Quantum computing uses quantum mechanics phenomena like superposition and entanglement to perform operations on quantum bits (qubits) and solve certain problems much faster than classical computers. One such problem is integer factorization, for which Peter Shor devised an algorithm in 1994 that a quantum computer could solve much more efficiently than classical computers. While quantum computing is still in development, it has the potential to break popular encryption systems like RSA and simulate quantum systems. Practical implementations of quantum computing include ion traps, NMR, optical photons, and solid-state approaches. Quantum computing could enable applications in encryption-breaking, simulation, and cryptography, among other areas.

1. Quantum Computing
Meghaditya Roy Chaudhury
BCSE – IV
Roll – 000810501052
Jadavpur University

2. Overview
Definition of Quantum Computing.
Why Quantum Computing is necessary?
Advantages over Classical Computation
Quantum Algorithm: Shor’s Algorithm
Current Developments and Future Prospects

3. What is Quantum Computing?
A quantum computer is a machine
that performs calculations based on
the laws of quantum mechanics,
which is the behavior of particles at
the sub-atomic level.

4. Why Quantum Computing?

5. Moore’s Law
Moore's law was a statement made in 1965 by
Gordon Moore, one of the founders of Intel.
Moore noted that the number of transistors
that could be squeezed on to a silicon chip was
doubling every year. Over time, this has been
revised to doubling every 18 months.
This has held true …….. So far

6. Stretching the limits: But how far?

7. Problems
At current rate transistors will be as
small as an atom.
If scale becomes too small, Electrons
tunnel through micro-thin barriers
between wires corrupting signals.

8. Quantum Computing Timeline
The story of quantum computation started as early as
1982, when the physicist Richard Feynman
considered simulation of quantum-mechanical objects
by other quantum systems
1985 when David Deutsch of the University of Oxford
published a crucial theoretical paper in which he
described a universal quantum computer.
In 1994 when Peter Shor from AT&T's Bell
Laboratories in New Jersey devised the first quantum
algorithm.

9. Nobody understands Quantum Mechanics
“We always have had a great deal of difficulty
in understanding the world view that
quantum mechanics represents ”
- Richard Feynman
("Simulating physics with computers" ,1982)

10. Representation of Data - Qubits
A bit of data is represented by a single atom that is in one of two states denoted by
|0> and |1>. A single bit of this form is known as a qubit
A physical implementation of a qubit could use the two energy levels of an atom.
An excited state representing |1> and a ground state representing |0>.
Light pulse of
frequency λ for
Excited time interval t
State
Nucleus
Ground
State
Electron
State |0> State |1>

11. Properties Of Quantum Mechanics
Quantum Superposition
Quantum Entanglement

12. Representation of Data -
Superposition
A single qubit can be forced into a superposition of the two states
denoted by the addition of the state vectors:
ψ
|ψ> = α 1 |0> + α 2 |1>
α
Where α 1 and α 2 are complex numbers and |α 1| 2 + | α 2 | 2 = 1
A qubit in superposition is in both of the
states |1> and |0> at the same time

13. Relationships among data -
Entanglement
Entanglement is the ability of quantum systems to exhibit
correlations between states within a superposition.
Imagine two qubits, each in the state |0> + |1> (a superposition
of the 0 and 1.) We can entangle the two qubits such that the
measurement of one qubit is always correlated to the
measurement of the other qubit.

14. Classical computation vs. Quantum Computation
Classical Computation Quantum Computation
Data unit: bit Data unit: qubit
= ‘1’ = ‘0’ =|1〉 =|0〉
Valid states: Valid states:
x = ‘0’ or ‘1’ |ψ〉 = c1|0〉 + c2|1〉
x=0 x=1 |ψ〉 = |0〉 |ψ〉 = |1〉 |ψ〉 = (|0〉 + |1〉)/√2
0 0
1 1

15. Classical computation vs. Quantum Computation
Classical Computation Quantum Computation
Measurement: deterministic Measurement: stochastic
State Result of measurement State Result of measurement
x = ‘0’ ‘0’ |ψ〉 = |0〉 ‘0’
x = ‘1’ ‘1’ |ψ〉 = |1〉 ‘1’
|ψ〉 = |0〉 + |1〉 ‘0’ 50%
√2 ‘1’ 50%

16. Quantum Algorithm:
Shor’s Algorithm
Shor's algorithm is a quantum algorithm for
factoring a number N in O((log N)3) time and
O(log N) space, named after Peter Shor.
The algorithm is significant because it implies
that RSA, a popular public-key cryptography
method, might be easily broken, given a
sufficiently large quantum computer
Like many quantum computer algorithms,
Shor's algorithm is probabilistic

17. Quantum Algorithm:
Shor’s Algorithm
Shor's algorithm consists of two parts:
A reduction, which can be done on a classical computer, of
the factoring problem to the problem of order-finding.
f(x) = axmod(N)
A quantum algorithm to solve the order-finding problem
The algorithm is dependant on
Modular Arithmetic
Quantum Parallelism
Quantum Fourier Transform

18. Quantum Algorithm:
Shor’s Algorithm
In 2001, Shor's algorithm was demonstrated by a group at IBM,
who factored 15 into 3 × 5, using an NMR implementation of a
quantum computer with 7 qubits
with a classical computer
# bits 1024 2048 4096
factoring in 2006 105 years 5x1015 years 3x1029 years
factoring in 2024 38 years 1012 years 7x1025 years
factoring in 2042 3 days 3x108 years 2x1022 years
with potential quantum computer
# bits 1024 2048 4096
# qubits 5124 10244 20484
# gates 3x109 2X1011 X1012
factoring time 4.5 min 36 min 4.8 hours

19. Quantum computing in
computational complexity theory
The class of
problems that can be
efficiently solved by
quantum computers
is called BQP, for
"bounded error,
quantum, polynomial
time".

20. Practical Implementations
Ion Traps
Nuclear magnetic resonance (NMR)
Optical photon computer
Solid-state

21. Applications
Factoring – RSA encryption
Quantum simulation
Spin-off technology – spintronics,
quantum cryptography
Spin-off theory – complexity theory,
DMRG theory, N-representability
theory

22. Thank You