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The Mathematics of Star
Trek
Lecture 12: Quantum
Computing
Topics
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Security of RSA
Thomas Young’s Light Experiment
A Modern Version of Young’s Experiment
Superposition
Many-Worlds
The Quantum Computer
Applications of Quantum Computing
Drawbacks to Quantum Computing
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Thomas Young’s Double-Slit
Light Experiment
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Imagine two ducks
swimming alongside each
other in a pond.
As each duck passes
through the water, a trail of
ripples will form behind the
duck.
The two sets of ripples fan
out and interact - canceling
out when a peak meets a
trough, forming a higher
peak when two peaks
meet, or lower trough when
two troughs meet.
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Thomas Young’s Double-Slit
Light Experiment (cont.)
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Starting in 1799, English
physician and physicist Thomas
Young (1773-1829) performed
a series of experiments with
light, including one in which a
partition with two narrow
vertical slits is placed between
a light source and a screen.
Young expected that there
would be two bright stripes on
the screen.
Instead, he found that the light
fanned out from the two slits
and formed a pattern of several
light and dark stripes on the
screen.
Handout of Young’s Experiment
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Thomas Young’s Double-Slit
Light Experiment (cont.)
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Assuming that light was a form of a wave,
Young concluded that the light coming out of
each slit was behaving like the ripples in the
water behind the ducks.
The dark and light stripes were caused by the
same sort of interactions as the ripples in the
water.
Light stripes were caused by two “peaks” or
two “troughs” of the light waves interacting.
Dark stripes were caused by the interaction of
a trough and a peak of the light waves.
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Thomas Young’s Double-Slit
Light Experiment (cont.)
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We now know that light does act like a
wave (or a particle), but at the time of
Young’s experiment, this was not well
known.
Young published his ideas on the nature
of light in the classic paper “The
Undulatory Theory of Light”.
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A Modern Version of Young’s
Experiment
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Light can be thought of a wave or made up of
particles, called photons.
Modern technology allows us to reproduce
Young’s experiment with a light source capable
of emitting single photons of light, at rates such
as one photon per minute.
As each photon travels towards the partition, it
may pass through one of the slits.
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A Modern Version of Young’s
Experiment (cont.)
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For this experiment, we use a “screen” made up of
special photo detectors that can record each photon
that makes it through the partition.
Over a period of several hours, we will get an overall
picture of where the photons are hitting the screen.
Since individual photons are passing through the
slits, we wouldn’t expect to see the same striped
interference pattern as we do for a regular light bulb.
Handout of Modern Version of Young’s Experiment.
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A Modern Version of Young’s
Experiment (cont.)
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Amazingly, we see the same pattern of light
and dark strips as for Young’s original
experiment, which means the individual
photons are somehow interacting!
This weird result defies common sense and
there is no way to explain what is going on in
terms of classical physics.
Since photons are very small particles, we can
try to use the ideas of quantum mechanics to
explain what we see.
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A Modern Version of Young’s
Experiment (cont.)
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It turns out that even experts in quantum
mechanics cannot agree on what is
happening!
Right now there are two competing
theories that are used to explain what is
happening in the modern version of
Young’s experiment.
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Superposition
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The first way to explain what is going on is via
superposition.
First of all, we only know two things for certain about an
individual photon:
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It leaves the light source.
It strikes the screen.
Everything else is a mystery, including if the photon
passed through the left slit or the right slit.
Since the exact path of the photon is unknown, we
assume it passes through both slits simultaneously, which
would allow the photon to interfere with itself, creating the
pattern we see on the screen!
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Superposition (cont.)
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Here is how superposition
works:
Each photon has two
possible slits to pass
through - left and right.
We call each possibility a
state and since we don’t
know which state the photon
is in, we say it is in a
superposition of states.
One way to understand the
idea of superposition is via a
famous example suggested
by Erwin Schrödinger (18871961)!
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Superposition (cont.)
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Suppose we have a (living) cat, a box,
and a vial of cyanide.
There are two possible states for the cat
- dead or alive.
Initially, the cat is in one of the two
possible states, namely alive.
Put the cat and vial of cyanide in the box
and close the lid.
Until we open the lid, we cannot see or
measure the state of the cat.
Quantum theory says the cat is in a
superposition of two states - it is both
dead and alive.
Superposition occurs when we lose sight
of an object and is a way of describing a
period of ambiguity.
Once we open the box, and look at the
cat, superposition disappears and the cat
is forced into one of its possible states.
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Many-Worlds
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The other way to describe what is
going on with the modern Young’s
experiment is via the many-worlds
interpretation!
Once the photon leaves the light
source, since it has two possible
slits to pass through, the universe
splits into two universes.
In one universe, the photon goes
through the left slit.
In the other universe, the photon
goes through the right slit!
These two universes somehow
interfere with each other and
produce the striped pattern of light
and dark stripes.
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Many-Worlds (cont.)
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In the many-worlds theory, any
time an object has the potential
to enter one of several possible
states, the universe will split
into many universes, one for
each potential state.
The huge number of universes
produced is called the
multiverse.
In the Star Trek Original Series
episode “Mirror, Mirror”, we see
an example of this
phenomenon - there are two
different universes - one where
the Federation is “good” and
another where the Federation is
“bad”.
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Uses of Quantum Mechanics in
the World Today
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Although strange and counterintuitive, many
phenomena in the world owe their
understanding or existence to quantum theory!
Examples include:
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Describing the modern version of Young’s experiment.
Calculating consequences of nuclear reactions in
power stations.
Explaining how DNA works.
Understanding how stars such as our Sun work.
Designing lasers for CD players.
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The Quantum Computer
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Another consequence of
quantum mechanics is the
possibility of a quantum
computer!
In 1985, British physicist David
Deutsch published a paper
outlining how a computer might
work according to the laws of
quantum mechanics instead of
classical physics.
Such a computer would have to
work at the level of fundamental
particles for the quantum
effects to manifest themselves.
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The Quantum Computer (cont.)
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So how would a quantum
computer differ from a classical
computer (i.e. the kind we use
right now)?
Suppose we have two versions
of a question, say version 1 and
version 2.
To answer the question with a
classical computer, we would
have to perform the following
sequence of operations:
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Input version 1 and wait.
Get an answer.
Input version 2 and wait.
Get an answer.
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The Quantum Computer (cont.)
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For a quantum computer, we
do the following:
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Combine the two questions
as a superposition of two
states, one for each
question.
Input this superposition to
the computer, which causes
the computer to enter a
superposition of two states,
one for each question, and
wait.
Get the answer to both
questions at the same time!
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The Quantum Computer (cont.)
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As an illustration of how powerful a quantum computer could be,
suppose we wish to answer the question: Find the smallest positive
integer whose square and cube use up all the digits 0-9 once and only
once.
For a classical computer, we’d need to do the following:
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1 => 12 = 1 and 13 = 1
2 => 22 = 4 and 23 = 8
3 => 32 = 9 and 33 = 27
…
69 => 692 = 4,761 and 693 = 328,509
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YES
If each computation took one second, it would take 69 seconds to
arrive at this answer.
For a quantum computer, instead of testing one integer at a time, we
could test many at once via superposition of states (or computation in
many universes)!
Thus, assuming one second per computation, we could get our answer
69 times faster with a quantum computer!
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The Quantum Computer (cont.)
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In order to ask many questions at once, we need a way to
represent data at a quantum level.
Just as with classical computers, we can use 0’s and 1’s the
represent numbers in binary.
One way to do this is via the spin of a fundamental particle
(such as an electron).
Many fundamental particles possess an inherent spin - they
spin either “west” (clockwise) or “east” (counterclockwise).
Thus, we could identify westward spin with 0 and eastward spin
with 1, so for example, seven particles with spins (in order):
east, east, west, east, west, west would represent the binary
number 110100 (decimal number 104).
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The Quantum Computer (cont.)
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With seven particles, we then would be able to represent
any number between 0 and 127.
Using a classic computer, we would have to enter each of
the numbers (one at a time) as a string of seven spins
(i.e. strings of 0’s or 1’s).
For a quantum computer, we would enter all 128
numbers at once as a superposition of the 128 different
states (one per number).
A natural question to ask is: “How do we achieve this
superposition?”
The key to achieving superposition is that until we
observe a spinning particle, it could be spinning east or
west, so it is in a superposition of the two states.
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The Quantum Computer (cont.)
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Suppose we observe a particle and it
is spinning west.
We can change its spin by adding a
sufficient amount of energy to the
particle.
If we add less energy, then the
particle may change spin or may stay
the same.
Using the idea of Schrodinger’s cat,
put the west spinning particle in a box,
close the lid, and add a little bit of
energy to the particle.
Until we open the box, we won’t know
the particle’s spin, so the particle has
entered a superposition of the states
east and west.
By performing the same operation
with seven particles, we will achieve a
superposition of the 128 possible
states!
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The Quantum Computer (cont.)
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In a traditional computer, a 0 or a 1 is
called a bit, which is short for “binary
digit”.
Since a quantum computer deals with a
superposition of a 0 and a 1, we call
such an object a qubit, which is short for
“quantum bit”.
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Applications of Quantum
Computing (cont.)
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To give an even better idea of how powerful a quantum
computer could be, suppose we were able compute with
250 qubits.
Using the Fundamental Principal of Counting, the number
of states in a superposition of these 250 spinning
particles would be 2250 which is about equal to 1.8 x 1075
different states (more than the number of particles in the
universe)!
Thus a quantum computer could perform over 1075
simultaneous computations in a short amount of time!
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Applications of Quantum
Computing (cont.)
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With this much computing
power, one big application
would be to factor large
numbers quickly.
In 1994, Peter Shor of AT&T
Bell Laboratories figured out
how to program a quantum
computer to factor a number
larger than a 129 digit
number in a short amount of
time (approximately 30
seconds).
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Applications of Quantum
Computing (cont.)
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Here is why this is significant:
In 1977, Scientific American
columnist Martin Gardner wrote an
article entitled “A New Kind of
Cipher that Would Take Millions of
Years to Break” that announced the
discovery of RSA cryptography to
the world.
In this article, he published a
message encrypted with a 129 digit
public key and offered a $100 prize
to the first person to decrypt the
ciphertext.
In 1994, 17 years later, a team of
600 volunteers, using
supercomputers and workstations,
announced that they had found the
factors of the public key and were
able to decipher the message!
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Applications of Quantum
Computing (cont.)
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Thus, a quantum computer can be used to very quickly
crack RSA, which is what many people throughout the
world rely on for transmitting messages securely!
Another cryptography-related use of quantum computers
is to search lists at high speed.
Currently, another cryptographic scheme in use is DES,
which relies on keys that, checking possible keys at a rate
of one million per second, would take over 1000 years to
crack.
In 1996, Lov Grover, also at Bell Labs, found a way to
program a quantum computer to find a DES key in about
four minutes!
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Drawbacks to Quantum
Computing
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So, if quantum computers are so great, what could
possibly be wrong with them?
One major issue is that we don’t know how to make one!
A lot of money has been invested into quantum computer
research by government agencies, such as DARPA
(Defense Advanced Research Projects Agency), but as
Serge Heroche of the University of Paris IV put it (in
1998):
Based on what we know about quantum computer
technology, building one right now would be like carefully
building the first layer of a house of cards and assuming
that the next 15,000 layers are a mere formality!
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References
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The majority of this talk is based on material from Chapter 8 of
The Code Book by Simon Singh, 1999, Anchor Books.
http://www.psd267.wednet.edu/~kfranz/Science/WaterHabitat/p
hotojrnlmar00.htm
http://micro.magnet.fsu.edu/optics/timeline/people/young.html
http://www-groups.dcs.stand.ac.uk/~history/PictDisplay/Schrodinger.html
http://en.wikipedia.org/wiki/Mirror,_Mirror_(Star_Trek)
http://www.qubit.org/people/david/
http://math.mit.edu/~shor/
http://www.csicop.org/si/9803/gardner.html
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