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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:
a|100> + b|011> decoded to a|111> + b|011> = (a|1> + b|0>)|11>
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:
a|001> + b|110> decoded to a|001> + b|101> = (a|0> + b|1>)|01>
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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