quantum computing Fundamentals and Applicaiton

QUANTUM COMPUTING
A PRESENTATION BY
ARCHANA M (3rd Year ECE A)
PAVITHRAA VK (3rd Year ECE A)

AGENDA:
1. What is Quantum Computing?
2. How does a Quantum
Computer work?
3. Basic Concepts of Quantum
Computers
4. Mathematical Applications in
Quantum Computers
5. Quantum—-Computing
Algorithms and Case Studies
6. IBM Qiskit Demo

Just one! Your own.
(With a little help from your smart phone)
Tip
Remember. If something
sounds like common
sense, people will
ignore it.
Highlight what is
unexpected about
your topic.

WHAT IS QUANTUM COMPUTING ?
Quantum computing uses specialized technology including
computer hardware and algorithms that take advantage of
quantum mechanics to solve complex problems that
classical computers or supercomputers can’t solve, or
can’t solve quickly enough.
QUANTUM COMPUTERS:
A computer that uses laws of quantum mechanics to
perform massively parallel computing through
superposition, entanglement, and decoherence is called as
a quantum computer.

CLASSICAL COMPUTING VS QUANTUM COMPUTING

HOW DOES QUANTUM
COMPUTER WORK?

• Quantum states, represented by Dirac’s ket, | 𝜳>, evolve in time
according to the Schrödinger equation:
which implies that time evolution is described by unitary transformations:
where |𝝋> is the quantum state (wavefunction) and H is Hamiltonian
This theory, which has been extensively tested by experiments, is
probabilistic in nature. The outcomes of measurements on quantum
systems are not deterministic.
• Between measurements, quantum systems evolve according to linear
equations (the Schrödinger equation). This means that solutions to the
equations obey a superposition principle: linear combinations of solutions
are still solutions.
QUANTUM MECHANICS:

Unitary Transformation as Quantum Computing:
PREPARATION:
The initial preparation of the state defines a
wave function at time t0=0.
STATE EVOLUTION:
Evolved by a sequence of unitary operations
…..
MEASUREMENT:
Quantum measurement is projective.
Collapsed by measurement of the state
Quantum Computing
Approach using Flowchart

Quantum Computing Approach using Quantum Circuit.
On a quantum computer, programs are executed by unitary evolution of
an input that is given by the state of the system, |n > , which can in
either 0 or 1 state. Since all unitary operators are invertible, we can
always reverse or ‘uncompute’ a computation on a quantum computer.

QUANTUM BIT
(QUBIT):
● A quantum bit is any bit
made out of a quantum
system, like an electron or
photon.It is the smallest
unit of information in a
quantum computer.
● It represents the state of
the wavefunction |> in
Schrödinger equation.
● Just like classical bits, a
quantum bit must have two
distinct states: one
representing “0” and one
representing “1”.

The general form of a qubit state can be represented by:
𝜶0|0› + 𝜶1|1›
where 𝜶0 and 𝜶1 are complex numbers that specify the probability
amplitudes of the corresponding states.
Normalization condition: |𝜶0|^2 + |𝜶1|^2 = 1

CLASSICAL BIT VS QUBIT:
Classical Bits : It’s a single unit of information that has a value of
either 0 or 1 (off or on, false or true, low or high).
Quantum Bits : A qubit is a two-state quantum-mechanical system,
one of the simplest quantum systems displaying the peculiarity of
quantum mechanics.

CLASSICAL BIT QUBIT
It can be in two distinct
states, 0 and 1.
It can be in state |0> or in state |1>
or in any other state that is a linear
combination of the two states.
It can be measured
completely.
It can be measured partially with
given probability.
They are not changed by
measurement.
They are changed by measurement.
It can be copied and can be erased. It cannot be copied and cannot be
erased.

BASIC CONCEPTS OF
QUANTUM COMPUTING
1. Superposition
2. Entanglement
3. Decoherence
4. Measurement

SUPERPOSITION:
Every quantum state can be represented as a sum of two or more other
distinct states.Mathematically, it refers to a property of solutions to the
Schrödinger equation; since the
Schrödinger equation is linear, any linear combination of solutions will
also be a solution.
A single qubit can be forced into a superposition of the two states denoted
by the addition of the
state vectors:
|𝜳> = 𝜶0|0>+𝜶1|1>
Where 0 and 1 are complex numbers and |0|2 + | 1|2 = 1
A qubit in superposition is in both of the states |1> and |0 at the same time

EXAMPLE
If this state is measured, we
see only one or the other state (live
or dead) with some probability.
The classic example of
superposition is Schrödinger’s Cat in
a black box. Since both a living and
dead cat are obviously valid
solutions to the laws of quantum
mechanics, a superposition of the
two should also be valid.
Schrödinger described a thought
experiment that could give rise to
such a state.
Consider a 3-qubit register. An equally
weighted superposition of all possible states
would be denoted by:
|𝜳>=1/√8|000>+1/√8|001>+...+1/√8|111>

ENTANGLEMENT:
When a pair particles is generated, interact, or share spatial proximity in a
way such that the quantum state of each particle of the pair or group cannot
be described independently of the state of the others, even when the
particles are separated by a large distance.

• An entangled pair is a single quantum system in a superposition of
equally possible states. The entangled state contains no information
about the individual particles, only that they are in opposite states.
• If the state of one is changed, the state of the other is instantly adjusted
to be consistent with quantum mechanical rules.
• If a measurement is made on one, the other will automatically collapse.
• Quantum entanglement is at the heart of the disparity between
classical and quantum physics:entanglement is a primary feature of
quantum mechanics lacking in classical mechanics.
• Entanglement is a joint characteristic of two or more quantum particles.
• Einstein called it “spooky actions at a distance”

Suppose that two qubits are in states:
The state of the combined system is their tensor product is:

DECOHERENCE:
• Quantum decoherence is the loss of superposition, because of the
spontaneous interaction between a quantum system and its
environment.
• Decoherence can be viewed as the loss of information from a system
into the environment.

MEASUREMENT:
If a quantum system were perfectly isolated, it would maintain coherence
indefinitely, but it would be impossible to manipulate or investigate it.
• A quantum measure is a decoherence process.
• When a quantum system is measured, the wave function | collapses to a
new state according to a probabilistic rule.
• If |> = 0|0> + 1|1>, after measurement, either | = |0 or | = |1,
and these alternatives occur with certain probabilities of |0
|2 and |1|2 with |0|2 + |1|2 =1.
• A quantum measurement never produces | = 0|0> + 1|1>.
• Example: Two qubits:|> = 0.316|00›+0.447|01›+0.548|10›+0.632|11›
The probability to read the rightmost bit as 0 is |0.316|2+ |0.548|2= 0.4

QUANTUM GATES:
A quantum gate (or quantum logic gate) is a basic quantum circuit
operating on qubits.
• They are the building blocks of quantum circuits, like classical logic
gates are for conventional digital circuits
• Due to the normalization condition every gate operation U has to be
unitary:U*U = I
• The number of qubits in the input and output of the gate must be
equal; a gate which acts on n qubits is represented by 2n x 2n unitary
matrix
• Unlike many classical logic gates, quantum gates are reversible.
MATHEMATICAL APPLICATIONS IN
QUANTUM COMPUTING:

UNITARY MATRIX
In linear algebra, a complex square matrix U is unitary if its
Conjugate transpose U*is also its inverse, that is,
U*U =UU*= I
if where I is the identity matrix.
Unitary transformations are linear transformations that preserve vector norm;
In 2 dimensions, linear transformations preserve unit circle (rotations and
reflections).

SINGLE QUBIT STATE:
PAULI X GATE:

Acting on pure states, the Pauli X gate becomes a classical NOT gate.,

● After a qubit in state |0 or |1 has been acted upon by a H gate, the state
of the qubit is an equal superposition of |0 and |1. Thus, the qubit goes
from a deterministic state to a truly random state, i.e., if the qubit is now
measured, we will measure |0 or |1 with equal probability.
● We see that H is its own inverse, that is, H−1 = H or H2 = I.
Therefore,by applying H twice to a qubit we change nothing. This is
amazing!
● By applying a randomizing operation to a random state produces a
deterministic outcome.

TWO QUBIT STATE:
CONTROLLED NOT GATE (CNOT GATE):
● If the control qubit is set to 0, target qubit is the same
● If the control qubit is set to 1, target qubit is flipped

QUANTUM ALGORITHMS:
• Quantum parallelism: by using superpositions of quantum
states, the computer is executing the algorithm on all possible
inputs at once.
• Dimension of quantum Hilbert space: the “size” of the state
space for the quantum system is exponentially larger than the
corresponding classical system.
• Entanglement capability: different subsystems (qubits) in a
quantum computer become entangled, exhibiting nonclassical
correlations.

CASE STUDIES:
CASE STUDY 1 : QUANTUM CRYPTOGRAPHY

CASE STUDY 2 : SHOR ALGORITHM OF FACTORING

CASE STUDY 3: VARIATIONAL QUANTUM EIGENSOLVER

QUANTUM PROGRAMMING:
QCL (QUANTUM COMPUTATION LANGUAGE):

quantum computing Fundamentals and Applicaiton

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