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- Quantum Computing

Meghaditya Roy Chaudhury

BCSE – IV

Roll – 000810501052

Jadavpur University - Overview

Definition of Quantum Computing.

Why Quantum Computing is necessary?

Advantages over Classical Computation

Quantum Algorithm: Shor’s Algorithm

Current Developments and Future Prospects - 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. - Why Quantum Computing?
- 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 - Stretching the limits: But how far?
- 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. - 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. - 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) - 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> - Properties Of Quantum Mechanics

Quantum Superposition

Quantum Entanglement - 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> = α |0> + α |1>

1

2

Where α

2

2

1 and α2 are complex numbers and |α | + |

1

α | = 1

2

A qubit in superposition is in both of the

states |1> and |0> at the same time - 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. - 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〉 = c |0〉 + c |1〉

1

2

x = 0

x = 1

|ψ〉 = |0〉

|ψ〉 = |1〉

|ψ〉 = (|0〉 + |1〉)/√2

0

0

1

1 - 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% - 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 - 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 - 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 - 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". - Practical Implementations

Ion Traps

Nuclear magnetic resonance (NMR)

Optical photon computer

Solid-state - Applications

Factoring – RSA encryption

Quantum simulation

Spin-off technology – spintronics,

quantum cryptography

Spin-off theory – complexity theory,

DMRG theory, N-representability

theory - Thank You