Q1L1a.txt

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Let us begin with a description of quantum mechanics,
and a brief overview of elements of quantum circuits.
The basic elements we need for this course
are captured entirely by four postulates.
They cover states, gates, measurement, and composition.
Postulate 1 describes how states are
defined in the context of isolated physical systems.
Basically, we take any system such as a spin one half,
and represent it as a state vector within a Hilbert space,
which is an inner product vector space.
The state space may have an infinite number of dimensions.
For example, such as with a harmonic oscillator system.
You may also be quite familiar from quantum mechanics
with the use of continuous variables, such as position,
in describing the quantum states.
However, in our study of quantum information science,
we will largely be interested in discrete Hilbert spaces.
Keep in mind that these Hilbert spaces
are complex vector spaces with an inner product that
is defined.
States are thus vectors, such as this zero state being
1,0 and the one state being 0,1, two component vectors which
are spinors, or are otherwise known as qubits
in quantum information science.
So that tells us about states.
Let us now move on to Postulate 2, which talks about gates.
Gates describe the evolution of closed quantum systems, i.e.
isolated systems, and thus will be unitary transforms.
Specifically, the unitary transform
will govern how a state psi evolves from the initial time,
t1, to some final time, t2.
Psi 1 and psi 2 are thus related by a matrix U which
multiplies a state vector psi 1 to produce a state vector
psi 2.
This unitary transform has a property
that U dagger U is equal to identity,
namely that it is a unitary matrix.
Describing quantum gates is a core concept
for quantum circuits.
Thus, let us consider a few examples of quantum gates.
Specifically, let us look at the action of a few common quantum
gates on qubits.
Recall the state zero.
The X gate-- 0,1, 1,0, a two by two
matrix-- acts on the zero state to give the one state,
as is shown by this simple matrix multiplication.
Thus, this X gate may be considered the quantum analog
of the classical NOT gate.
A really uniquely quantum gate is this transform,
which is described by the two by two matrix 1,1, 1, minus 1.
This is the Hadamard gate.
The Hadamard gate acts on 0 to give
a superposition of 0 and 1, an equal superposition
with coefficients 1 over square root of 2.
That is, the two-component vector 1 over
square root of 2, 1 over square root of 2.
You can check for yourself that the Hadamard operator
is its own inverse.
OK.
The example we just finished was a gate
acting on a single qubit space.
Let us now look at a slightly more complicated example
of a gate acting on a slightly larger Hilbert space.
This is a state in a four-dimensional Hilbert
space with four coefficients, which are arbitrary.
Of course, subject to normalization.
We may also write the state as a sum over basis vectors, which
are labeled 0 through 3 with coefficient c sub x rather than
a through d. That's the same.
We may also write it as a four component
vector in this inner product Hilbert space.
The unitary transform we'll look at
is this four by four matrix with 0s and 1s.
Note that the columns and rows are
labeled by the same basis vectors as in the state
that we just described.
Thus, the first column and row act
on the top-most label, and the bottom right
acts on the bottom label.
We can therefore work out straightforwardly by a matrix
multiplication that the action of U on psi
is to flip the labels 2 and 3.
This should not be a surprise, because after all U
is a permutation matrix.
So it permutes the labels of the state.
OK.
Now that we have gone through two gate examples,
let us move on to Postulate 3.
This postulate describes measurements.
Measurements are going to be described
by a collection of measurement operators, M sub m.
And these operators act in the state space of the system being
measured with an index m.
The index describes some possible measurement outcomes
that may occur when the measurement is done.
M occurs with probability given by an inner product
of the state being measured and the measurement operator.
Measurement changes the quantum state.
If an initial state psi is measured,
then the post-measurement state is given by this expression.
Basically, a product of M with a state
divided by the probability of the measurement result being
obtained.
It is important to keep in mind that these Ms are
pretty arbitrary.
The only constraint is that the sum of M dagger M
must be identity.
The Ms need not be square.
For example, Ms may be projection operators.
The operator formed by the product of two Ms
may be familiar to you from a different context.
They are also known as positive operator-valued measures.
Measurements are something we will return to a great deal
later.
So for now, let us move on to the fourth postulate,
composition.
This is how we mathematically describe
how individual physical systems may
be described as a composite.
Mathematically, it is like digits.
These digits are tensor products for linear vector spaces.
For example, if we combine two qubit states,
there are four possible outcomes.
If we have n qubits, there are clearly
2 to the n possible outcomes.
And these are basis vectors for the new inner product Hilbert
space, which is the combination of the n qubits.
Let us look at some examples of composition.
Consider two qubits.
We may easily write them as two digits like this.
Sometimes, we'll explicitly write the tensor product
operator, as well.
A Hadamard operator acting on the two qubits written
like this represents the tensor product
of the Hadamard acting on the two individual qubits.
We may work out what that is explicitly simply
by taking the tensor product of the two superposition states.
Note that the tensor product distributes over addition.
Thus, this expression yields a superposition
over the four basis vectors.
Incidentally, it is common and convenient to represent
these binary vectors often by integers.
00 is 0.
01 is 1, et cetera, such that we may write this superposition
state also as a superposition of 0, 1, 2, and 3.
An expression common in quantum information science
is thus the use of n Hadamard operators acting on n qubits,
initially at 0, producing an equal superposition
over all basis vectors labeled from 0 to 2
to the n minus 1 written out as integers.
This is very useful.
x is just the set of all binary strings that have n bits.
All right.
Let's now also look at some examples
of tensor product structure described
with quantum circuits.
Here, I show a 0 qubit state going through a single Hadamard
gate, giving the equal superposition we've
previously seen.
We may do this twice, with two quantum circuits.
And this is simply described as the tensor product of the two.
Now, there is a much more interesting case
than two separate qubits undergoing gates like that.
Recall that we had looked at this unitary transform
on a four-dimensional Hilbert space.
We may also consider this unitary transform
as acting on a two qubit composite space.
So let's take two qubits, both in their zero state.
And we'll represent this transform
by the following circuit diagram.
That circuit symbol is known as a controlled NOT gate.
The first qubit controls whether a NOT,
an X gate, happens to the second qubit, the one on the bottom.
If we replace the first part of this circuit
with a Hadamard acting on a zero state,
then we actually get something far more non-trivial.
Recall that the Hadamard gives an equal superposition state.
And what we find is the output of this controlled NOT gate
gives a 2 qubit state, which is much more interesting.
This is known as an entangled state.
And we'll come back to this later in the course.
OK.
That ends our brief journey through the four
basic postulates of quantum mechanics
which we'll use regarding states, gates, measurement,
and composition.
Basically, this is all we need about the fundamentals
of quantum mechanics for quantum information science.