Download Quantum Computers - Computing Sciences

Quantum Computers
By
Andreas Stanescu
Jay Shaffstall
Quantum
Computers:
Overview
*
*
*
Quantum Mechanics
Quantum Algorithms
Future Applications
QC: Quantum mechanics
*
*
*
*
Max Planck first described light quanta
Light travels as a wave but arrives as a particle
Feynman's QED
His key observation is that we must consider
every path from A to B!
QC: Experiment with two slits
*
*
*
*
Understanding the experiment and the results
Light interference
The probability of a photon going through every
single path is resolved (or collapsed) once the
photon is observed
Niels Bohr first described the quantum behavior
using his famous Copenhagen Interpretation
QC: another experiment
*
*
*
Called delayed-choice experiment
Instead of counting photons at the slits, place
detectors behind the slits but before the screen
Look to see if the photons are behaving like
particles or like waves after they had passed the
slits but before they hit the far screen
QC: another experiment (cont.)
*
*
The behavior of the photons is changed by how
we are going to look at them, even when we
haven't made up our minds how we will be looking
at them!
It is as if photons know in advance exactly how
the world will be when the light makes it there
QC: Einstein's puzzles
*
*
*
Heisenberg proved the Uncertainty Principle
We can either measure the position or the
momentum of a quantum object but not both at
the same time
"I cannot believe that God plays dice with the
Universe."
QC: Einstein's puzzles
*
*
*
*
*
Polarization
Entanglement
Non-locality principle
"Spooky action at a distance."
Communication at speeds greater than the speed
of light is not allowed by the Law of Relativity
QC: Quantum Algorithms
*
*
General Concept
Shor’s Factorization Algorithm
QC: Quantum Algorithms
*
General Concept
A single bit in a normal computer can hold a 0 or
a 1.
A single quantum bit can hold a 0, a 1, or both at
the same time. This is called superposition.
QC: Quantum Algorithms
*
We indicate the value of a superposition like this:
|0> + |1>
This notation just means that the quantum bit
contains the values 0 and 1 at the same time.
QC: Quantum Algorithms
*
*
Just like in the two slit experiment, where
measuring the photons collapsed the
probabilities, measuring a quantum bit collapses
the bit into only one value.
Which value is measured is random. So
measuring our quantum bit that contains the
values 0 and 1 will result in the bit taking on the
value of either a 0 or a 1.
QC: Quantum Algorithms
*
*
Using entanglement, we can create a memory
register containing multiple quantum bits.
Entanglement allows the bits to interact so that
some values are excluded. For example, a two
bit register might contain the values 1 and 3 at
the same time, but not the values 0 or 2.
QC: Quantum Algorithms
*
So we can imagine a quantum register that
contains the values 2 and 5 at the same time. If
the register is a 3-bit register, we would represent
that superposition as:
|010> + |101>
*
Entanglement ensures that those are the only
combinations of bits that can occur.
QC: Quantum Algorithms
*
Now, imagine subtracting one from the value in
our quantum register. The register contains the
values 2 and 5 in superposition, so by subtracting
one we would get the values 1 and 4 in
superposition:
|001> + |100>
QC: Quantum Algorithms
*
We have, in effect, performed two calculations for
the cost of one.
Now imagine having a quantum register that
contains thousands of values in superposition,
and performing a thousand calculations for the
cost of one.
QC: Quantum Algorithms
*
*
Quantum interference is the last piece to what
makes quantum algorithms possible.
Interference creates a relationship between two
quantum memory registers.
For example, consider our memory register that
contains the values 2 and 5. We again subtract
1 from that register, but now we place the result
in a second register.
QC: Quantum Algorithms
*
*
*
Register 1 still contains 2 and 5, while register 2
now contains 1 and 4.
If we measure the value in register 2, the
superposition will collapse randomly into either a
1 or a 4.
Let’s say that it collapses into the value 4.
QC: Quantum Algorithms
*
*
*
Quantum interference will cause register 1 to
take on a subset of its values that are consistent
with the measured value in register 2.
So in our example, register 1 would now contain
the value 5, even though we did not measure
register 1.
We’ll see more of this in the factorization
algorithm.
QC: Quantum Algorithms
*
*
That is the basic idea of how a quantum computer
can do so much work in a short period of time.
It’s like having a powerful supercomputer that can
add cpus whenever it needs them.
Parallel algorithms for supercomputers are,
however, easy to understand compared to
quantum algorithms.
QC: Quantum Algorithms
*
Shor’s Factorization Algorithm
The problem is to determine the prime factors of a
large number. With a normal computer, this is an
exponential algorithm. Parallel programming
helps make the running time faster, but the
problem is still exponential.
QC: Quantum Algorithms
*
*
In April 1994, Peter Shor at Bell Labs in New
Jersey discovered a way to use a quantum
computer to factor large numbers in polynomial
time.
Why do we care? If large numbers can be
factored easily, data encryption can be broken.
Data encryption depends on the difficulty of
factoring large numbers.
QC: Quantum Algorithms
*
Shor’s algorithm depends on some math.
If N is the number we want to factor, and X is a
prime number that is not a factor of N, we can
write a function:
F(a) = Xa mod N
QC: Quantum Algorithms
*
*
This function is periodic, which means that if you
put successive values of a into the function, every
r values you’ll get the same result.
Shor discovered that if we can find the period r of
the function, we can find the factors of N.
QC: Quantum Algorithms
*
*
Finding the period of a function is, unfortunately,
also an exponential problem. So this insight does
not do normal computers any good.
Shor also came up with a way to use quantum
computers to find the period of a function. That is
the heart of Shor’s factorization algorithm.
*
*
Shor’s algorithm uses two quantum memory
registers. These registers will be as big as is
needed for the size of the number to be factored.
We place into register 1 a superposition of all
possible values for the register. This represents
the values of a we want to test in the formula:
F(a) = Xa mod N
QC: Quantum Algorithms
*
For example, let’s say that register 1 is two qbits.
It will hold the values 0, 1, 2, and 3 all in
superposition.
QC: Quantum Algorithms
*
Now we use register 1 as the value of a, and
calculate Xa mod N.
*
The results of this calculation are placed into
register 2.
QC: Quantum Algorithms
*
Back with our example values, let’s say that the
number we wish to factor, N, is 25, and X is 3.
Performing the calculation with all values of a
results in:
1, 3, 9, 2, and 6
*
All these values are now in register 2
QC: Quantum Algorithms
*
*
Now it starts to get complicated. We measure the
contents of register 2.
This results in register 2’s superposition
collapsing into a single value, because any time
you measure a quantum superposition you get
only one of the values out of it.
QC: Quantum Algorithms
*
The quantum interference effect also changes the
contents of register 1 to be consistent with the
value we measured in register 2. This means that
if we measured some value V in register 2,
register 1 will now contain all the values of a for
which Xa mod N = V.
QC: Quantum Algorithms
*
Remember that the function is periodic, with a
period of r, so if the first value of a that results in
the value V is represented as C, register 1
contains all values of a equal to C, C+r, C+2r,
C+3r, etc.
QC: Quantum Algorithms
*
*
That’s the heart of Shor’s factorization algorithm.
Now that we have the contents of register 1, we
can perform some mathematical tricks to come up
with the value of r, the period of the function.
Some additional mathematical tricks with r will
result in calculating the factors of N.
QC: Quantum Algorithms
*
*
*
The key to Shor’s algorithm is that he discovered
a way to perform a specific exponential problem
in polynomial time on a quantum computer.
This allows us to factor extremely large numbers,
such as those used in public key cryptography
systems.
Assuming we had a quantum computer.
QC: Future Applications
*
*
We are still years away from practical quantum
computers, but many scientists are already
coming up with applications for quantum
computers.
Within the next thirty years, here are some of the
applications for which we may be using quantum
computers.
QC: Future Applications
*
Quantum Cryptography
Since a quantum computer can break any
conventional cryptography scheme by factoring
the key, it only seems fair that a quantum
computer can provide a different way of
encrypting data.
QC: Future Applications
*
*
The essence of quantum cryptography is a way of
transmitting a cryptography key without allowing
anyone else to listen on the line.
Because of the nature of quantum superposition,
if a spy was measuring the key while it was being
transmitted, the receiver would be able to tell that
someone else had looked at the key first.
QC: Future Applications
*
The sender and receiver could then abandon the
key and try a different one (after locating the spy,
of course).
QC: Future Applications
*
Quantum Teleportation
This does not mean transmitting people from place
to place (it may eventually, but we are far more
than thirty years away from that).
QC: Future Applications
*
*
Quantum teleportation does mean transmitting a
quantum superposition without sending the
quantum bit through a network.
This allows information to be safely transmitted
from point to point without any possibility of
another person listening.
QC: Future Applications
*
*
Quantum teleportation does require a
conventional network, since both sides of a
communication must exchange some information.
But the quantum information itself simply moves
from the sender to the receiver without using the
network.
QC: Future Applications
*
Quantum Simulation
Imagine having a computer that you can ask to
simulate any physical process exactly. You could
determine what the results of an experiment
would be without actually running the experiment.
QC: Future Applications
*
*
Quantum simulation is what physicists are most
interested in. Conventional computers simply
cannot simulate physical processes realistically,
but a sufficiently large quantum computer could.
It probably won’t allow you to predict the stock
market, but it may very well allow you to know
what the weather will be for the next month.
QC: Future Applications
*
More importantly, it will allow physicists to
simulate experiments that would be impossible to
actually conduct.
QC: Future Applications
*
These have been the more realistic applications
we can expect from quantum computers. As
scientists continue working in this field, we can
expect more and more futuristic applications to
arise.
QC: Summary
*
*
*
Quantum Mechanics
Quantum Algorithms
Future Applications
QC: Presentation
*
This presentation is available at:
http://cs.franklin.edu/~shaffsta/quantum.ppt
QC: Bibliography
*
*
*
*
Brown, Julian (2000) Mind, Machines, and the
Multiverse: The Quest for the Quantum
Computer; Simon & Schuster
Hayward, Matthew (1999) Quantum Algorithms;
http://www.imsa.edu/~matth/cs299/node19.html
Gribbin, John (1984) In Search Of Schrodinger's
Cat; Bantam Doubleday
Gribbin, John (1996) Schrodinger's Kittens And
The Search For Reality; Little, Brown, &
Company
QC: Bibliography
*
*
*
Graham P. Collins (1999) QUBIT CHIP; Scientific
American,
http://www.sciam.com/1999/0899issue/0899scici
t5.html
A. Barenco et al (1996) A Short Introduction To
Quantum Computation; Centre For Quantum
Computation,
http://www.qubit.org/intros/comp/comp.html
Artur Ekert (1995) What Is Quantum
Cryptography; Centre For Quantum Computation
http://www.qubit.org/intros/crypt.html