"Quantum Computing" is among those terms that are widely discussed but often poorly understood. The reasons of this state of affairs may be numerous, but possibly the most significant among them is that it is a relatively new scientific area, and it's clear interpretations are not yet widely spread. The main obstacle here is the word "quantum", which refers to quantum mechanics - one of the most counter-intuitive ways to describe our world. But fear not! This is not a course on quantum mechanics. We will gently touch it in the beginning and then leave it apart, concentrating on the mathematical model of quantum computer, generously developed for us by physicists. This doesn't mean that the whole course is mathematics either (however there will be enough of it). We will build a simple working quantum computer with our bare hands, and we will consider some algorithms, designed for bigger quantum computers which are not yet developed. The course material is designed for those computer scientists, engineers and programmers who believe, that there's something else than just HLL programming, that will move our computing power further into infinity. Since the course is introductory, the only prerequisites are complex numbers and linear algebra. These two are required and they have to be enough. Happy learning!
Characteristics of Computational Systems
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Из курса от партнера Saint Petersburg State University
The Introduction to Quantum Computing
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Intro
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Сысоев Сергей Сергеевичкандидат физико-математических наук
Кафедра системного программирования
In the last lecture
I told you that the source engine was never implemented.
Why is that? Why our modern computers are
not implemented as numerous source engines?
Why we choose to implement our computers in a different way?
Here on the table I have
two different computational systems, this and this.
This you can still buy in
a general store and this is not produced anymore.
Why? It can store the same amount or
approximately the same amount of information
and I believe that this can store it better.
Doesn't need batteries or light or anything like this,
and you just to keep it from children.
So, why humanity prefers this over this?
It appears that computational processes are not all the same.
They have some characteristics
that differ from process to process,
and some of these characteristics matter more than others.
The most important characteristics are probably these.
First, the information capacity of the system.
How much states it has and how much information it can store.
So, the bigger is the information capacity,
the more significant tasks we can solve.
Second is speed or the frequency of state switching.
So, the faster we change the states,
the faster we compute and the faster we obtain the result.
The third but not the last and the least is universality.
So, which tasks can be solved with a computational system?
What repertoire does it have?
Let's talk about this last one, the universality.
I understand why I advise you to read some popular science books.
For example, in the book of David Deutsch,
The Beginning of Infinity,
there was an example of a computational device
which illustrates this notion very well.
I mentioned a shepherd which doesn't know how to count.
The main function of a shepherd is to maintain
the same amount of sheep in the herd during the day.
So, he invents very particular computing device
to maintain this number of sheep.
He uses a rope like this one.
So, each morning when sheep leaves the stall,
he unwinds the rope.
For example, one piece of
rope per sheep equal to the length of his hand.
So, this is one sheep,
this is the second sheep, et cetera.
When the sheep return in the evening,
he winds it back.
So, here he takes the length of the rope according to his hand per
each sheep and he winds up this piece of rope.
At the end of this process if he has some rope left,
he understands that he has to go and to search for the lost sheep.
Now, the word sheep is
very good choice for
this example because it doesn't have a plural form.
So, actually that means that in
this case when a shepherd goes for his search for the lost sheep,
he actually doesn't know how many sheep are lost.
Maybe one or maybe more.
Still this computing device,
this computing system which contains two parts,
the rope and the shepherd,
it works and it even has some advantages.
The main advantage is that the rope length is continuous.
So, if the number of sheep in the herd grow
dramatically and this length of the rope is not enough,
the shepherd can choose to use
the shorter pieces of rope per sheep when he counts them.
Here he has no limit in doing
this in the shortening the piece of rope per each sheep.
The system which have some parameter which is continuous,
I usually represent the analogous computing systems,
and they have two main drawbacks which
obscure this advantage of continuous parameter.
These two drawbacks are,
the first, error correction.
Each time shepherd unwinds the rope,
he makes a small mistake and it's a big number of sheep,
these mistakes accumulate and they can
produce one sheep either missing or extra erroneous sheep.
Hence, there is no way the shepherd can
fix his drawback to fix this error.
The second drawback is universality.
This rope for shepherd can maintain
the same amount of sheep but he cannot subtract numbers,
add numbers, multiply them.
He can do better.
He can make nodes on this rope,
and now it should not represent one sheep.
He loses the continuous parameter,
the length of the rope, he digitizes it.
Now, here all these drawbacks fade away because he does
not have anymore of this errors.
The small errors which accumulate
and with this rope he can do more.
He can add numbers, subtract numbers,
multiply numbers and do many other things.
This is actually the quality of digital systems.
They have bigger universality.
They have larger repertoires.
We are going to discuss this notion of
universality more formally later,
but now we can get back to
this two first characteristics we have not yet discussed.
Now, if we put away universality,
we have only two main characteristics
left and these are the number of
states which the system can
accommodate and the speed of the state is switching.
How fast the system can switch the states.
The bureaus that both these characteristics depend on
the base element that we choose
to build our computational system on.
If we take a look at the computer generations,
we can see the evolution of these base element.
For some reason engineers try to make
this base element as small as possible,
from vacuum tubes to transistors,
then to even smaller transistors
on an integrated circuits, et cetera.
Why is that? Why we try to
make the base element
of computational process as small as possible?
Well, the answer is obvious and there are three reasons.
First, with smaller elements,
we can pack them all in the same volume.
So, we can have the bigger number of states in the same volume.
The second, the smaller elements have smaller inertia.
This electronic devices, this inertia is
called inductance and the smaller,
is conductor of the smaller inductance it
has and the faster it can change the states,
so the faster we can compute.
Again, with smaller conductors,
with smaller inductances, we can dissipate
less energy which is also an important issue.
So, now computer devices reach this limit,
then nanometers base elements they're p-n junction
and this is only 200 times bigger than a hydrogen atom.
This means that this miniature organization has a limit.
We can't implement the p-n junction on one atom or molecular.
So, we have to choose something
else if you want to grow our computer power,
something else than a p-n junction,
something else than transistor.
This something else has to be as small as atom or even smaller,
but at least scales we can disregard this quantum effects,
which means that new generation computers
must employ their quantum effects.
It appears that the next generation
of computers has no way but to be quantum.
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