The Mathematics of Star Trek

The Mathematics of Star Trek

Lecture 12: Quantum Computing

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Topics 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 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.) 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.) 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.) 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 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.) 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.) 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.) 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 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:• 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.) 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 (1887-1961)!

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Superposition (cont.) 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 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.) 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 Although strange and counterintuitive, many

phenomena in the world owe their understanding or existence to quantum theory!

Examples include:• 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 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.) 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:• Input version 1 and wait.• Get an answer.• Input version 2 and wait.• Get an answer.

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The Quantum Computer (cont.) For a quantum computer, we

do the following:• 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.) 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:• 1 => 12 = 1 and 13 = 1 NO• 2 => 22 = 4 and 23 = 8 NO• 3 => 32 = 9 and 33 = 27 NO• …• 69 => 692 = 4,761 and 693 = 328,509 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.) 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.) 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.) 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.) 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.) 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.) 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.) 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.) 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 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 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.st-and.ac.uk/~history/PictDisplay/Schrod

inger.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