quantum computers

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


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.


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.


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.


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.


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.


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>


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.


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.


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.


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.


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.


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.


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.


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


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.


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


For example, let’s say that register 1 is two qbits. It will hold the values 0, 1, 2, and 3 all in superposition.


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.


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


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.


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.


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.


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.


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.


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.


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.


The sender and receiver could then abandon the key and try a different one (after locating the spy, of course).


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).


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.


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.


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.


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.


More importantly, it will allow physicists to simulate experiments that would be impossible to actually conduct.


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/0899scicit5.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

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