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The ability to work with singular
quantum systems will be necessary if we are able to harness the power of
quantum mechanics and apply it to computational and information theories. (atom
trap – isolate a single atom and probe its behaviour)

Study of information processing
tasks that can be accomplished using quantum mechanical systems

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A classical computer is simply an
extravagant calculator, a device whose memory can store a sting of numbers, 0s
and 1s, known as bits. These devices can perform operations and building these
up gives us an algorithm. In 19** Somebody Moore presented a paper detailing

 

Information can be encoded

 

In 1965 Gordon Moore observed
that transistors, (info about what a transistor does here) were shrinking at
such a rate that every year double the number could fit onto a chip, ten years
later this slowed to doubling every two years, and since this result was stated
over 50 years ago the prediction has held true, until now as it begins to slow.
Moore’s company, Intel, now suggests that the currently used silicon
transistors will reach their limit in around five years. What does this mean
for computing as we know it? Over the last 50 years we’ve seen a computational
revolution, where previously (info about what we could do in the beginning with
a computer – army room sized) we can – and over x% of the worlds population –
now carry this around in their pockets. We have become accustomed to continual
improvement and development and see some exciting advances for the future, with
the advent of machine learning, self-driving cars and (another new use of
computing), will we be able to continue this without the ever shrinking
transistors and chip capability?

From Morse Code to Classical computing and the next step into the
quantum world.

How does a classical computer
work?

In Claude Shannon’s 1948 paper
the notion of information was defined and quantified. The smallest unit of
information, the binary digit, simply the bit, was defined and Shannon showed
it was possible to encode any message using only two values, 0s and 1s in a
string. A further way to see this is using the idea of the game 20 questions,
the only possible responses are ‘yes’ or ‘no’ and yet we can gain a huge amount
of information about the thing the ‘sender’ is trying to communicate. If we
extend this further and remove the limit of the number of questions we are able
to describe everything, simply using this string of 0s and 1s.

 

Throughout its vast development, communication
has been fraught with problems. From the days when fires were lit across
hilltops to transfer news in the fog and rain, to analogue systems gaining
noise and becoming distorted to the current state of classical computing
seemingly reaching its limit, what is the next step? In this paper we look at
the abilities of classical computers and their mathematical limits and how we
compare and contrast this with the developing quantum computers, considering
their possible uses and capabilities. Since Claude Shannon’s 1948 revolutionary
paper on information theory the developments of computation have been rapid.

 

A Classical Computer

Whilst computers have developed
to be able to do a whole host of tasks, they still remain relatively simple
machines operating via two key functions which use the transistor. Firstly, a
transistor can store a bit, a value of either 0 or 1, and with a string of just
8 bits we can encode around 255 characters, both the upper and lower case
alphabets, the numbers zero to nine and a number of the most commonly used
symbols. Secondly, logic gates are used to undertake computations using
transistors which are connected together, resulting in a new string of bits. It
is these two processes, storage and transmission of information, which
Shannon’s Mathematical Theories In Computation paper discuss, that underlie the
theory of classical information. In order to efficiently use the power of
transistors we need to compress the given data into as small a piece as
possible.

 

 In 1965 Gordon Moore observed that
transistors, were shrinking at such a rate that every year double the number
could fit onto a chip, ten years later this slowed to doubling every two years,
and since this result was stated over 50 years ago the prediction has held
true, until now as it begins to slow. Moore’s company, Intel, now suggests that
the currently used silicon transistors will reach their limit in around five
years. What does this mean for computing as we know it? Over the last 50 years
we’ve seen a computational revolution, where previously (info about what we
could do in the beginning with a computer – army room sized) we can – and over
x% of the worlds population – now carry this around in their pockets. We have
become accustomed to continual improvement and development and are on the edge
of some exciting advances for the future, with the advent of machine learning,
self-driving cars and (another new use of computing), but will we be able to
continue this once we reach the limit of ever shrinking transistors and chip
capability?

In order to consider the limits
which Shannon determined on both data compression and data transmission we
require a background of the various types of entropy defined. In this section
we will introduce the notion of entropy, joint entropy, conditional entropy as
well as mutual information from the classical point of view. From there we will
discuss Shannon’s Noiseless Channel Capacity Theorem and Shannon’s Noisy
Channel Capacity Theorem.

 

In Shannon’s 1948 paper the notion
of information as discrete data was presented, allowing the idea of entropy to
form, that is, how much information we gain by obtaining the outcome of an event.
A useful example of this idea is a coin toss. Suppose we have a fair coin,
where the likelihood of obtaining heads upon tossing the coin is 0.5, then we
gain the maximum about of information when we see the result, we could not have
predicted the outcome. Now suppose the coin is heavily biased, where obtaining
a head is 0.9, then we are not surprised when heads occurs, we have gained less
information.

Shannon required that for a
series of possible events with probabilities p1, p2, … pn to find a measure
H(p1, p2, …, pn) of uncertainty regarding the outcome

1.       The
respective probabilities must be continuous

2.       If
all the probabilities are equal pi = 1/n then n should be a monotonically
increasing function

3.       For
two sources of outcomes A and B, H(A,B) = H(A) + H(B|A) i.e. the associated
measure of uncertainty is the weighted sum of the two separately.

The only function satisfying
these properties is of the form H(Pi) = -K sum_{i=1}^n p_i log p_i where p_i
is the probability distribution of our series of possible outcomes. Moving
forwards we will work with log_2, which gives us the binary entropy and removes
the constant K which is a unit of measure.

Considering the entropy of a case
which has two possible outcomes with probabilities p and q = 1-p shows some
interesting properties of the binary entropy function. Plotting H = -p log p + q
log q gives (figure) which shows entropy to always be positive, and only equal
to zero when there is no uncertainty in an event happening. Further, H is
maximised when p = q, showing that uncertainty maximises entropy.

From here we can define the joint
entropy, suppose we have two sources X and Y with discrete outcomes, then
H(X,Y) = – sum p(x,y) log p(x,y), if the outcomes are perfectly independent
then we can describe the joint entropy as H(X,Y) = H(X) + H(Y), however, in general
H(X,Y) R$ ie the capacity of the channel is greater than
the rate of information transferred.

 

It is widely believed that all
classical systems can be described using quantum mechanics, and so it is
natural to view computing, which we consider in a macroscopic world, on a
microscopic scale and in terms of quantum mechanics. Just as a classical
computer is built from wires and transistors that perform computations a
quantum computer uses wires to transfer information and quantum gates to
manipulate this quantum information, and it is the behaviour of the qubit, the
smallest possible quantum unit, that allows us to explore new methods in
computing.

The most simple quantum system
can be described as a two dimensional Hilbert space spanned by the orthonormal
basis ${|0>, |1>} and alongside the basis states of $|0>$ and $|1>$,
analogous to $0$ and $1$, the system can have any other state which is a linear
superposition of $a|0> + b|1>$ with $|a|^2 + |b|^2 =1$. A useful way to consider
this is to view the pure states of a singular qubit as the points on a Bloch sphere,
where the classical bits would be the north and south pole, a qubit can take on
any point on the sphere. The fact that a qubit can take on any of an infinite
number of possible states, as opposed to the two classical choices, allows
quantum computing to be different to that of classical computing.

Many aspects of classical
information theory have analogous properties in the quantum world, there are,
however, other ideas which are impossible in the classical world which allow us
to view quantum computing as a different and more useful capability.

Von Neumann

 

 

this is not the case in the
quantum world, where a state can take the form of any linear superposition of
the form $a|0> + b|1>$

 

 

 

 

 

 

 

In classical computing it is
possible to divide a system into its subsystems allowing us to decode and thus
deconstruct the entire system, this is not the case in the quantum world.
Entanglement is the property that allows us to know everything about our full
system with no ability to decompose it to its parts, leaving us uncertain about
the individual components.

 

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