"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!
Quantum System Evolution. Computations. Part 1
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来自 Saint Petersburg State University 的课程
The Introduction to Quantum Computing
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Mathematical Model of Quantum Computing
与讲师见面
Сысоев Сергей Сергеевичкандидат физико-математических наук
Кафедра системного программирования
Okay. We have just launched
the mathematical model of quantum information.
We know that quantum information is represented by
qubits which are vectors in two-dimensional Hilbert space.
For a system with multiple qubits,
for example, with n qubits,
the vector will be in a space with dimensionality two in the
power n. But for computations, we need more.
We need not just to store these qubits,
we need also to modify them.
So, we need to modify these vectors in Hilbert space.
The physicist will tell us that
any such modification must be unitary.
So, any computations, any program for
a quantum computer must be represented by a unitary operator.
Let's recall this notion.
The unitary operator.
It is a linear operator for which it's adjoint operator.
It's also inverse.
The matrix of an adjoint operator is
the conjugate transpose of the matrix of original operator.
So, this is our definition of unitary operator.
These unitary operators also have a very important feature.
They preserve the length of the vectors.
So, norm of ux willl be equal to norm of x.
They also preserve the inner product,
so they preserve angles.
If we have xu, uy,
and this just x and y,
this among with linearity is
sufficient for an operator to be unitary.
So, if you have some square matrix and we
have to check if it implements some unitary operator,
we can either check this,
or we can check if this operator
maps an orthonormal basis to orthonormal basis.
Let's see some examples.
The orthonormal transform or the familiar name to us.
So, this is the matrix for the orthonormal transform,
probably the most used operator in quantum computation.
H star here is itself.
Because we don't have imaginary parts here,
so we don't have to conjugate and to transpose is
also changing places all this once.
Let's multiply our H by H, so it's one-half,
one minus one,
one multiplied by one plus one multiplied by one.
We have two here,
and one multiplied by one minus one multiplied by one,
zero, and again zero, two.
So, it is just matrix of the density operator.
So, okay, we can see that H is unitary.
Let's see how it acts on our basis zero and one.
So, H zero is square root of two,
one, one, one minus one, one,
zero which is one divided by square root of two,
one and one, which is just this vector.
The mark plus, and
maybe not a surprise to you that H acts on vector one.
It gives us other minus.
So, the other minus,
the lighter the other gets maps our traditional normal basis,
zero, one, two, the other mark basis plus minus.
Of course, since H is its own inverse,
H of plus equals to zero and H of
minus equals to one.
Another very useful operator which we call X,
the matrix you can see on the slide.
Let's see how it acts on basis one-zero.
So, X-zero is zero one, one zero,
one zero, and we have zero here and one here,
this one and X of one is again zero, one, one,
zero, zero one, and we
have one, zero, zero.
So, it maps our standard basis
zero one to itself, but it flips it.
So, this actually the quantum implementation of not operator.
Since it maps orthonormal basis to orthonormal basis,
we can be sure that it is unitary.
The next operator is CNOT,
all the previous operators,
they're matrices two-by-two, so they
acted on the space free of one qubit.
This operator acts on a space with
two qubits since it is in matrix four by four.
Let's see how it acts on our standard four-dimensional basis.
So, CNOT of zero, zero,
that's this matrix, and this vector.
It's actually itself.
So, it is again zero, zero.
It's zero, one.
It will have the same result.
It's mapped to zero, one.
You can easily check that one,
zero will be mapped to one,
one by CNOT, and one,
one will be mapped to one, zero.
So, again, this operator
maps orthonormal basis to orthonormal basis,
thus, it is unitary,
and we can see that it flips
the second qubit only if the first qubit is one.
So, it is a conditional NOT operator.
This bit, the highest bit,
is the control bit and
this bit is under the control of the controlled bit.
