Amr Kamel Ahmed

PHD Preparatory - 2014

Helwan University – Computer

Engineering

Quantum Computing & Quantum

Information

Introduction

Definitions A quantum computer is a computation system

that makes direct use of quantum-mechanicalphenomena (such as superposition andentanglement), to perform operations on data.[1]

Quantum superposition is a fundamentalprinciple of quantum mechanics that holds aphysical system — such as an electron — existspartly in all its particular theoretically possiblestates simultaneously; but when measured orobserved, it gives a result corresponding toonly one of the possible configurations

Definitions - Qubit

In quantum computing, a qubit or quantum bit is a

unit of quantum information — the quantum

analogue of the classical bit. A qubit is a two-

state quantum-mechanical system

Such as the polarization of a single photon: here the

two states are vertical polarization and horizontal

polarization.

In a classical system, a bit would have to be in one

state or the other, but quantum mechanics allows

the qubit to be in a superposition of both states

at the same time, a property which is fundamental

to quantum computing.

Qubit & Superposition

0 state

1 state

Superposition

Where α and ß could

be complex numbers

in general and

Classical Bits

Qubits

Opening the same door reads the same result

•Opening the complementary

door gives completely random

result.

•Opening the wrong door

destroys information.

•No quantum copying machine

for unknown state qubit

Multiple Qubit Superposition The number of superpositions of n qubits are 2^n

superpositions

So the number of superpositions of 300 qubits are 2^300 which is greater than the number of atoms of the appearing universe

It is really a huge number; however; the number of possible information also are much higher.

Note that the number of possible distinct Boolean circuits for n inputs and one output are 2^(2^n)

when n=8 number of circuits = 2^256

when n=16 number of circuits = 2^65536

Quantum Entanglement Quantum entanglement is a physical

phenomenon that occurs when pairs or groups ofparticles are generated or interact in ways suchthat the quantum state of each particle cannotbe described independently—instead, aquantum state may be given for the system asa whole. It thus appears that one particle of an entangled

pair "knows" what measurement has beenperformed on the other, and with what outcome,even though there is no known means for suchinformation to be communicated between theparticles, which at the time of measurement may beseparated by arbitrarily large distances.

Entangled Qubits

Although the measurement of the value of each

single qubit is completely random.

When they two qubits are entangled a correlation

is established between the two bits such that the

value of one qubit could be the same or the

opposite value of the other.

However; this correlated information couldn’t be

measured locally on single qubit. It should be

collectively measured (globally measured).

Entangled Qubits

Erwin Schrodinger:- “The best possible

knowledge of a whole does not necessarily

include the best knowledge of all its parts, even

though they may be entirely separated and

therefore virtually capable of being ‘best possibly

known’”

John Preskill:- “The whole is definite, the part is

random”

Why Entanglement is important

Each satellite will generate a constant stream of entangled pairs.

Each member of the pair will be sent to separate stations on the ground, where it will be stored in quantum memories.

Once the entanglement is stored on the ground, it can then be used as needed to send secure messages, or even sent locally across the quantum Internet using short optical fibers.

Quantum Logic

Cost of Information Loss

Fundamental logic dictates that energy must be

dissipated when information is erased

Energy dissppated = kT . Ln2 per bit erased

k:- Boltzman constant (1.3805*10-23 JK-1)

T :- Absolute temprature (in degrees Kelvin)

One way of suppressing this unwanted heat is by

modifying the chip design to use only reversible

logic gates

Reversible Gates

In reversible gates there is always a unique input

associated with a unique output and vise versa.

So reversible logic never erase any information

when they act

Unitary Gates Hadamard equation Makes superposition

|0>

|1>

Pauli-X gate

|0> |1>

|1> |0>

Pauli-Y gate

|0> i |1>

|1> -i |0>

Pauli-Z gate (Phase shift gate with )

|0> unchanged

|1> - |1>

Not & Swap gates

Not Gate Swap Gate

a a’

0 1

1 0

a b a' b'

0 0 0 0

0 1 1 0

1 0 0 1

1 1 1 1

CNOT – Controlled NOT Gatea b a' b'

0 0 0 0

0 1 0 1

1 0 1 1

1 1 1 0

Called

CNOT and also called

Feynman Gate or FG

Quantum Cost = 1

a’ = ab‘ = a xor b

Universal Reversible GatesTOFFOLIO (Controlled-controlled-

not)

a b c a' b' c'

0 0 0 0 0 0

0 0 1 0 0 1

0 1 0 0 1 0

0 1 1 0 1 1

1 0 0 1 0 0

1 0 1 1 0 1

1 1 0 1 1 1

1 1 1 1 1 0

Called also

CCNOT or TG

Quantum Cost = 5

a‘ = ab‘ = bc‘ = (a.b) xor c

Peres Gate

Peres Gate

a b c a' b' c'

0 0 0 0 0 0

0 0 1 0 0 1

0 1 0 0 1 0

0 1 1 0 1 1

1 0 0 1 1 0

1 0 1 1 1 1

1 1 0 1 0 1

1 1 1 1 0 0

Quantum Cost = 4

a‘ = ab‘ = a xor bc‘ = (a.b) xor c

Double Feynman Gate

Double Feynman Gate

a b c a' b' c'

0 0 0 0 0 0

0 0 1 0 0 1

0 1 0 0 1 0

0 1 1 0 1 1

1 0 0 1 1 1

1 0 1 1 1 0

1 1 0 1 0 1

1 1 1 1 0 0

a‘ = ab‘ = a xor bc‘ = a xor c

Called also

F2G

Quantum Cost = 2

Universal Reversible Gates

FREDKIN (Controlled-Swap)

a b c a' b' c'

0 0 0 0 0 0

0 0 1 0 0 1

0 1 0 0 1 0

0 1 1 0 1 1

1 0 0 1 0 0

1 0 1 1 1 0

1 1 0 1 0 1

1 1 1 1 1 1

Also called F gate

Quantum Cost = 5

a‘ = ab‘ = not(a).b + a.cc‘ = a.b + not(a).c

Quantum Circuits

SR Latch CircuitConventional Cross

Coupled Design – NAND

SR Latch

Pres Gates based

design without Enable

S R Action

0 0not

allowed

0 1 Q = 1

1 0 Q = 0

1 1No

Change

SR Latch Circuit

Other design of

SR Latch

This design is

gated design

including

“Enable” input

Gated D Latch

E/C D Q QComme

nt

0 XQpre

v

Qpre

v

No

change

1 0 0 1 Reset

1 1 1 0 Set

JK Latch Design

J K Qnext

Commen

t

0 0 QNo

change

0 1 0 Reset

1 0 1 Set

1 1 Q’ Toggle

Quantum Neural Networks

(QNN)

Research on Quantum Neural

Networks

Reversibility & Dissipation Hopfield network is an

example of ANN.

Hopfield network is used as an associative memory.

In associative memory multiple patterns is mapped to single pattern.

Associative memory is irreversible circuit

This raises an important question about how 100 billion neurons processes information with energy dissipation.

Non Linear Activation Non-Linear activation

functions is an important characteristic of neural networks.

It is a source for non-linear properties of neural networks

Sigmoid function is the most famous example

One open issue of Quantum Neural Networks is how to incorporate non-linear functions in quantum systems which is linear

Qubit Neurons (Qurons)

A quron is a qubit in

which the two levels

stand for active an

resting neural firing

states.

This allows for neural

network to be in a

superposition of firing

patterns.

Ideas for Interpreting Step Function

as Measurement KAK intoduced the idea of

quantum neural computation

In a Hopfield like network he interpreted the necessary conditions for a stable states as an eigen value equation of a quantum system

Updating a network corresponds to quantum measurement selects the eigenstates of the system

Meneer & Narayanan the many-universe interpretation of quantum mechanics to look at superposition of networks each storing one pattern instead-of one network storing several patterns

ZAK & Willimas do not consider single neurons as quantum objects but introduce a unitary walk between quantum network basis states.

This approaches using nonlinear dissipative and irreiversible transformation as a trial for playing the role of a natural quantum sigmoid function

Interacting Quantum Dots The green function by

Feynman sums all possible paths prpagatingthe system

Elizabeth Behrman noticed the nonlinearity of the equation.

Instead of different Qurons the network is realized by propagation of one Quron only

Its state after each of N different time slices simulates the states of N virtual neurons.

Synaptic weights are engineered by interaction of the Quron with the environment

Behraman proposes to implement this “time-array neural network” as quantum dot molecule interacting with phonons of surrounding lattice & external field

The network can be trained by back-propagation rule

References1. "Quantum Computing with Molecules" article in

Scientific American by Neil Gershenfeld and Isaac L. Chuang, 1998

2. “Quantum Computation and Quantum Information” book by Micheal A.Nilesen & Isaac L. Chuang, Cambridge University Press, 2000

3. “Quantum Computing & the Entanglement Frontier” public lecture at California Institute of Technology, by John Preskill

4. “Schrodinger’s Phelosophy of Quantum Mechanics”, book by Michel Bitbol, volume 188, 1996

5. “How Entanglement-Generating Satellites Will Make the Quantum Internet Global” article in MIT Technology Review, October 30 2014

6. “Explorations in Quantum Computing” book by Williams, C.P, published by Springer, 2011

References

7. “Introduction to Reversible Logic Gates & its

Applications”, Report by Prashant R. Yelekar &

Prof. Sujata S. Chiwande, 2nd National

Conference On Information and Communication

Technology 2011

8. “Design of Reversible Sequential Circuits

Optimizing Quantum Cost, Delay, and Garbage

Outputs”, Report by Himanshu Thapalyal &

Nagarjan Ranganathan, University of South

Florida