Quantum Computing

Meghaditya Roy ChaudhuryBCSE – 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 Feynmanconsidered 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>.

Excited State

Ground State

Nucleus

Light pulse of frequency λλλλ for time interval t

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:

|ψψψψ> = αααα |0> + αααα |1>

Where αααα and αααα are complex numbers and |αααα | + | αααα | = 1

1 2

1 2 1 22 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 superpositionof 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

Data unit: bit

x = 0 x = 1

0

1

0

1

Valid states:x = ‘0’ or ‘1’ |ψ⟩ = c1|0⟩ + c2|1⟩

Quantum Computation

Data unit: qubit

Valid states:

|ψ⟩ = |0⟩ |ψ⟩ = |1⟩ |ψ⟩ = (|0⟩ + |1⟩)/√2

=|1⟩ =|0⟩= ‘1’ = ‘0’

Classical computation vs. Quantum Computation

Classical Computation

Measurement: deterministic

x = ‘0’

State Result of measurement

‘0’

x = ‘1’ ‘1’

Quantum Computation

Measurement: stochastic

|ψ⟩ = |0⟩

|ψ⟩ = |0⟩ + |1⟩

State Result of measurement

|ψ⟩ = |1⟩

√2

‘0’

‘1’

‘0’ 50%

‘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 cryptographymethod, 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

# bits 1024 2048 4096factoring in 2006 105 years 5x1015 years 3x1029 yearsfactoring in 2024 38 years 1012 years 7x1025 yearsfactoring in 2042 3 days 3x108 years 2x1022 years

with a classical computer

# bits 1024 2048 4096# qubits 5124 10244 20484# gates 3x109 2X1011 X1012

factoring time 4.5 min 36 min 4.8 hours

with potential quantum computer

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

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-representabilitytheory

Thank You