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Simulation of Quantum Computation on

Intel-Based Architectures

Yan Pritzker

ypritzker@openqubit.org

http://www.openqubit.org

Revision A

...trying to nd a computer simulation of physics, seems to me to be an excellent

program to follow out...and I’m not happy with all the analyses that go with just the

classical theory, because NATURE ISN’T CLASSICAL, dammit, and if you want to

make a simulation of nature, you’d better MAKE IT QUANTUM MECHANICAL, and

by golly it’s a wonderful problem because it doesn’t look so easy.

-Richard P. Feynman (1981) (International Journal of Theoretical Physics, Vol. 21,

p. 486)

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Contents

1 Introduction to Quantum Mechanics

3

1.1 The Wave-Particle Duality of Matter . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.1.1 Young’s Double-Slit Experiment . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.1.2 The Discovery of the Electron . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.1.3 Blackbody Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.1.4 The Photoelectric Eect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.1.5 The Wave-Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.1.6 The Double-Slit Experiment, with Electrons . . . . . . . . . . . . . . . . . . .

7

1.2 Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.3 Philosophical Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.3.1 The Copenhagen Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.3.2 Many Worlds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.4 Dirac Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Quantum Computers

12

2.1 Inside the Quantum Computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.1.1 The Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.1.2 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.1.3 Quantum Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.2 Quantum Computing Pros/Cons . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.2.1 Usefulness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.2.2 Feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.2.3 Quantum Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.3 Quantum vs. Classical Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.3.1 Eciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.3.2 Shor’s and Grover’s Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Quantum Computer Simulation

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3.1 OpenQubit Model Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.1.1 OpenQubit Main Class Diagram . . . . . . . . . . . . . . . . . . . . . . . . .

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3.1.2 Sample Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 OpenQubit Documentation

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About the OpenQubit Project

This project began as a hobby but quickly got out of hand. The idea for this project came to me

after I had read several books on quantum physics several months ago. I was always interested in

the subject, and my honors chemistry class sophomore year gave me some foundation to understand

it. Several of the books mentioned quantum computers and simulation thereof. I was disappointed

to nd out that quantum computers wouldn’t be realized physically for several decades, but the

prospect of simulation of such computers was very exciting.

I didn’t have the mathematical or physical background to even begin to understand quantum

computing, so I decided to nd help in the Linux community. Linux is a free operating system which

has been historically used by mostly programmers.

I founded the OpenQubit organization on December 9, 1999 when I posted an article to a large

news site on the Internet (slashdot.org, which gets more than a million hits a month), stating my

intentions to create a quantum computer simulation. I was immediately ooded with requests for

more information. I organized a mailing list which in the rst day gained eighteen members, and

within a week was well over 100. Currently, there are 183 subscribers to the list.

The list became a place for physicists, computer scientists, hobbyists, and laymen to discuss

a multitude of topics. After talking with some of the physicists, I began to understand quantum

mechanics and quantum computing much more. I started writing some basic code for the simulator,

and within several days released it to the mailing list. Several other programmers on the list helped

me with some other code. Everyone worked on a piece of the simulator. The computer scientists

(myself and about ve others) would write some code, post it to the list, and then the quantum

physicists would tell us where we were wrong and where the code didn’t make sense. On February

9, 1999 we nally had a working, stable simulator. It could run Shor’s factoring algorithm, which

was one of my original goals for the project. Two days later, we released the code to the Linux

community through another news site (freshmeat.net).

Currently, the OpenQubit software is in its third revision (series 0.3.x). An article about quan-

tum computing, which will mention OpenQubit, will be published in PC World in May of 1999. The

OpenQubit web page is located at

http://www.openqubit.org

(Denmark mirror at

http://dk.openqubit.org

)

and the latest source code can be downloaded from

http://www.openqubit.org/code/devel

.

Although the goal of this paper is to introduce the concepts of quantum computing, a rm

understanding of the basic concepts of quantum theory is necessary in order to comprehend the

nature of quantum computers. A kind of summary of quantum theory is packed into the rst ten

or so pages in the rst section of this paper.

1 Introduction to Quantum Mechanics

Anyone who is not shocked by quantum

theory has not understood it.

-Niels Bohr

Quantum theory is, without question, the one greatest achievement of 20th century physics. It is far

more signicant, direct, and practical than perhaps its rival in greatness, the theory of relativity. Yet

it makes some very strange predictions. So strange, in fact, that even Albert Einstein found them

incomprehensible and refused to accept the implications of quantum theory, spending a large part of

his life trying to nd a way around it. Einstein, like many physicists of the time thought quantum

theory to be nothing more than a mathematical trick which happened to explain the workings of

atomic and subatomic particles, but had no real physical signicance (Gribbin, 1984).

In the world of quantum mechanics, nothing is real until an observation is made, and one cannot

make statements about things while one is not observing them. Events are governed by probabilities;

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a radioactive atom, for example, might decay or it might not. One cannot say what will and will

not happen, but can only talk about probabilities of these events occurring. Even Schroedinger, one

of the founders of quantum theory, was upset at its implications. He tried to show the absurdity of

quantum theory by setting up a thought experiment which would tie together the microscopic world

of quantum mechanics with the macroscopic world (Gribbin, 1984).

In this experiment, a radioactive atom which had exactly a fty percent chance of decaying or

not was placed in a room with a radioactivity detector (a Geiger counter) which when set o would

break a vial of poison, releasing it into the room which contained a live cat. Quantum theory says

that while the atom is not observed, it has neither decayed nor not decayed, and thus the detector

has neither been set o nor not set o. Consequently, the cat neither lives nor dies. If one accepts

quantum theory and all of its implications, then the cat in the room is neither dead nor alive until

someone looks into the room and observes it. Thus, nothing is real until it is observed. This may

be a bold claim to make, but this theory nonetheless accurately explains scientic observations.

The next several sections will introduce the reader to the concepts of quantum mechanics, and why

scientists today believe that quantum theory is a valid explanation for everyday events (Gribbin,

1984).

1.1 The Wave-Particle Duality of Matter

Newton thought that light was made of particles, and he had good reason to believe so. Rays of light

were observed to travel in straight lines, as particles do. Light also bounces o of a mirror much

like a ball from a wall. However, a contemporary of Newton, Dutch physicist Christiaan Huygens

developed the idea that light was not made of particles, but was rather a wave, moving through a

substance he called the \lumeniferous ether." This wave theory explained refraction and reection

just as well as the particle theory did. However, certain observed eects could only be explained

by the particle theory, so the wave theory was forgotten and discarded. By the early nineteenth

century, however, the status of the two theories had become almost completely reversed (Gribbin,

1984; Guillemin, 1968).

1.1.1 Young’s Double-Slit Experiment

In 1801, British Physicist Thomas Young designed an experiment that would test whether or not

light propagates in the same manner as a wave of, say, water does. Let us examine the nature of

water waves in order to understand his experiment better. Suppose one drops two stones into a pond.

The stones produce tiny waves, or ripples, in the water. When they are dropped close together, the

ripples of one stone interact with those from the other. Because of the oscillating nature of a wave,

it happens sometimes that two crests meet and create a bigger crest (or two troughs to create one

trough equal in amplitude to the sum of the two). Other times, a crest from one wave meets the

trough of the other. In this case, the amplitudes of the waves cancel leaving no wave in their place.

This eect of adding and canceling amplitudes is known as

interference

(Gribbin 1984; Guillemin,

1968).

Young decided to test the wave nature of light by setting up an experiment much like the dropping

of the stones in the water. He placed a wall with two tiny slits in front of a detector screen. At this

wall, he directed a light beam. Because of the eects of diraction, light on the other side of the wall

would spread from each slit in semicircles. Young found that the pattern observed on the detector

screen when light was passed through the slits was an interference pattern much like that of the

rocks in the water, which appeared on the detector screen as a series of light and dark bands. Thus,

Young proposed that light must be a wave since a clear interference eect was observed between the

light coming from the two slits (Gribbin, 1984).

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There was a remaining question however, if light was a wave, what was the medium through which

it propagated? For example, water waves use water as a medium, and radio waves propagate through

air. The wave theory seemed to be at last completed when James Clerk Maxwell established the

existence of waves as uctuating electromagnetic elds. This answered the question of what exactly

is \waving" in a beam of light. It was the electromagnetic uctuation that gave light its wave-like

characteristics (Gribbin, 1984).

1.1.2 The Discovery of the Electron

In the late 1800s, studies were being done on the discharge of electricity through rareed gases.

During these studies, it was discovered that when very high voltage was put through gases, a beam

of something appeared to travel from one end of the gas chamber to another. These beams were

named

cathode rays

because they appeared to originate at the cathode, the negative electrode in the

gas chamber

.

At the time, no one really knew what these rays were, but merely that they existed.

Some scientists thought that the rays may be a form of light. However, when magnetic elds were

brought near the spot the beam made on one end of the chamber, it was deected. This suggested

that cathode rays could be charged particles. The direction of their deection was consistent with a

negative charge. Cathode rays soon became recognized as beams of particles which today are called

electrons

(Guillemin, 1968).

1.1.3 Blackbody Radiation

In the early 1900s, many scientists were attempting to explain

blackbody radiation

which is basically

the phenomenon of hot objects glowing with dierent colors. At this time, the best scientic view of

the world required that material objects must be described in terms of particles, but electromagnetic

radiation, including light, must be described in terms of waves. Scientists sought to unify the wave

and particle theories by looking at how radiation interacts with matter. But instead of unifying,

classical physics broke down almost completely (Gribbin, 1984).

A hot object radiates electromagnetic energy, and the frequency of the radiation emitted increases

with temperature. Thus, a warm piece of iron emits infrared radiation, invisible to the eye, which

gradually increases towards the frequency of red light (at which point we call the object red-hot)

and then towards blue and ultraviolet. The electron had only been recently discovered, but it was

quite easy to see how a charged part of the atom such as the electron could vibrate and produce a

stream of electromagnetic waves. At higher temperatures, it would oscillate more rapidly and thus

produce higher frequency waves. The only problem with this theory, a combination of the best of

classical theories{statistical mechanics and electromagnetism{was that it predicted a very dierent

form of radiation than was actually observed coming from hot objects (Gribbin, 1984; Guillemin,

1968).

The classical theory predicted that the radiation given o by hot objects would approach innity

as the temperature of the object increased and drop o at zero at extremely low frequencies (low

temperatures). This was not the case however; instead of the radiation becoming more and more

ultraviolet at high frequencies, it dropped to zero after a certain cuto point. Thus the predictions

of the theory were incorrect and this dilemma is sometimes called the \ultraviolet catastrophe"

(Gribbin, 1984).

The theory was not a complete failure, however, as it could explain things at low frequencies. At

least the classical theory was half right. The reason for radiation dropping o to zero at high fre-

quencies was somewhat of a mystery. This puzzle attracted many physicists including Max Planck,

a conservative scientist whose primary interest was thermodynamics. In 1900 he made a mathemat-

ical discovery through desperation, luck, insight, and a slight misunderstanding of the mathematical

tools he was using. Planck’s formula explained the curve of blackbody radiation, but no one could

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explain the physical assumptions that went with the formula. In other words, the solution to the

problem was known, but nobody knew why it was the solution. After some more work, Planck

found the reason behind the \ultraviolet catastrophe" in what became known as

Planck’s quantum

hypothesis

(Gribbin, 1984).

Planck said that energy inside the atom could only be emitted in lumps of a certain size, called

quanta, which could be described quite nicely by the equation

E

=

h

where

h

is what we now call

Planck’s constant and is the frequency. Planck explained that for very high frequencies, the energy

needed to emit one of these energy packets was very large, and thus only a few of these quanta were

emitted. This, of course, explains why at high frequencies there was not a lot of ultraviolet radiation

given o, but rather the radiation level dropped to zero (Gribbin, 1984).

1.1.4 The Photoelectric Eect

In 1899, Phillip Lenard showed that electrons can be produced by light shining onto a metal surface

in a vacuum. This was called the

photoelectric eect.

Lenard looked at the way the intensity of the

light aected the way electrons were stripped from the metal. He hypothesized that if he used a

brighter light, more energy would shine on the surface of the metal, and thus electrons should be

knocked o the metal more rapidly and y with a greater velocity. However, he found that as long

as the wavelength of the light remained unchanged, the electrons ew o with the same velocity.

Only the number of electrons ejected increased (Gribbin, 1984; Guillemin, 1968).

Einstein realized that there was a simple way to explain this, provided that Planck’s equations

were taken as physically meaningful. Einstein simply applied the equation

E

=

h

to electromagnetic

radiation. He said that light was

not

a wave, contradicting scientic thought for hundreds of years.

He said that light comes in packets of energy (quanta) just as Planck proposed the oscillations inside

the atom do. Every time this quanta of energy hits an electron, it gives it the same amount of energy

and therefore the same velocity. Increasing the brightness of the light means simply increasing the

amount of light quanta emitted and thus increasing the ejected electron count. These light quanta

are known today as

photons.

This was the work for which Einstein received his Nobel Prize in 1921

(Guillemin, 1968).

1.1.5 The Wave-Particle

So, there seemed to be two conicting theories. Young’s experiments seemed to show that light

was a wave, and yet Einstein found that light was made of particles, or quanta. For some time,

many scientists struggled with this problem but no one could explain how it could be possible that

light was sometimes a wave and sometimes a particle. Physicists nally gave up and came to the

conclusion that this

wave-particle duality

simply exists and it must be accepted (Gribbin, 1984).

Niels Bohr proposed the famous

principle of complementarity

which states that one cannot

observe both the wave and particle nature of an object at the same time. If one is performing an

experiment to test the wave nature of light, he or she will verify it to be a wave. If, on the other

hand, one is testing for particle properties, those are the only properties he or she shall observe

(Gribbin, 1984).

Louis deBroglie later generalized that all matter has both wave and particle properties and came

up with mathematical ways of describing the wavelength of any object. Even macroscopic objects

such as tables and chairs have a wave nature; their wavelength is just so tiny that we do not observe

them \waving." However, with microscopic objects such as electrons and other small particles, the

wave nature is quite visible (Gribbin, 1984).

Although some scientists seemed bothered by the idea that light should behave sometimes like

a particle and sometimes like a wave, there really is nothing to worry about. Nature seems to have

no problem with light being both a particle and a wave. It is only because of the way we visualize

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things that we are confused when we are told that light is both particle and wave in one. There is

no reason for light, electrons, or anything in nature to be like anything that we can visualize, or any

model that we can invent. So we must simply accept this existence. In fact,

electrons do not really

exist, but they could exist, and it is this potential for existence that we call \an electron

" (Gribbin,

1984).

1.1.6 The Double-Slit Experiment, with Electrons

Now, as a conclusion to the wave-particle duality principle we shall examine what happens when

we perform Young’s double slit experiment with electrons. This experiment really demonstrates

quantum theory in its full glory, and Feynman noted that if one comes to terms with the mysteries

of this experiment, then every other situation in quantum mechanics can be explained by referring

back to this simple, yet interesting experiment (Feynman, 1965).

Recall Young’s double-slit experiment discussed in (1.1.1). First let us imagine what happens

when we re bullets, as suggested by Mr. Feynman, at a double-slit setup. Bullets, are of course

particles and thus can travel through either one hole or the other. We will place a wall on the other

side of the slits, and a detector box on this wall that will count how many bullets land inside it.

Thus, if we count the total bullets that passed through the hole and the bullets inside the box, we

can calculate the probability that a bullet will land in the detector box (Feynman, 1965).

When we cover one of the holes, we can determine the probability

P

1

that the bullet will go

through the uncovered hole and hit the detector. Doing the same with the other hole, we discover a

probability

P

2

for the bullet to hit the detector by going through the second hole. If we now open

up both holes, we nd that the probability of nding a bullet in the detector is simply the sum of

the individual probabilities:

P

12

=

P

1

+

P

2

(1)

Now we will set up a similar experiment with electrons instead of bullets. An electron gun will

be the source of electrons. In front of the detection screen we place an electron detector that will

make a \click" on a loudspeaker, for example, when an electron arrives. We will also place a detector

box as in the bullet experiment to count the probability of an electron landing in there (Feynman,

1965).

When we run this experiment we notice two things. First, we always hear a click when an electron

arrives. We always hear a full click, and no half-clicks, for example. We conclude that whatever

arrives at the detection screen comes in lumps, or quanta so it must be indivisible, consistent with

the particle theory (Feynman, 1965).

Now we close one hole, obtain the probability of the electron striking the detector box and do

the same with the other hole. When we open both holes we notice that something strange has

happened. The overall probability

P

12

is

not

the sum of the individual probabilities! This eect

comes from interference. If we treat the electron as a wave instead of a particle, then we can no

longer talk about the probability of the electron arriving at the detector. Instead, we can talk about

the amplitude of the probability wave. In quantum mechanics, this probability wave is called the

\wavefunction" of a particle. Standard convention is to label the amplitude of the wave as (\psi")

or (\phi"). To obtain the probability of the electron arriving at the detector, we must square the

wavefunction. Thus,

P

1

=

j

1

j

2

and

P

2

=

j

2

j

2

. Now, the sum of the probabilities is the sum of

the squares of the amplitudes:

P

12

=

j

1

+

2

j

2

=

2

1

+

2

2

+ 2

1 2

(2)

There is a third term introduced, 2

1 2

which is the term that causes the interference. One can

see that if one of the waves is negative (at a trough), the probability sum will be decreased, and if

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two troughs or two crests meet, then the probability will increase at that point. How could it be,

though, that the electron \clicked," on the detector, registering as a particle, yet still managed to

interfere with itself like a wave? (Feynman, 1965).

The answer to this is simple, yet strange, and that is the world of quantum mechanics that

we must accept. The electron

sometimes appears as a particle, and sometimes appears as a wave

.

When we hear it click, we are measuring the particle nature of the electron. When we observe the

interference pattern, we see that the electron traveled as a probability wave through both holes.

This apparent state of being in both holes at once is called a \superposition of states" (Feynman,

1965; Gribbin, 1984).

Perhaps there is a way to \trick" the experiment. What if we place some sort of detector at

the holes so that we can actually observe which hole the electron goes through. We may place a

light source at the holes so that we may \see" the electrons coming through. We nd, much to our

surprise, that the electron now acts just as an ordinary particle, and does not form an interference

pattern. It acts like a bullet instead, and the resulting probability distribution on the detection

screen is the simple sum of the individual probabilities. What we have done by observing the

electron is collapse its wavefunction

,

restricting its probability wave to a certain subspace. In this

case, we have determined the hole that it went through, and thereby turned the wave into a particle

because there are no more choices to be made. The reason for this quantum trickery is called the

indeterminacy

or

uncertainty

principle (Feynman, 1965; Gribbin, 1984).

1.2 Uncertainty

Quantum theory places a limitation on the accuracy of measurement. Werner Heisenberg originally

stated that if we know an object’s momentum

p;

we cannot know its position,

x

more accurately

than

x

=

h= p;

where

h

is a constant called Planck’s constant and equals approximately 6

:

63

10

34

J s

. The units (Joule-seconds) seem strange but they are known as \actions" and are not

everyday features of classical physics. An action has an interesting property{it is absolutely constant

and has the same size for any observer (this property is shared by the speed of light). The signicance

of actions became apparent only after Einstein’s theory of relativity because it turns out that actions

are conserved in space-time. Heisenberg’s uncertainty principle talks about the uncertainty of any

action measurement (such as that of momentum and position, or energy and time) (Gribbin, 1984).

Basically, Heisenberg said that if we know an object’s position with innite accuracy, we have

absolutely no idea about its momentum (i.e. how fast it is moving). Similarly, if we are very sure of

its momentum, we have no idea where it is. The physical reason behind this theory is this: in order

to \observe" something we need photons to bounce o of this object and enter our eye. For example,

I only see a book on my table because millions of photons are bouncing o of it. On the macroscopic

level, this seems to have no implications. However, on the atomic level, a photon striking an electron

will knock it o course. Thus, if one wishes to measure the position of an electron, the photon we

send at it will strike it and modify its momentum (Gribbin, 1984).

Heisenberg’s uncertainty principle, in a sense, protects quantum mechanics. Heisenberg realized

that if it were possible to measure position and momentum at the same time, quantum mechanics

would collapse. We would be able to, literally, nd out which hole the electron went through and still

observe an interference pattern. Thus he proposed that it was impossible. Then, many physicists

tried to imagine creative ways to defeat the uncertainty principle, but no one could do it (Gribbin,

1984).

Imagine for example that we allow the wall through which the electrons pass in the double-slit

experiment to move. When an electron comes in, it will deect from a certain part of one of the

hole and thus the wall will get a kick and move in the opposite direction of the deection. Thus,

by not touching the electrons at all and merely by observing the wall we can apparently gure out

where they hit (Feynman, 1965).

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However, to do this we will need to know accurately the momentum of the wall. The uncertainty

principle says that if we measure the momentum of the wall with great accuracy, we will lose accuracy

in its position. Thus, an electron ying through will knock the plate o to one side and we will be

able to tell either where it is or how fast it is moving, but not both. And we need both to accurately

judge where the electron went through (Feynman, 1965).

As a nal example, let us see why atoms do not collapse. Although many of us take atoms for

granted, the idea of atoms is completely illogical. If a negative charge is orbiting a positive one, it

would spiral into the nucleus and the atom would collapse. Atoms are completely impossible from

the classical point of view. However, if an electron were to fall into the nucleus, we would then know

its position with innite accuracy, and we would know its momentum at the same time, and this, is

quantum-mechanically impossible (Feynman, 1965).

1.3 Philosophical Implications

1.3.1 The Copenhagen Interpretation

The view of quantum mechanics presented above, utilizing wavefunctions and observers collapsing

them is known as the Copenhagen Interpretation, named after the home city of Niels Bohr. The

Copenhagen Interpretation says that unlike, in the classical world, nothing exists in a determined

state until we observe it. In fact, the observer interacts with the system being observed to such an

extent that the system cannot be thought of having an independent existence. We cannot say what

a particle does, or even if it exists while we are not looking at it. When we perform an experiment

we get a set of readings or properties which suggest to us that a certain type of particle exists.

Almost every time we repeat this experiment, we get similar readings. But as John Gribbin put it,

\the interpretation in terms of particles is all in the mind, and may be no more than a consistent

delusion." If we cannot say what a particle does when we are not looking at it, then it is even

reasonable to say that the electron did not

exist

before the late 1800s when it it was discovered

(Gribbin, 1984).

In quantum physics, one cannot describe the position of a particle in space-time precisely, so in

this sense even the theory of relativity is regarded as a \classical" theory. Newton thought that if

we could ever measure all the properties of all the particles in the universe, we could determine the

future state of the universe because they would follow classical laws.

1

In quantum physics, we cannot

perform such a measurement in principle because by measuring position we disturb momentum, and

by measuring energy we lose precision in the time the measurement took place. Thus, quantum

theory says that the future of the universe is undetermined, a thought that was deeply unsettling

for Albert Einstein whose view on the subject was: \I cannot believe that God would choose to play

dice with the universe"(Gribbin, 1984).

1.3.2 Many Worlds

However strange the Copenhagen Interpretation is, it seems to be the most widely accepted inter-

pretation, for the second interpretation of quantum physics is perhaps even more unsettling. This

second view involves parallel universes existing alongside ours, and although that may sound unusual

it is in fact less strange than the Copenhagen Interpretation if looked at closely (Gribbin, 1984).

One of the features of the Copenhagen Interpretation that doesn’t make much sense is the concept

of the collapse of the wavefunction by the observer. What happens when an observer enters the room

where Schroedinger’s cat is residing? The wavefunction of the cat is collapsed, but now this observer

becomes part of the wavefunction. If he chooses not to tell anyone about what he saw, to the outside

world the room with the cat remains in a superposition of states. If he calls up his friend and tells

1

i.e. objects in motion would remain in motion unless aected by an external force, etc.

9

---

Page 10

him the news, now they are both part of the wavefunction. And this spreads and spreads until, say,

the entire universe knows about the cat. But who shall collapse the wavefunction of the universe?

By denition, the universe contains everything, so there cannot be an outside observer. Indeed,

the Copenhagen Interpretation seems to imply that there is scientic reason to believe in a God!

(Gribbin, 1984).

The many worlds theory gets rid of wavefunctions and the need for an observer. There is no such

thing as a dead-alive cat paradox because the cat is both dead and alive. It is simply alive in one

universe and dead in another. The many worlds theory proposed by Hugh Everett in the 1950s says

that wavefunctions do not collapse. All of the possibilities in a quantum state are real all at once,

and each exists in its own part of \superspace" (and supertime). When we make a measurement

at the quantum level, we are forced by the process of observation to select one of the alternative

universes which then becomes real for us (Gribbin, 1984).

Let us visit Schrodinger’s cat one more time. The conventional Copenhagen Interpretation tells

us that both possibilities of the cat being alive and dead are equally unreal, and only one of them

becomes a reality when we collapse the wavefunction of the cat inside the box. Everett’s many

worlds theory says that both possibilities are equally

real

and that both exist in their own worlds.

There is both a live cat and a dead cat, a decayed atom and an atom that never decayed; but they

are located in dierent worlds. Faced with the decision of whether or not the atom should decay,

the whole universe split into two versions of itself, identical completely except for the fact that in

one the atom decayed and in the other it did not. It sounds like science ction, but it is really only

the consequence of taking the equations of quantum mechanics literally (Gribbin, 1984; Guillemin,

1968).

1.4 Dirac Notation

Without going too deep into the mathematical reasons behind it, Paul Dirac proposed a simple

system of notation for describing quantum systems called

bra-ket

(read \bracket") notation. An

initial state in a quantum system is represented in bra-ket notation by the symbol

j

somestate

i

which is called a

bra

and a nal state is represented by the symbol

h

somefinalstate

j

which is called

a

ket.

When these two are put together we get a braket, which represents the phrase \the amplitude

that" (Feynman, 1965).

So, if we wanted to describe, in the double-slit experiment with electrons, the amplitude that an

electron would arrive at detector

x

after leaving its initial position

s

we could write it as.

h

Particle arrives at

x

j

Particle leaves

s

i

It is often convenient to simply abbreviate this to single letter notation such as

h

x

j

s

i

. When we talk

about describing a quantum system, we are talking about describing the wavefunction of the system

which is the sum of all of the amplitudes for every outcome in the system. Quantum states can be

treated as vectors. A completed bracket denotes a dot product so it is is just a number. Because of

the nature of waves, this number must be complex

2

(Feynman, 1965).

Let us describe Young’s double-slit experiment with electrons using this notation. If we wish to

describe a specic route the electron takes, for example from the initial position to hole one to the

detector, we can write it as the product of the amplitude to get from the initial position,

s

to the

rst hole and from the rst hole to the detector at position

x

:

h

x

j

s

i

via hole

1

=

h

x

j

1

ih

1

j

s

i

In general, we can describe the complete amplitude of going from

s

to

x

by using the sum of all

the individual routes, conveniently represented with summation notation:

2

The real part describes the amplitude of the wave, while the imaginary part describes the phase.

10

---

Page 11

h

x

j

s

i

=

X

i

=1

;

2

h

x

j

i

ih

i

j

s

i

(3)

The vectors represented by

i

are referred to as base states (base vectors). A base state is one

from which all future behavior of the state is determined. i.e. we can say that the base states in

our system are the state of the electron arriving at hole one and the state of the electron arriving at

hole two. It does not matter where the electron started because whether it arrives at the detector

only depends on which hole it went through. Base states must always be

orthogonal

, i.e. their dot

product must be zero. This means that we cannot jump from one base state to another (Feynman,

1965).

As shown above, we can represent the wave function of any system by listing the amplitudes

from going from any initial state to each of the base states. So, to generalize from (Eq. 3):

h j i

=

X

i

h j

i

ih

i

j i

(4)

Notice, that since this equation uses only closed brackets, it is dealing entirely with complex

numbers and no vectors. This is the equivalent of re-writing the all-too-familiar vector equation for

force:

F

=

ma

into real number form by taking the dot product of both sides by an arbitrary vector

C

:

C F

=

C

(

ma

)

but since this is true for any vector

C

there is hardly a use for writing the

C

. Going back to

our wavefunction equation, we see that both sides are multiplied by the vector representing the nal

state

h j

:

So we can simply discard that term from (Eq. 4) to result in the equation:

j i

=

X

i

j

i

ih

i

j i

Also, observe that there is a complete bracket, which is the amplitude that is in the base state

i

which is also referred to as the coecient for the base state

i

. This coecient can be written simply

as

a

i

. Thus, our nal general formula for describing a quantum system is:

j i

=

X

i

a

i

j

i

i

(5)

Remember, that the probability of an outcome is the square of the magnitude of the amplitude

so the probability of outcome

i

is

j

a

i

j

2

. So, to represent Young’s double-slit experiment, once more,

with the electron having equal probability of arriving at either hole 0 or 1, denoted by the states

j

0

i

and

j

1

i

, respectively:

j i

=

1

p

2

(

j

0

i

+

j

1

i

)

And that concludes our excursion into quantum physics (Feynman, 1965).

11

---

Page 12

2 Quantum Computers

Quantum Computation is a very new eld in science. It relates as much to quantum physics as

it does to computer science. The concept of quantum computers is currently at a very theoretical

stage, as working quantum computers may not be realized for another couple of decades. However,

there have been several very promising proposals on their implementation, namely the

linear ion

trap

and

NMR

(

N

uclear

M

agnetic

R

esonance) architectures, which will be discussed later. Today,

quantum computing is approached largely from the standpoint of simulation, which allows for the

development of programs for a quantum architecture on

classical

hardware.

Quantum computers will be radically dierent from the computers of today. Classical computers,

as the computers of today are referred to in the quantum computing eld, interpret the movement of

thousands of electrons, or lack thereof, as ones and zeros or

bits

. A quantum computer, on the other

hand, may function using only several quantum objects such as atoms, ions, or photons, interpreting

their states, or polarizations as ones, zeros, and superpositions of ones and zeros(i.e. it is neither

one or zero until observed; or, if one follows the many-worlds theory, it is

both

one and zero until

observed). Either way, each quantum bit, or

qubit

can exist in a superposition of states with dierent

probabilities of seeing a one or a zero when one observes the qubit (Steane, 1997).

A quantum computer, operates in a memory and eciency space exponential to its resources.

In other words, while a standard computer with three bits can store the numbers 0 through 7, a

quantum computer can store all of these numbers in a coherent superposition. The power of such

computers comes from the idea of

quantum parallelism.

Classically, we can decrease the time of

computation of a certain process by increasing number of processors working in parallel. However,

in order to achieve exponential parallelism, we need to add an exponential amount of processors. In

a quantum computer, parallelism increases exponentially with the size of the system. Thus, a linear

increase in physical space causes an exponential increase in computation power (Steane, 1997).

2.1 Inside the Quantum Computer

If one wishes to keep a superposition of states, one must not observe the state he/she is working

with. Thus, it is impossible, for example, to set up a superposition of the numbers one through ve,

and add it to a superposition of the numbers six through ten, and expect as a result a superposition

of the results of each individual addition. This is not possible because by observing the outcome, we

will only get

one

of those values. It would seem that this fact removes all superiority from quantum

computers, but there is a way around this limitation. Sometimes, we do not need to know the \sum"

in order to solve a problem; there may be some other thing that we can observe and use it indirectly

to solve the problem at hand. This is the technique employed by Shor’s polynomial-time factoring

algorithm which shall be discussed shortly. Another technique is to transform the state in order to

\amplify" the likelihood of certain outcomes of interest. This technique is used by Grover in his

database searching algorithm (Steane, 1997).

2.1.1 The Qubit

The basic unit in a quantum computer is the

qubit,

or

qu

antum

bit

, analogous to the bit of classical

computers. The qubit can be represented by any two state system such as the ground and excited

state of an ion, the spin of an electron, or the polarization of a photon. For the purposes of quantum

computing, the base vectors (states)

j

0

i

and

j

1

i

in a two dimensional vector space are used to encode

the classical bit values 0 and 1, respectively. However, this is where all analogies to classical systems

cease, for quantum bits may exist in superpositions of states such that

=

a

j

0

i

+

b

j

1

i

where

j

a

j

2

+

j

b

j

2

= 1

12

---

Page 13

If such a superposition is measured, one will get

j

0

i

with probability

j

a

j

2

and

j

1

i

with probability

j

b

j

2

:

Thus, a qubit can be in an innite number of superpositions, as long as the probabilities of the

states add to one (Steane, 1997).

Although a qubit can be in an innite number of states, it can still only encode a zero or a

one. The limit is because one must still measure it to obtain a value. Thus, if one needs to store a

number larger than a zero or one, he/she must use several qubits. In a classical physical system of

n

particles whose states can be described by a vector in two dimensional space{basically, a system

with only two outcomes such as one or zero{forms a vector space of 2

n

dimensions. In other words,

if there are two classical bits, they can each alone represent only the state zero or one. But when

they are put together, they can represent the numbers 00 to 11, which in decimal are zero to three.

Thus, two particles can represent four states (Steane, 1997; Rieel and Polak, 1998).

In a quantum mechanical system, though, the vector space grows

exponentially

with the size of

the system. Thus, a three qubit system has an eight dimensional vector space. This system can store

all of the numbers zero through seven in a coherent superposition. To understand this better, think

of a macroscopic objects that suddenly breaks into pieces. It is possible to describe this object by

describing each of its pieces. In a quantum system, this is not true. It is possible to have states that

are not describable by their individual particles, which are called

entangled states

which have no

classical counterpart whatsoever and which contribute to the exponential space (Rieel and Polak,

1998).

2.1.2 Measurement

The process of measurement of one or more of the particles in a quantum system results in the

vector space of the system being reduced to a subspace which is restricted to the measured value.

For example, suppose that there is a two qubit system such that

=

a

j

00

i

+

b

j

01

i

+

c

j

10

i

+

d

j

11

i

This is a coherent superposition of all of the numbers zero through three. If one measures the

rst (rightmost) qubit of this system, the system will reduce to one of the following superpositions:

=

a

j

00

i

+

b

j

10

i

or =

a

j

01

i

+

b

j

11

i

The vector space is reduced because the measurement has restricted the value of the rst qubit

to either zero or one. Thus the system has gone from being a superposition of four states, to a

superposition of only two states. Another measurement will result in the complete collapse of the

wavefunction into one value in the set

fj

00

i

;

j

01

i

;

j

10

i

;

j

11

ig

(Rieel and Polak, 1998).

Measurement can be used to show the entanglement between particles. For example, the rst

state in this section was entangled because measuring one part of it aected another (i.e. measuring

the rst qubit to be zero caused the probability of measuring zero in the second qubit to increase

from 1

=

4 to 1

=

2. On the other hand, neither of the two states resulting from the measurement

is entangled because measuring a one or zero in the rst qubit of these states does not aect the

probability of obtaining a one or zero in the second qubit (Rieel and Polak, 1998).

2.1.3 Quantum Gates

It is possible to transform quantum states without measuring them. The mathematics of such

transformations is complex and will not be discussed, but the ideas will be. Any transformation

on a quantum state must be unitary. One can think of a unitary transformation as simply the

rotation of the complex vector space. Another requirement is that quantum transformations must be

reversible which is due to laws of thermodynamics (classical systems dissipate heat when performing

13

---

Page 14

computations, but a quantum system must maintain its superposition and thus cannot lose any

heat, and operations must therefore necessarily be reversible). This is not important to understand

in order to accept (Barenco et al., 1995).

Adriano Barenco, et al. showed that all quantum transformations can be built from a basic set of

gates which consisted of one bit gates (such as negation and qubit rotation) and the two bit exclusive-

or gate (also known as the controlled-not, or CNot).

3

The one bit gates suggested by Barenco, et.

al. are the qubit rotation (

R

y

), the phase rotation (

R

z

), and the phase shift (

Ph

). These operations

are particularly nice because they are easily implemented using a laser, for example. Using these

fundamental operations it is possible to perform any imaginable quantum transformation. It was

also noted that any unitary transformation may be represented by a matrix (Barenco et al., 1995).

2.2 Quantum Computing Pros/Cons

2.2.1 Usefulness

Building a quantum computer will be a great technical challenge; are the advantages of building such

a computer great enough to face this challenge? One of the primary goals of a quantum computer will

be to act as a factoring engine, using Shor’s polynomial-time factoring algorithm to quickly factor

large numbers. Currently, the best known algorithm for factoring numbers runs in exponential time

(i.e. the time to factor increases exponentially with the size of the input) (Preskill, 1996).

Shor’s algorithm reduces this time to polynomial time.

4

This means that large numbers that took

years to factor before may be factorable in hours or even minutes. The factoring of large numbers is

used as a base for todays most secure encryption algorithms including the infamous Triple DES and

RSA. A quantum computer will make cracking codes as easy as multiplying numbers. A quantum

computer can also be used to create a cryptographical system which is completely uncrackable by

any methods but has certain limitations (Preskill, 1996).

Another application, proposed by Grover is a database search algorithm for unsorted data which

runs in a time proportional to the square root of the input size. Conventional approaches require a

number of comparisons equal to the number of items in the database. This type of algorithm does

not have an exponential speedup over the classical search, but it is a large speedup nonetheless.

There has not been much more proposed for the use of quantum computers, but as they become

more and more developed, new algorithms are bound to show up (Preskill, 1996).

2.2.2 Feasibility

Because a superposition of states may only exist while it is not being observed, a quantum computer

must be completely isolated from its environment. If it does interact with its environment, the

quantum state is said to

decohere

, or lose its meaning and information. Because of this, it has

proven very dicult to build a true quantum computer physically. Optimistic scientists speculate,

however, that such a computer may be built within several decades, when technology is sucient

(Preskill, 1996).

Entangled states are especially vulnerable to decoherence since an error in one of the qubits can

aect the whole state. In order for a quantum computer to function, it must employ very eective

error-correcting techniques. These error-correction techniques must be eective enough that not

even one qubit fails during the course of a computation. A problem with error-correcting is that it

requires enormous

overhead

(extra resources) both in terms of the number of qubits involved and

3

The exclusive or is dened as follows: For bits

a

and

b;

if

a

is set, then

b

is ipped; otherwise

b

is not changed.

The exclusive-or leaves the state of bit

a

intact, and thus this is a reversible operation because one can always deduce

the inital state from the nal state.

4

Specically, the time to factor increases only as the cube of the input size.

14

---

Page 15

the number of quantum gates needed. This increase in resources, by itself causes more errors to

appear. Thus, error-correction will, at best, be very dicult to implement (Preskill, 1996).

2.2.3 Quantum Hardware

There are currently two very promising architectures for physical quantum computers, known as

linear ion trap

and NMR, respectively. Both have shown results that are impressive, but both have

their intrinsic limitations. Quantum hardware of the future, will probably not be similar to these

proposed architectures (Preskill, 1996).

Simply put, the linear ion trap uses trapped calcium ions (positively charged atoms) and manip-

ulates them by sending photons (in the form of a laser pulse) at them. The frequency and duration

of the laser pulse determines how the ion will be aected. Through careful use of the laser, it is

possible to perform a controlled-not operation, which has been shown to be universal for quantum

computers. The speed of such a computer, however, is limited by the frequency of the laser, which

at high frequencies causes instabilities and errors (Preskill, 1996).

NMR, or Nuclear Magnetic Resonance, quantum computer uses the fact that nuclei in a molecule

act as tiny magnets when placed in an electromagnetic eld and detects their spin (aligned or

antialigned to the eld) as a logical one or zero. Because this signal from an individual molecule is

very small, NMR has to use a group of molecules which are presumed to be in a similar state, and

uses \majority rules" to decide which way the atoms are spinning. The problem with NMR is that it

is dicult to make sure that all the molecules start in the same state, but two separate solutions to

this problem were published in 1997, giving a great push to NMR architecture development (Preskill,

1996).

2.3 Quantum vs. Classical Algorithms

2.3.1 Eciency

An algorithm is called \ecient" if it runs in polynomial time or faster. What does this mean? Let

us examine the task of nding the factors of a number. Consider the following problem:

? ? = 29083

Using paper and pencil this problem will take approximately one hour to solve. However, to

solve the reverse problem:

127 229 =?

Takes less than a minute. This is because an ecient algorithm for multiplication is known, but

there is no such algorithm for factoring. In order to classify problems by eciency, they can be

analyzed by the way the time to perform increases with the size of the input. For example, adding

another digit to the factoring problem will increase the work to several hours, while another digit

added to each of the factors in the multiplication problem will only increase the work by less than a

minute. Multiplication is a polynomial-time algorithm; factorization runs in exponential time (Shor,

1996).

2.3.2 Shor’s and Grover’s Algorithms

Shor’s algorithm, published in 1994, solves the problem of factorization by using a method which

runs in polynomial time. There is one catch, however: this polynomial speedup can only be achieved

on a quantum computer. Shor reduces the problem of nding the factors of a number to nding the

period of a certain function. This is an \indirect" approach which measures a property of a function

15

---

Page 16

in order to solve another problem. The mathematical details of this algorithm are beyond the scope

of this paper, but in brief Shor uses the exponential parallelism of quantum systems to compute the

period of a certain function. Classically, this period computation would take an exponential number

of steps by itself, and thus the algorithm would run in exponential time. A quantum computer allows

for this computation to take place in polynomial time. This period is then obtained with fairly high

probability (if it is not, the algorithm is simply repeated). This period can then be used to factor the

number at hand. Thus, Shor’s algorithm is exponentially faster than any known factoring algorithm

for classical computers (Shor, 1996).

Grover’s algorithm, presented in 1996, applies to searching an unsorted database. It provides

only a polynomial speedup, but this speedup is very important nonetheless. Grover sets up a

superposition of all of the entries in the database and then makes several rotations and special

operations so that the output is the item that is being looked for with a probability of at least 0

:

5

(Steane, 1997).

3 Quantum Computer Simulation

Although quantum computers, at this point in time, are not technologically feasible, this minor

setback does not prevent the development of quantum algorithms, i.e. programs that will run on

a quantum computer once it is built. In order to develop such algorithms, a quantum computer

simulator

5

is necessary.

The OpenQubit core development team, consisting of the author and several colleagues around

the world has developed such a simulator. It is currently in the stages of its infancy, but there

is a working prototype of Shor’s algorithm, and very recently proposals for the implementation

of Grover’s database search algorithms have been submitted. At this time, the stable version of

OpenQubit is 0.2.0 and this is the version that was used in the eciency tests. There is also a test

version in development (0.3.x, a.k.a. NewSpin) which has a better model, and so this will be the

version that is discussed.

3.1 OpenQubit Model Overview

OpenQubit uses an object-oriented model

6

to describe the workings of the quantum computer. The

main class in the QState class which describes the quantum state of the machine. For an

n

qubit

machine, it stores 2

n

amplitudes as complex numbers in order to describe all the possible outcomes

of the state. This class provides memory management facilities for the class QRegister, which is a

substate, addressing several or all the qubits in the QState. The basic set of operators (

R

y

,

R

z

,

Ph

, CNot) is provided through a set of classes. Because the act of applying an operator is the same

for any gate class

7

, this application process has been abstracted away into a separate class, and

all the gates

inherit

8

from this class. A detailed description of the software follows. It consists of

documentation generated from source code and a diagram representing the class structure.

5

A simulator or emulator is a program which runs on top of classical hardware and provides a working environment

like that of a quantum computer. Thus, the user may then develop a program as if the computer he was using was of

a quantum nature.

6

Briey, object oriented programing (OOP) centers heavily on the modeling of physical entities. Traditional

programming methods concentrate on algorithms and procedures, but OOP concentrates on describing objects and

their interactions. Each object is described by a set of properties and a set of methods (functions that it can

perform). The basic unit of object oriented programming is the

class.

For example, a person might be modelled by

a class containing properties to describe the person’s hair color, etc, and methods such as walk, talk, and eat.

7

In order to apply the gate, just multiply the quantum state by the matrix which represents the gate.

8

An OOP term. If a class inherits from another class, then it will gain all the properties and methods of that class

as its own, and more can be added as seen t by the designer.

16

---

Page 17

The OpenQubit software was developed by the OpenQubit core development team consisting of

Yan Pritzker, Ralf Podeszwa, Peter Belkner, Chris Dawson, Igor Kofman, Jonathan Blow, and Joe

Nelson. It was developed for the Linux operating system. OpenQubit is an organization which I

founded on December 9, 1998 as an attempt to bring together physicists, computer scientists, and

laymen together in order to create a quantum simulation API (Application Programmer Interface).

At the time, I had little experience with quantum physics, and approached the matter from largely

a computer science standpoint.

The OpenQubit software library is released under the GPL2 (GNU Public License Version 2)

which, in brief, requires the software to always include source code and to always be distributed

free of charge. A copy of this license can be obtained at

http://www.gnu.org/copyleft/gpl.html

.

OpenQubit is a work in progress, so certain parts of documentation or diagram may be inaccurate

by the time this paper is read.

3.1.1 OpenQubit Main Class Diagram

QState

-_nQubits: static int

-_nStates: static int

-_Available: static int

-_qArray: vector<Complex>

-_RNG: RandGenerator<_RNGT_>*

+Initialize(nQubits:int=1): static void

+Print(): static void

+Save(filename:char[]): static void

+Read(filename:char[]): static void

#_Collapse(): static int

#_CollapseBit(bit:int): static int

#_CollapseMask(mask:int): static int

#_CollapseRange(start:double,end:double): static int

#_Allocate(numbits:int): static int

#_Coef(index:int): Complex&

#_Reset(): static void

-_Normalize(): static double

-_Reset(): static void

QRegister

+QRegister(bits:int=1)

+SetState(c:vector<Complex>): void

+Measure(): int

+Measure(bit:int): int

+Measure(start:int,end:int): int

+Coef(bit:int): Complex&

+Reset(): void

QRegister is a relatively indexed

interface to QState. Thus, bit 0 in

a QRegister may be bit 5, say, of

the entire QState.

OpenQubit 0.3.x [NewSpin] Class Diagram

3.1.2 Sample Output

Provided here is the output from a sample run of Shor’s factoring algorithm using OpenQubit.

OpenQubit version 0.2.0, Copyright (C) 1999 OpenQubit.org

Shor’s algorithm for factoring numbers

Enter number to factorize: 15

Enter a number from 1..14: 4

Preparing equal superposition in the first register

Modular exponentiation

Fourier transformation of the first register

Measured state is:

1.000000 |000100000001>

The result is 128

17

---

Page 18

Fourier domain is 256

Extracting the period using continued fraction expansion...

Period guess is: 2

Period guess is probably correct

Factors found! 15 = 5 * 3

4 OpenQubit Documentation

The documen+tation following this paper was generated using the PERCEPS documentation genera-

tor from OpenQubit source code. The full source code to OpenQubit is available at

http://www.openqubit.org.

18

---

Page 19

References

[1] Barenco, Adriano et. al. (1995, December 18).

Elementary gates for quantum computation.

xxx.lanl.gov e-Print Archive.

[2] Feynman, R., Leighton, R. B., Sands, M. (1965).

The Feynman lectures on physics.

London:

Addison-Wesley.

[3] Gribbin, John (1984).

In search of Schroedinger’s cat.

New York: Bantam Books.

[4] Guillemin, Victor (1968).

The story of quantum mechanics.

New York: Charles Scribner’s Sons.

[5] Preskill, John (1996, December 17).

Quantum computing: pro and con.

xxx.lanl.gov e-Print

Archive.

[6] Steane, Andrew (1997, September 24).

Quantum computing.

xxx.lanl.gov e-Print Archive.

19