0


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Quantum Dot Wire

By placing two cells adjacent to eachother and forcing the first cell into a certain

state, the second cell will assume the

same state in order to lower its energy

ee

ee

ee

ee

ee

ee

ee

ee

The net effect is that a 1

has moved on to the next cell

By stringing cells together inthis way, a pseudo-wire can

be made to transport a signal

In contrast to a real wire,

however, no current flows


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Quantum Dot Majority Gate

Logic gates can be constructed withquantum dot cells

The basic logic gate for a quantum dot cell is

the majority gate

in

in

in

out

in

in

in

out


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Quantum Dot Inverter

Two cells that are off center will invert asignal

in

out

in

out

in

out


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Quantum Dot Logic Gates

AND, OR, NAND, etc can be formed fromthe NOT and the MAJ gates

0

A

B

A and B

1

0

1

A

B

A or B

0

1

0

1

0

A

B

A nand B

1

1

0


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Quantum ErrorQuantum Error

CorrectionCorrection


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Quantum Errors

PROBLEMPROBLEM: When computing with a quantum

computer, you cant look at what it is doing

You are only allowed to look at the end

RESULTRESULT: What happens if an error is

introduced during calculation?

SOLUTIONSOLUTION: We need some sort of quantumerror detection/correction procedure


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Classical Error Codes

In standard digital systems bits are added to a dataword in order to detect/correct errors

A code is e-error detectinge-error detectingif any fault which causes atmost e bits to be erroneous can be detected

A code is e-error correctinge-error correctingif for any fault whichcauses at most e erroneous bits, the set of all correctbits can be automatically determined

The Hamming DistanceHamming Distance, d, of a code is the minimumnumber of bits in which any two code words differ

the error detecting/correcting capability of a code depends onthe value of d


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Parity Checking

PROCESS: Add an extra bit to a word before

transmitting to make the total number of bits even orodd (even or odd parity) at the receiving end, check the number of bits for even or

odd parity

It will detect a single bit error Cost: extra bit

Example: Transmit the 8-bit data word 1 0 1 1 0 0 01 Even parity version: 1 0 1 1 0 0 0 1 0

Odd parity version: 1 0 1 1 0 0 0 1 1


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Quantum Schemes

In 1994 the first paper on Quantum errorcorrection was presented at a conference

in England

It required the quantum computer to runsimultaneous copies of a calculation

If no errors occurred all the separate copies

would produce the same answer

Using a inefficient procedure a wrong answercould be restored


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Improvements

In 1995, Peter Shor developed a better

procedure using 9 qubits to encode a

single qubit of information

His algorithm was a majority vote type of

system that allowed all single qubit errors

to be detected and corrected


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Example

A 3-bit quantum error correction schemeuses an encoder and a decoder circuit as

shown below:

EncoderEncoder DecoderDecoder0

0

Input qubit Output qubit

OperationsOperations

& Errors& Errors


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Encoder

The encoder will entangle the tworedundant qubits with the input qubit:

a|0> + b|1>

|0>

|0>

If the input state is |0> then

the encoder does nothing sothe output state is |000>

If the input state is |1> then

the encoder flips the lower

states so the output state is|111>

If the input is an superposition state, then the output

is the entangled state a|000> + b|111>


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Decoder

Problem: Any correction must be done without looking

at the output

The decoder looks just like the encoder:

Corrected output

Measure: if 11 flip the top qubit}

If the input to the decoder is |000> or |111> there wasno error so the output of the decoder is:

Input Output

|000> |000>

|111> |100> (the top 1 causes the bottom bits to flip)

Error free flag


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Example

No Errors:

a|000> + b|111> decoded to a|000> + b|100> = (a|0> + b|1>)|00>

Top qubit flipped:


So, flip the top qubit = (a|0> + b|1>)|11>

Middle qubit flipped:a|010> + b|101> decoded to a|010> + b|110> = (a|0> + b|1>)|10>

Bottom qubit flipped:



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Decoder w/o Measurement

The prior decoder circuit requires themeasurement of the two extra bits and a

possible flip of the top bit

Both these operations can be implemented

automatically using a Toffoli gate

If these are both 1

then flip the top bit}


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Possible Capstone

For a senior project, work out examples of

quantum error correction schemes and

compare them to digital error correction

Implement a Quantum Dot simulator and

construct Quantum Dot circuits


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Possible Quiz

Remember that even though each quiz is worthonly 5 to 10 points, the points do add up to a

significant contribution to your overall grade

If there is a quiz it mightcover these issues:

What is a quantum dot?

Why are errors a problem with quantum systems?

What does a controlled NOT gate do?


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Summary

Quantum Logic Gates II

Quantum Dots

Quantum Error Correction

Documents

Lecture 33: Quantum Computing 2