"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!
Qubit Measurement. Part 1
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來自 Saint Petersburg State University 的課程
The Introduction to Quantum Computing
38 個評分
從本節課中
Mathematical Model of Quantum Computing
與講師見面
Сысоев Сергей Сергеевичкандидат физико-математических наук
Кафедра системного программирования
[MUSIC]
In the last lecture we have chosen the auto
normal buzzes in the two dimensional qubit space.
This shows seems to be qubit array and in practice,
it is needed for the measurement of the quantum bit.
And it is defined by the particular parameters of the quantum hardware.
Now, why would we want to measure the qubit?
As you remember, quantum systems,
which we employ for computing, are isolated systems.
So we reside in some kind of big black box.
As you remember from the previous week and if you don't know the state
of the system and we want to know it then we have to introduce
some external part to the system.
Some measurement device or some observer.
And as you remember, this observation does not alter the state of the system.
Instead, it records information about the system into the observer and
alters the wave function of the observer.
Like this.
Here, yeah.
They're not there quantum state of the observer.
And, here we have this quantum bit, it's a quantum system and
that observer chooses to distinguish two different states.
Like this.
And after observation we have two different copies of observer.
Each of them observing different state of the system.
So this how measurement looks like from them.
Multiverse point of view.
Now, what we can tell about this measurement process.
First, we understand that this observer can choose
the states he is going to distinguish.
So it is the choice of the basis here or
the choice of observable operator as code.
And after the measurement,
observers subjectively obtains one of these states.
So one of these by these vectors, and this is a very important
point when you choose buses for
this measurement for observation after observation.
You will always get get one of the vectors of these buses.
As your measurement outcome and since for one qubit,
there are only two possible such measurement outcomes
like two vectors because the space is two dimensional.
That means that after measurement,
after observation you will receive one beat of information.
And for you after measurement subjectively.
The system will be in this state that you have update as the measurement outcome.
So subjectively it will look like you have altered the state of the system.
Like you destroyed it plus some combination of vectors before measurement.
Like this.
We have vector phi here.
But after the measurement,
the system will be either in the state 0 or in state 1.
And the probability to obtain the stat 0 as
the measurement outcome equals to alpha modulo squared.
And the probability to get outcome 1 equals to beta model squared.
So the closer is vector here to 1 of the basis vectors.
The bigger is probability top 10.
This basis vector is the measurement outcome.
Actually, we can compute, for example, this dot product.
To see that this is
alpha 0 plus beta 1.
We choose alpha.
Remember, we have linearity on this second argument, alpha 00,
which is one plus beta 0, which is 0.
And it is just alpha.
And if take the absolute value
then we see that basis the cosine,
Of this angle theta here and,
If you compute, This dot product,
It will be beta, it'll be sine of that so
probability of obtaining vector
0 is cosine squared of angle theta and
probability of obtaining vector
1 is sine squared theta.
Well I told you that this measurement observation does not
alter the very function of the system.
Instead it alters the very function of the observer.
But I also told you that the observer here can choose the basis of the states.
It, he choose to distinguish.
And after observation, each particular copy
of that observer entangles with one of these states,
and subjectively observes this, as the measurement outcome.
So the system for this observer here is now in state one.
How's that possible, for example, for vector phi here?
We will have observer which observes 0 and
the copy of observer which
subjectively observes 1.
And these are now the real states of the system.
But we remember that the real state of the system was this vector phi.
And of course this observer can choose another basis and
in this case, they'll be another vectors here.
How's that possible if you don't change the system.
From the interpretation from the multiverse point of view,
when we say that this system is in a state v,
that means that among all the snapshots of the Universe,
among all this corpus of the universe which contains this state.
The mean value of this state is phi.
So it doesn't mean that in every universe, this state is exactly vector phi.
We understand that is mean value over
all the corpus of the universe.
So there are Corpus of the universe
where the state is 0 and where the state is 1 and
the shares of this corpus in the whole multiverse.
Is defined by these coefficients.
So they are proportional.
So the squares of the models of these coefficients.
And then we choose some basis here.
If you choose, for example, to measure our qubit in the basis zero, one.
Then, we consider this system only in the universes.
Which contain these vectors here.
So let us sum up everything that we've learned about
the measurement of the qubit.
To perform the measurement,
first we have to choose the basis or the observable operator.
And the measurement outcome is going to be one of the vectors of this basis.
Second, the measurement is a probabilistic process.
And the probability of obtaining each of the vectors of the basis
are defined by the decomposition of the system vector in this basis.
Felt subjectively, measurement is destructive.
Because after measurement, the system for us flips or
jumps to the vector of the basis that we obtained as a measurement of outcome.
And the last one after the measurement we have then only one vector
of two possible vectors.
So the basis will be obtained one beat of classical
information from one qubit of quantum information.
And now some of you may probably ask yourself, is there a way to obtain more?
Actually to obtain these coefficients here while the measurement process.
Well, if we run measurement once, we obtain only one bit.
We don't obtain this two complex values, alpha and
beta, which will be extremely nice to know in some cases.
And if we perform measurement twice, we obtain always the same result because
for example, if we obtained one as the measurement outcome,
the second and all other measurements of the system will give us one.
Again, because the probability of obtaining one is one.
And the probability of obtaining
zero is zero for this example.
But maybe, we can make, for example,
thousandth, or million corpus of this state.
And to measure all these states to make the estimation
of the alpha and beta but of the models.
Which will give us much more information than just one bit.
And unfortunately, we cannot do even this because the theorem
proved by.
1982, tells us that
coping of unknown quantum states is impossible.
So, To conclude,
both these systems, this classical system here on the left and
this quantum system on the right have many possible values like
we distinguish here on the vertical and horizontal,
but there are much more other directions of polarization.
And here, we distinguish on the left and the right part of the camera.
But there are a lot of different actual states of this system.
And from the informative point of view.
This system here, and these systems here contain only one bit.
The process of reducing the number of states on the left is called digitization.
And on the right for the quantum system.
We call this process measurement.
