Quantum computers are incredibly powerful machines that take a new approach to processing information. Built on the principles of quantum mechanics, they exploit complex and fascinating laws of nature that are always there, but usually remain hidden from view. By harnessing such natural behavior, quantum computing can run new types of algorithms to process information more holistically. They may one day lead to revolutionary breakthroughs in materials and drug discovery, the optimization of complex manmade systems, and artificial intelligence. We expect them to open doors that we once thought would remain locked indefinitely. Acquaint yourself with the strange and exciting world of quantum computing.

1. Quantum Computing
Davide Nardone
Parthenope University of Naples

2. Outline
 Introduction and History
 Data representation and storing
 Operations on Data
 Computing power
 Problems and limitations of Quantum computers
 Innovations key
 Conclusion and open questions

3. What’s a quantum computer?
A quantum computer is a machine that performs calculations based
on the laws of quantum mechanics, which makes use of the
quantum state of sub-atomic particles to store information.
Why quantum computer ?
In the history, one of the most efficient solutions for improving the
performance of a classical computer has been the reduction in size
of transistors used in modern processors. However, this continuous
reduction (physically) seems to show itself as a fundamental
limitation to the advancement of our technology, therefore, lately
has been seeking a new computing systems that can overcome the
computational speed of the current using systems, and one of the
most exiting and promising ideas of the past decade is the
Quantum Computing (QC).
Introduction

4.  1982 – Richard Feynman proposed the idea of creating a car
based on the laws of QC rather than the laws of classical
physics.
 1985 – David Deutsch developed the quantum Turing
machine, showing that the quantum circuits are universal.
 1994 – Peter Shor (Bell Labs) invented a quantum algorithm
to factor very large numbers in polynomial time.
 1997 – Lov Grover developed a quantum search database
algorithm with a time complexity of O(√N).
Cont.

5. Outline
Introduction
 Data representation and storing
 Data operation
 Computing power
 Problems and limitations of Quantum computers
 Innovations key
 Conclusion and open questions

6.  A qubit is the basic unit of information in a quantum computer and it’s
represented by a single atom that can exist in two states (simultaneously or
at different times) denoted by |0> e |1>.
 A qubit is typically a microscopic system such as an atom, a nuclear spin
or a polarized photon.
 A physical implementation of a qubit could use 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 λ for time
interval t
Election
State |0> State |1>
Qubit – Data representation

7.  An important distinguish feature between a qubit and a classic bit is that
multiple qubit can present a quantum entanglements
 An important distinguishing feature between a qubit and a classical bit is
that multiple qubits can present a quantum entanglement.
 The Entanglement is the ability of a quantum system to present a
correlation between states within an overlap, and is an essential element for
any quantum computation that can not be efficiently achieved on a classical
computer. Imagine two qubits, each in a state | 0> + | 1> (an overlap of 0 and
1). It is possible to link two qubits such that the measurement of a qubit is
always correlated with the measurement of the other.
Relationship between the Data - Entanglement

8. A single qubit can be forced into a superposition of two states denoted by the
addition of state vectors:
where α e β are complex numbers and |α|2
+ |β|2
= 1
Nota: An overlapping qubit is in both states | 1>
and | 0 at the same time!
Impulse of light of frequency
λ for a time interval t/2
State |0> Sate |0> + |1>
Data representation - Overlapping

9. Data storing – Quantum register
 A collection of n qubits is called the quantum register of size n (also
known as qregister).
 A n-qubit register can represent 2n
numbers simultaneously, as long as it is
not processed (computed, read), in fact, if we try to recover the value
represented in an overlap, the latter collapses randomly to represent only one
of the original values.
 How much information is stored?
- Theoretically infinite, since there are infinite possible different overlaps of
the values 0 and 1.
- 9-since when the information is processed the value of the qubit collapses
on 0 and 1- a qubit can not contain more information of a classic bit.

10. A mathematical description of a quantum register is obtained using the tensor
product of qubits in the following notation bra-ket or ket:
Supposing the information is stored in binary form, then for example, the
number 6 is represented by a register in the state:
A quantum register can store individual numbers such as 3 or 7:
but, it can also store both values simultaneously.
Data storing – Quantum register (Cont.)

11. Outline
 Introduction and History
 Data representation and storing
 Operations on data
 Computing power
 Problems and limitations of Quantum computers
 Innovations key
 Conclusion and open questions

12.  Given the nature of quantum physics, the destruction of information in
a logic port would generate heat, which could therefore destroy the
overlap of the qubit states.
A B C
0 0 0
0 1 0
1 0 0
1 1 1
Input Output
A
B
C
In these 3 cases, the
information has
been destroyed.
Ex.
The AND Gate
This type of logic port can not be used. It is necessary to use
quantum ports.
Operations on qubits - Reversible logic

13.  The quantum logic gates are similar to the classical logic ports, however,
their outputs are irreversible (i.e., their original input states can not be
derived uniquely from their output states). They must be reversible.
 This means that a deterministic calculation can be performed on a
quantum computer only if it is reversible. (Demonstrated by Charles H.
Bennett - 1973)
 Quantum ports are represented by unit matrices that are reversible
objects. The most common quantum ports operate on a space of one two
qubits, just as the classical logic ports operate on one or two bits.
Quantum gates

14.  The simplest and most common quantum port involving only one qubit is
called the Hadamard Gate. It is used to perform a unit transformation
(known as the Hadamard transformation) which determines the overlap
of a qubit. It is defined as:
Note: Two Hadamard gates used in succession can
be used as a NOT gate
H
Stato
|0>
Stato
|0> + |1>
H
Stato
|1>
Quantum gates - Hadamard

15.  In order to "bind" two or more qubits (quantum register) it is
necessary to extend the concept of quantum gates to two qubits.
 A gate that operates on two qubits is called the Controlled-NOT
(CN) Gate. It inverts the second (destination) qubit if the first (control)
qubit is | 1>.
A B A’ B’
0 0 0 0
0 1 0 1
1 0 1 1
1 1 1 0
Input Output
Note: The CN gate has a similar behavior to
the XOR gate with some extra information to
make it reversible.
B - Destinazione
A - Controllo
B’
A’
Quantum gates - Controlled NOT

16. 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 1 1 1
1 0 0 1 0 0
1 0 1 1 0 1
1 1 0 1 1 0
1 1 1 0 1 1
Input Output
A - Target
B – Control 1
C – Control 2
A’
B’
C’
 A logic gate that operates on three qubits is called Controlled
Controlled-NOT (CCN) Gate.
 It inverts the third (destination) qubits if both (control) the line qubits
are | 1>.
Quantum gates - Controlled Controlled NOT

17. Besides the Hamard-Gate, CN and CCN there are other functional logic
ports such as:
 Pauli-X gate
 Pauli-Y gate
 Pauli-Z gate
 Phase shift gates
 Swap gate
 Toffoli gate
 Fredkin gate
 ed altre
Universal quantum gates

18. Outline
 Introduction and History
 Data representation and storing
Operations on Data
 Computing power
 Problems and limitations of Quantum computers
 Innovations key
 Conclusion and open questions

19.  Although a quantum computer does not present "directly" advantages
over a classic computer, its computing power increases exponentially.
Classic Computing
For example, suppose you have to
apply a classic 2-bit operator, and
have to calculate all the possible
outputs: how do you proceed? It is
necessary to consider all the
combinations (00, 01, 10 and 11), and
operate on each of them.
Quantum Computing
How many qubits would take to do the
same calculation? Each qubit is
simultaneously 0 and 1, and then in a
system of 2-qubits all the four
combinations coexist, it is therefore
sufficient to operate on this system to
obtain the result; in fact, only 2 units of
information were used, instead of 22
.
Nota: It is easy to demonstrate that a quantum processor that can operate
on n qubits has the computing power of a processor operating on 2n
bits: an
exponential increase!
Computing power

20. What a classical computer can or can not do?
IT problems are classified according to how many computational steps are
performed by an algorithm (known) to solve a problem. Three of the most
important classes of computational problems are listed here:
 P PROBLEMS: That set of problems that a computer can solve
efficiently in polynomial time.
 NP PROBLEMS : Those problems whose solutions are easily verifiable
(polynomial time) through a calculator.
 NP COMPLETE: These are all problems for which an efficient solution
could provide an efficient solution to all NP problems.

21. In computational complexity theory, BQP (Bounded-error Quantum
Polynomial time) is a class of decision problems solvable by a quantum
computer in polynomial time, with a probability of error of almost 1/3 for
all instances.
In other words, there is an algorithm for a quantum computer that solves
decision problems with a high probability, guaranteeing the solution in
polynomial time. In any execution of the algorithm, there is a probability of
at most 1/3 that it will give a wrong answer.
Where quantum computers perform well ?

22. The BQP class includes all the P problems and very few of the NP
problems like:
 Factoring
 Discrete logarithm
Most of the other NP problems and all NP Complete problems are thought
not to belong to the BQP class, thus implying that not even a quantum
computer would be able to solve them in less than polynomial Ω (n2
).
Some of these problems are:
 Hamiltonian cycle,
 Salesman traveler,
 and many others.
Where quantum computers perform well ? (Cont.)

23. Here it’s shown how the
class of problems that
quantum computers
would solve efficiently
(BQP) could be related
to other fundamental
classes of
computational
problems.
Where quantum computers perform well ? (Cont.)

24.  On the basis of the considerations just made, a quantum computer
(theoretically) would be able to solve (in polynomial time) only a small
subset of problems that a classic computer would never be able to solve in
an acceptable time.
 Therefore, assuming that a real quantum computer can be built (flying
over all the problems connected to it for a moment) some of the areas in
which the quantum computing would have a significant impact are:
Encryption (RSA Breaks, DSA, etc.).
 Military applications
 Machine Learning (Clustering, PCA, regression and classification)
 Quantum simulations (Chemical systems, scientific material)
Financial analysis
 and others.
Quantum computing applications

25. Outline
Introduction and History
Data representation and storing
Operations on Data
Computing power
 Problems and limitations of Quantum computers
 Innovations key
 Conclusion and open questions

26. What has been seen so far seems promising, but huge obstacles still need to
be overcome. Some of the problems related to quantum computing are:
Interference – During the calculation phase, the minimum perturbation in a
quantum system causes the entire calculation to collapse, a process known as de-
coherence. Therefore, a quantum computer must be totally isolated from any
external interference during the calculation phase. Some good results have been
obtained with the use of qubits in intense magnetic fields (ions).
Connection error– Since the isolation of a quantum system has proved to be a
difficult challenge to solve, systems of error correction for quantum calculations
have been developed. Since qubits are not digital bits, they can not use conventional
error correction methods, and given the nature of quantum computing, error
correction is a very sensitive matter (one mistake in a calculation can cause the
validity of the entire calculation). Over time, however, there has been considerable
progress in this area, with a correction algorithm that uses 9 qubits (1 calculation
and 8 correction) and another that uses 5 qubits (1 calculation and 4 correction).
Problems and limits of quantum computers

27.  Output remarks – The outputs (results) that a quantum computer gets to some
questions (problems) are in a probabilistic form. In other words, they could be
wrong and therefore should be checked. If a given solution is wrong, the
calculation must be repeated until the correct answer is obtained (a flaw that
paradoxically slows down the speed that these calculators offer compared to the
classic devices).
 In an example of a quantum calculator with 500 qubits, you have 1 chance over
2500
to get the correct result (if we quantify the output). Therefore, what is
needed is a method that ensures that, as soon as all the calculations are
performed and the act of reading the data has been completed, the observed
value will correspond to the correct answer. How can this be done? It was
obtained from Lov Grover and its database search algorithm, which is based on
the special waveform of the probability curve inherent in quantum computers
that guarantees (once all the calculations have been performed) the act of
measurement of the quantum state in the correct answer.
Problems and limits of quantum computers (Cont.)

28. Outline
Introduction and History
Data representation and storing
Operations on Data
Computing power
Problems and limitations of Quantum computers
 Innovations key
 Conclusion and open questions

29. Several key innovations have been made in the field of quantum computing in
recent years. Here are some of them listed:
 1998 – Researchers from Los Alamos and MIT have succeeded in spreading a
single qubit through three nuclear spins in each molecule of a liquid solution
(alanine).
 2000 – Los Alamos laboratory scientists announced the development of a 7-
qubits quantum computer within a single drop of water.
 2001 – Scientists from IBM and Stanford University successfully demonstrated
the Shor algorithm on a quantum computer.
 2005 – The institute of quantum optics and quantum information at the
University of Innsbruck announced that the scientists were able to create the first
qubyte using ionic traps.
 2006 – Scientists in Waterloo and Massachusetts have devised methods for
quantum control on a 12-qubits system.
 2007 – A new Canadian D-wave company has illustrated a quantum computer of
16 qubits. The computer solves a sudoko puzzle and other pattern matching
problems.
Key innovations

30. Outline
Introduction and History
Data representation and storing
Operations on Data
Computing power
Problems and limitations of Quantum computers
Innovations key
 Conclusion and open questions

31.  Quantum computers may someday replace classic computers, but for now, quantum
computing is still in its early stages of development, and many computer scientists
believe that the technology needed to create a practical quantum computer is years
away; in fact the most advanced quantum computer does not go beyond the
manipulation of more than 16-qubits, implying that we are very far from practical
applications (in fact these must have at least several tens of qubits to be able to solve
the problems of the real world) .
 Plus, much research in quantum computing is still very theoretical.
 However, the potential remains, and the fact that one day these computers can
perform, quickly and easily, calculations that are incredibly expensive (time) on
conventional computers is very exciting.
Note: If functional quantum computers could be built, they would be valuable in the
factoring of large numbers, and therefore extremely useful for coding and decoding
secret information. On the other hand, however, no information on the Internet would
be safe - our current methods of cryptography are simple compared to the possible
complicated methods of a quantum computer.
Conclusions and open questions

32.  What will be the next algorithm that will be discovered?
 Will a quantum computer ever solve NP Complete problems in polynomial
time?
 How would this power be handled in terms of security?
 Can the real calculation application of quantum computers open new
horizons for quantum mechanics?
Conclusions and open questions (Cont.)

33. Thank you for your attention!