"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 2
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來自 Saint Petersburg State University 的課程
The Introduction to Quantum Computing
36 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
So, if you don't know anything about the quantum state,
then measurement gives us very little information about it.
But if you do know something,
then we can attune our measurement process to obtain more.
Let's see an example.
Imagine that we know from
a trusted source that
our qubit state is either this vector or this vector.
This will go phi,
phi is the sum of zero and one.
Remember, here we have vector zero and the vertical vector is
vector one on this graph.
This we got t and t
is one divided by square root of two, one minus zero.
So, we know that our quantum state is either this or this.
Then, measurement of this qubit in
the standard basis will give us
zero additional information because for both these states,
the probability of outcome one is
the probability of outcome zero they're equal and it is one half.
So, if you obtain one as measurement outcome,
it gives us nothing because it has the same probability for
both vector psi and phi and the same is for zero.
But we can choose another basis
to obtain more information from this state.
I'm going to choose the basis,
we shall draw in green here.
So, these basis has
also two vectors of course and the first vector we'll call plus,
this vector and this actually the same as vector phi
and vector minus which is here is this.
This basis is called the Hadamard
Basis in the name of French mathematician Jacques Hadamard.
Now, these two vectors are orthogonal,
they are unitary, so they form the orthonormal basis of course.
If you measure all the system in this basis,
then result plus will tell us that the system
was and still is in the state phi
and minus outcome tells us
that the system was instead psi, before the measurement.
So, choosing the correct basis is
crucially important for obtaining information from quantum system.
Though we don't have the state of the art quantum hardware here,
we can still illustrate the measurement process.
Remember I told you about electromagnetic wave that we can carry
qubits coded by the polarization of photons.
The polarization is the direction of
the wave oscillation which is orthogonal to
the direction of wave propagation.
Fortunately, we have crystals
which can detect and alter this polarization,
this characteristic of a photon and on molecular level,
these crystals are organized like this.
They contain dipoles which are all oriented along one axis.
You see these little springs here on the picture,
actually this picture is not exactly correct,
there are no springs,
there on the molecular level but the idea of spring there exist.
If we apply an electromagnetic field
alternating electromagnetic field
with the vector of oscillation parallel to this axis,
then these charged particles pluses and minuses begin
oscillate around the positions of the minimus of potential energy.
As you know, when charged particles oscillate they produce waves,
so this another wave,
secondary wave is depicted here, in blue.
It's not hard to show that this wave,
this secondary wave will be shifted half of the period to
the primary wave if this primary wave is for example sinusoidal.
Here on this slide,
you can see the picture of the fallen wave here,
we see this crystal from the side.
So, the following wave falls into the crystal,
it passes through, it is depicted in orange.
Here we can see the secondary wave in
blue and of course it propagates in both directions.
This we shall draw in green,
it is a reflected wave and
this wave which is orange is primary and blue is secondary,
they're shifted half of the period so they
adopt resulting and no wave here.
So, the wave doesn't pass the crystal if it
is oriented along this axis.
On the contrary, if you have the wave polarized like this,
then it does not cause
oscillations of this charged particles so it doesn't
produce the secondary wave and the wave easily passes the crystal.
So this is why some materials
are not transparent for some wavelength.
If inside of the material there is some charged particles which
can oscillate with a frequency of this falling wave,
then these charged particles produce secondary waves
and they self-destroy the secondary wave,
self-destroy with the primary wave.
So wave does not
pass through this material and this fully reflected.
This is why all metals are not transparent
for almost all frequencies of the electromagnetic waves because
of free electrons there and also this is why
they are so shiny because of these reflected waves here.
This is probably another surprise to you.
So if you have this axis which shows the waves to pass,
it's called the optical axis of the crystal.
So, I draw here the optical axis,
this is the axis for which all waves polarized along this axis.
They are allowed to pass through the crystal.
This axis is orthogonal and
all waves which are oriented like this polarized like this,
they don't pass the crystal.
It is actually the line along which all the dipoles are oriented.
If you have a wave polarized like this with
some angle theta with the optical axis,
then it is not probably a surprise to you that
the wave passes through the crystal
with probability cosine square Theta and it doesn't pass
with probability sine square Theta.
So these crystals can perform for us a measurement
of qubits which are encoded by the photons polarization.
These crystals, linear polarizers are easy to get.
They are used in many modern devices.
For example, the surfaces of
almost all LCD screens have several polarizing films on them,
but if you want to get something this clear and transparent,
then you probably have
to go to a shop where photographers buy their stuff.
The reason why photographer use these things,
these linear polarizers is this: Many surfaces when they reflect,
the light also polarize it.
So you can either enhance
this polarized light with linear polarizer or to reduce it.
For example, if you want to take a picture of a blue blue sky,
so we want to enhance the blue color of the sky,
you're going to use this thing to enhance this blue color because
the light reflected from
different layers of atmosphere is polarized.
If you want to take a picture of
a shallow pond or the bottom of a shallow pond,
then you probably would have a problem
with the rays of light which are reflected from
the surface of the water because they mix with
the light which is reflected from
the bottom of the pond and they spoil the image.
So with this polarizer,
you can also restrict this reflection
from the surface of the water because it slightly polarizes it.
If you have two of these things,
then we can do even more because you understand that the light
which pass through the polarizer
is polarized along its optical axis.
So if you cross the optical axis of polarizers,
then the light which came from the first polarizer is
restricted by the second polarizer like this.
So now I hope you don't see me.
Now you do.
Aside of these useful application of hiding me from students,
there are many more.
Almost all transparent materials alter the light polarization.
So if you put something
transparent between these two crossed polarizers,
then the light which came through
this transparent material can now
pass the second restriction polarizer
because its polarization slightly changed.
If you have some higher densities in this transparent material,
then the light there is shifted,
the light polarization is shifted more.
So on these areas where we have higher densities,
we will have more light
which is not restricted by the second polarizer.
Here on this slide, you can see the picture of
ordinary glasses which is put between two polarizers.
So, it is a picture in the polarized light and you can see
some tensions here for example on
the place where glass is attached to the glass screen,
there are more light because there are some tensions
which the glass has in this area,
and thus we can analyze transparent materials.
We can see the defects in them which we don't see in normal light.
So, we can now understand that this polarizers can perform
the measurement process for us for the light polarization and
the choice of the basis is performed by rotation of the polarizer.
So, the choice of the basis is rotation of polarizer and
the light which pass through the polarizer
is now polarized along its optical axis and
the light which is fully reflected is
polarized perpendicularly through its optical axis.
