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Quantum Computers: An Analysis of the Next Step of
Moore ’s Law and How it Impacts Us
James Brown
Research Paper
Writing 202
York College of Pennsylvania
April 7, 2011
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Abstract
Quantum computing is the use of subatomic particles to carry information instead of a micro-circuit. The
use of these particles allows for speed of light transmission, continuous bits and almost unbreakable
computer security. By using infinitesimally small particles to carry data instead of electric circuits, the
processing speed jumps to incredibly high amounts. But more than that, it can perform many algorithms
much faster than traditional transistor computers. Also, by using qubits (quantum bits), observing data
while it is being transmitted is almost impossible. Besides being based on a probabilistic model—where
the qubits are probabilistically placed in positions, but the true position can never actually be known—
and being unobservable by traditional methods, the qubits are affected by the Heisenberg Uncertainty
Principle. This states that observing something changes it, and since qubits behave as electrons and can
be located in several discrete positions at the same time, not only does observing it serve no purpose,
but the observation can be noted and tracked. The development of this technology and its ramifications
are yet unknown, but it has the potential to change the world forever, and will effect cyber security
systems the world over.
Introduction
Moore’s Law states that every two years, the number of transistors able to be efficiently put on
a chip will double. Since processing speed and computer size are strongly linked to this, these properties
will grow exponentially as well. This trend has held approximately true ever since 1965, when Moore
wrote his paper. Quantum Computing will be the end of this law. By using parallel computing planes and
entanglement (more on this later), quantum computers will be the final step in Moore’s Law, as the laws
of Physics will not allow anything smaller or faster that is remotely possible in the visible future. The
time notation for a quantum computer to solve a large prime factorization, such as the RSA algorithm
that protects most banks, is log N, where N is the number of calculations performed1. This doesn’t
sound too impressive, until it actually sinks in that it would only take 12 seconds to do 1 trillion
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factorizations. To put that into perspective, most RSA keys are from 1028 to 4096 bits long. The largest
number ever factored by conventional computers was 768 bits, and that took days. A quantum
computer could factor these numbers in seconds.
Basics of Quantum Computing
So how does this work, and how come there aren’t quantum computers on sale in Best Buy™
right now? Quantum computers work by taking the unique way subatomic particles behave to transmit
data in large quantities, using a closed system of neutral particles, or by using light itself. Scientists at
the University of Geneva in Switzerland have successfully managed to use a photon as a hard data
source, and read the data they saved off of said photon2. By using subatomic particles, the behavior of
the computer is radically different from anything that Newtonian Physics even allows as possible.
Current computers operate with bits, which are binary in nature. A 1 represents a transistor is on, and a
0 means that it is not. Quantum computers are different. Instead of bits, they have qubits, which can be
a 1, a 0, or a combination of both 1 and 0 with probabilities assigned to each state. How is this possible?
Well, photons, which are currently the favored particle, behave as subatomic particles. If the particle
encounters a split it will travel both paths simultaneously. When scientists tried to prove this with
detectors, the light was only registered hitting one3, which brings us to another property of quantum
computing that greatly increases security. If one state of the particle is observed along the way to its
target, the other states are disturbed and are actually changed. Observing subatomic particles changes
how they behave, and forces the particles into a state that follows Newtonian physics models and rules,
instead of quantum rules, and acts on a Euclidian plane instead of a Hilbert space (Appendix 1). A
Euclidean plane is actually a vector space, but it is only one. A quantum computer would operate on
many of these running in parallel. The physics behind this is extraordinarily complex, but to sum it up,
both the location and velocity of a subatomic particle such as a photon or electron can never be known.
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If the location is known, velocity is impossible to find, and vice versa. Thus by, setting up something to
observe a particle, the base state is changed. There are several other benefits to using quantum
computers.
In quantum physics, particles can become entangled, which will give them special properties
that can lead to a large system of qubits traveling over huge geographical distances without worrying
about interferences. Entanglement is essentially the shared properties of two or more quantum
particles over a distance. It is caused by sharing wave functions, which links the system of particles to
share properties. This can be purified by refining the wave functions by adding or releasing energy and
changing how much of the properties are shared. This is extremely useful, because a data containing
particle can be “placed” near an entangled particle, and the interaction can be recorded at both ends. 11
In fact, a recent experiment involved sending a data-containing photon between 2 entangled photons
across the Danube River, almost 600m, without anything actually “moving”.12
A Quantum Computer operates on parallel Hilbert planes, which for the sake of simplicity can be
defined as parallel planes of computing, though they have also been described as parallel universes.
Each individual qubit exists on 2 such universes. Once multiple qubits are introduced to the system, the
growth of universes is exponential. For example, a 3 qubit system needs 8 universes, or 23.4 This is
known as superposition. Nobel Prize winner Richard Feynman was the first to acknowledge what
superposition and parallelism can do. In short:
For example, a system of 500 qubits, which is impossible to simulate classically,
represents a quantum superposition of as many as 2500 states. Each state would be
classically equivalent to a single list of 500 1's and 0's. Any quantum operation on that
system … would simultaneously operate on all 2500 states. Hence with one fell swoop,
one tick of the computer clock, a quantum operation could compute not just on one
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machine state, as serial computers do, but on 2500 machine states at once! Eventually,
however, observing the system would cause it to collapse into a single quantum state
corresponding to a single answer, a single list of 500 1's and 0's, as dictated by the
measurement axiom of quantum mechanics. The reason this is an exciting result is
because this answer, derived from the massive quantum parallelism achieved through
superposition, is the equivalent of performing the same operation on a classical super
computer with ~10150 separate processors (which is of course impossible)!!
3
This is mind boggling. In the time it takes for 1 bit to be processed and computed in serial computers,
the quantum computer does 2500. Of course a system of 500 qubits is far off, as scientists are still
struggling with systems of 2 or 3 qubits. This can be mainly attributed to the fact that there are
problems with the quantum computer, which is why it still remains an idea and not a reality. The major
problem is known as decoherence.
Decoherence is based off of Heisenberg’s Principle (observing a particle changes it). Adding any
form of “noise”, or interference of any form, to the quantum systems is equivalent to observing it, as
the particles are so small, the slightest wave would be like getting hit with a basketball. Formally,
decoherence is the degeneration of the qubit to a classical state, that is a state measurable by classical
physics. As the parallelism of Hilbert states is essential to the function of a quantum computer, the
particle must remain in its coherent state to be of any use. Therefore, any form of outside interference
that is not controlled will send the entire system reeling4. Until scientists find a way to enclose the entire
system entirely, quantum computers are unlikely. Some, like Michael Dyakonov of the University of
Montpellier in France, think that achieving a fully quantum computer is, “akin to achieving perpetual
motion”—in other words a pipe dream5. Most scientists think that it is a definite possibility. Andrew
Steane of the University of Oxford is casually dismissive of Dyakonov’s misgivings, saying, “It is true that
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the quantum computing community should be cautiously optimistic rather than confident, but his
arguments are largely misleading.”5 Though this is one of the biggest problems, it is far from the only.
Measuring the quantum states is a problem unto itself. There are two types of measurements
being considered for either bounded or unbounded Hilbert spaces. The discussion of complete spaces,
vector space, Banach spaces and Cauchy sequences is too complicated for this small essay, so for the
sake of simplicity, a Hilbert space is a parallel vector space where all rules of Euclidian geometry apply in
parallel to the main dimension of existence. Operations in Hilbert spaces can be seen by projected
values, which project themselves into a usually observable subspace, and follow the operations in
Hilbert space, updating after every finite step in the process. The question of when to measure depends
on the operation and the type of space. Either multiple measurements can be done after every discrete
step or asymptotically after the entire algorithm is complete. Because of the intricacies of Hilbert
computing, the quantum state will either reject or accept the measurement through the observable and
an identity. Since this is probabilistically based, multiple measurements may be missed, skewing the
observable calculations and the entire value could be missed in a single step measurement. These
problems greatly slow the advancement of quantum computing into widespread use. So until these
computers become a reality, the community at large must prepare to deal with the repercussions of
these super fast computers.
The Repercussions of Quantum Computing: National
Throughout the world, world powers are run almost entirely on computers and also
cryptography, and the encryption of important data. Modern day cryptography is advanced and
practically unbreakable, especially for high level functions such as defense and infrastructure.
Computers control everything from industry to national finances. Of course these are highly secure and
encrypted more ways than most people can fathom. With current technology, hacking at national levels
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is relatively in check. Everyday hundreds of thousands of sophisticated attacks probe the United States’
infrastructure, but for the most part the Department of Homeland Security, the Pentagon, and the new
US Cyber Command have managed to defeat and even retaliate to cyber attacks through an evolving
and aggressive cyberspace denial of entry7. However, such endeavors are costly. The Pentagon actually
was forced to create an entire other security Bureau to deal with cyber attacks and defense, and
declared cyberspace as its own domain of warfare, along with air, sea, and land. But all the preparations
and training and strategy are all irrelevant when suddenly the opponent is thousands of times faster
than the defense. Institutions previously thought to be 100% secure would be wide open to attack.
Financial institutions, nuclear missile codes, everything is laid bare. The billions the United States spends
on cyber security would be wasted chasing something that can never be caught. An attacker with a
quantum computer system and the knowledge to use quantum cryptography would be able to break
into any system and get whatever they want, and leave without being followed. The attacker can throw
up walls that cannot be breached without quantum computers, and effectively stop all administrators
and moderators from tracking them or even identifying them. In order to adapt to the threats that grow
and mutate in cyberspace, the United States must have the technology first, otherwise be open to all
attacks, and not even know that an attack has occurred. Right now, this is not a problem. The encryption
keys used, both secret and public defy external cracking with the sheer amount of calculations or
possible permutations. But this will not always be the case. A quantum computer will be able to easily
break these defenses, and reverse them against the original owner.10
A great example is the RSA key and other such public key encryptions. Cracking these algorithms
means cracking a huge number of password-based defenses such as email and bank accounts. Going on
from these, a good engineer could easily reverse the process to create a system that would be almost
entirely unbreakable without inside information. The development of quantum computing would make
whoever created it first, much like the atomic bomb, far out ahead of their enemies and competition,
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making them invulnerable to attack. Imagine a force that is protected entirely from cyber attacks, with
the ability to quickly and reliably take down all enemy computer controlled forces and defenses. They
would be unstoppable. In fact, conventional means of warfare would quickly go obsolete, as the force
with the computing power could launch attacks with unmanned vehicles and missile strikes that the
opposing force would be helpless to defend against. Quantum computing is the Manhattan Project of
today, with the potential to overwhelm any countries, entities or states without it. Quantum computing
will completely revolutionize foreign relations and domestic infrastructure, much the way computers did
when they were first introduced. It will do the same for domestic parties, only much slower.
The Repercussions of Quantum Computing: Citizens
Before quantum computers are even distributed to the public, the United States cyber system
will have to receive a complete overhaul. Banks will have to update to quantum computing and
cryptography, the military will renovate all its technology to run on this technology and so on. But the
citizens are then left floundering. The computer this is being displayed on right now may be top-of-theline, liquid-cooled hex-core with a pedabyte of memory and the fastest processing available, and
military grade encryption, being run by Steve Jobs. It might as well be a Commodore 64 with a 2-bit
encryption policy and a technophobe at the helm for all a quantum computer is concerned. A complete
novice could write a program that uses brute force to factor out every possible password from a public
key or run through every possible number combination ever, and take over everything in a reasonable
amount of time. Every password, online transaction, email sent, tweet tweeted, wall post, document,
every download that wasn’t supposed to go public is available to this attacker. Putting technology of this
power commercial would require some major technology overhauls for the entire country. Anyone
missed would be at risk from every kind of identity theft and privacy violation that the Internet can
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throw at it. Assuming this integration goes smoothly, however, there are several other foreseeable
problems. The effect on employment and job availability will be huge.
Most industry jobs will be unaffected, as production must be kept at finite speeds for realitys
sake. Service jobs, however, are another story. The evolution into subatomic computers could make all
non-physical jobs automated, and even lead to a true A.I—artificial intelligence—with the adaptive
properties of a human that is infinitely smarter, faster , and more organized than human beings. The
technology boost would be similar to the technology boom that came with computers, only
exponentially larger and much more impressive. Instead of moving from typewriters and Turing
machines to keyboards and calculators, we move from jerky automatons and tiny transistors to a
computer that will do more calculations in the time it takes light to travel across a room than the human
mind will do in a lifetime. The effects of a pure quantum computer are unpredictable, but they are sure
to be interesting and life changing.
Conclusion
Quantum computers represent the theoretical limit of computational growth. Any faster is
physically impossible, because they will violate the known laws of physics. Quantum computers are
extraordinarily complex, and this explanation of their workings is highly simplified and generalized. The
use of Hilbert planes and entanglement to send and process information at the speed of light and faster
will be invaluable to making computing as fast as possible. Though there are implications of gaining this
technology, and especially in gaining this technology second, the implementation of this technology will
render all competing technology obsolete, and revolutionize technology as we know it. It will lead to
better lives, better medicine, better computers in general. Everything run by computers now will
become much more efficient, much faster, and much more reliable. This step up in computing could be
mankind’s next giant leap.
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Appendix 1: Hilbert Spaces7
A Hilbert space is essentially a parallel vector space, where a vector space is essentially a plane
comprising of two operations, addition and scalar multiplication. These must follow a specific set of he
rules, with an inner product (f, g) , or for those familiar with vector math, a generalized dot product. This
inner product must follow 4 rules:
1.
.
2.
3.
4.
.
.
and equal if and only if
.
The inner product for finite spaces with definite boundaries is merely the bounds of the function and is
made of real numbers. An infinite Hilbert space however has the inner product of (f, g) = ∫f(x)g(x)dx,
with upper bound infinity and lower bound negative infinity. The inner product must have a vector norm
such that |f| = √ (f, f). This makes the space a complete metric space, and makes it a Banach Space as well,
though not every Banach space is a Hilbert space. A Banach space is any vector space, and the vector
norm does not have to represent the inner product.
Appendix 1.1: Vector Spaces
A vector space is a closed set of values that is bounded by the basic rules of addition and scalar
multiplication. In order for a vector space to exist mathematically, all values x, y, z on the plane of V (the
vector space)—abbreviated X, Y, Z є V—must obey all the rules of the Euclidean plane we operate on in
Newtonian physics and conventional mathematics. These include the commutative property of both
addition and multiplication, identity properties, zero properties, etc.
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Appendix 2: Entanglement12
Entanglement was first discussed by a three person team of Einstein, Podolsky, and Rosen. It
was observed that preparing two particles in similar states and then moving them apart did not yield
different results. The position of one could be accurately measured by the position of the other and
momenta worked in the same fashion. Furthermore, when position was measured and then a
subsequent momenta calculation was performed on both, the momenta were different. This defied all
logic, as the particles could not encompass a complete quantum state (where both have all the same
properties) as assigning labels to correlating values would assign a correlation to the two that didn’t
actually exist for all cases. However, Schrödinger later stated that it was complete, and that the two
particles shared an infinite number of dynamic states, seemingly with absolutely no interaction besides
initialization. Schrödinger described the particles as having left imprints of themselves on each other,
which is supported by the Heisenberg Uncertainty Principle, that interacting with these particles
changes them. Both particles essentially become reflections of each other, and therefore entangled. He
also may have observed that the entanglement decreases over distance. His observations appeared to
suggest that if the distance and time light traveled between the two particles was negligible, then
entanglement stayed pure. Otherwise, they began to mix with other states. However, John Bell in the
80s reported findings that supported entanglement over any distance and that, since the differences in
the two particles do not have in common are not predictable; the correlations must have a common
cause. There is still much debate about the subject, but it remains that entanglement will be highly
useful in quantum computing for long distance data transfers.
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