"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!
Measuring the Multiple Qubits Systems
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來自 Saint Petersburg State University 的課程
The Introduction to Quantum Computing
37 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
As I already told you, the most amazing fact
about the quantum system description we just learned,
we just introduced, is that for any vector like this,
as you can see on the slide.
Any unitary four-dimensional vector.
They exist.
Quantum system or is can be physically implemented.
The quantum system which is described
by this state, by this vector.
But not any vector like this can be constructed,
the way we did before by
this tensor product in the previous lecture.
What would you say about these set of states?
There are famous representatives of a set of
states which are called the Bell states
for each alpha and beta
plus minus one divided by square root of two.
Well, indeed it is some unitary four-dimensional vector.
So describes some state of a system onto separate particles.
But what can we say about the states of those particles?
Can we write the states down?
No, we can't. It appears that there's
no such thing as a state of a separate particle here.
I mean, the separate particles that
exist they can be taken to the pocket
and put to the pocket and taken
to the different parts of the universe.
But these particles still don't have a separate state.
They are in some sense entangled.
This is what is called,
this phenomenon, the quantum entanglement.
Einstein did not like these entangled states because,
and the reason for that was that,
actually for any system with multiple qubits,
you can measure these qubits separately.
Because these qubits are carried by
separate particles and the measurement process
is when you do something with the particle.
So the particles are separate and you can measure them separately.
For the state like this for example,
you can measure only the first qubit.
Imagine that you did it and you obtained zero,
so the first qubit is zero.
So the first qubit,
the state of first qubits is collapsed but the second is not.
The second is still alive.
So you have to remove
all the vectors which contain one on the first place.
We will have this state.
You only have to normalize it,
so the vector must be unitary.
So this is the description of our state
after the measurement of the first qubit.
So the first qubit between nodes is
now zero but we still don't know the second cubit
and this measurement did not alter or change the second particle.
Now, imagine that you have two entangled photons which
describe it by this state.
Imagine that these photons travel in the opposite directions.
They do it for quite a long time for example for one year.
So this very large distance between these two photons.
Eventually, one of these photons meet linear polarizer on its way.
Suppose that it passes the polarizer.
Now, it is in state one.
Now what is the state of the whole system,
the system of two photons?
Call it photon [inaudible]. The first one.
So it is described by the first qubit for example.
Now, we don't have this part in our state
because on the first place here stays zero,
and we're now pretty sure that the first qubit is one.
So only this state
stays and the whole system is now in the state 1, 1.
But please note that the second photon this one.
Which is very far from the
first did not meet any linear polarizer.
Was not altered in any way at all.
But for some reason,
immediately after the measurement of the first photon,
it also changes its state.
Since it goes immediately,
Einstein called it some spooky action at a distance.
But if we employ
the multiverse interpretation of quantum mechanics,
then we can explain this situation without any notion of action.
For example for the state like this.
If you describe it in the multiverse.
There are four types of universes in multiverse.
Each differ only with the state of
this system for the first type of universe.
This system will be in state zero, zero seconds.
Will be the set zero, one, etc.
The share of universes of each type will depend on
the coefficients before each state.
So the share of this will be alpha squared, beta squared etc.
Now, when we measure this state.If we have
some observer which is eager to know which state is it.
They all learned that
this measurement process splits the wave function of the observer.
So now, we have four different types of observers.
Each witnessing different states.
Now imagine that this observer,
it measures only the first qubit.
So he is now sure,
for example that this first qubit is zero.
Then we have actually two types of observers.
This observer here, which is sure
that the first qubit for him it is subjectively zero.
The copy of this observer for which the first qubit is one.
These both copies of the observer are still not sure,
they still don't know the value of the second qubit.
So the second qubit for them are still,
is still in the superposition.
Now, if you look at this state here,
it has only two possible values.
Only two basis vectors employed.
So when our observer
measures only vectors first or only the second qubit,
this measurement precisely defines the type of the universe.
We've reached this observer now is entangled.
So this measurement of the first qubit defines the type of
the universe for this copy of an observer.
There is no action at all.
This is an explanation of this effect without any action.
Now, this is the state which
can not be easily implemented on the flipping coins.
If you have for example two flipping coins,
there's no way this- I don't know the way.
How you can arrange this state and this flipping coins.
Because for quantum states,
they really exist in some places in the multiverse.
They actually exist and they can compute for the flipping coins.
The only measurement outcome is real.
