![\"\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9781482238129\/files\/bg1e.png\")
<\/p>\n\n\n\nInformation as accessed by the human mind, and the myriad ways of process-<\/p>\n\n\n\n
ing it, is what has set humankind apart in the animal world. The ability to use<\/p>\n\n\n\n
the physical systems around us to encode and then to process information has<\/p>\n\n\n\n
been evolving in leaps and bounds ever since the dawn of our race! The earliest<\/p>\n\n\n\n
form of computation was probably in the form of account-keeping by counting<\/p>\n\n\n\n
pebbles or on the \ufb01ngers. This developed into the abacus, writing symbols,<\/p>\n\n\n\n
the computing machine, and now, a laptop or a supercomputer: everywhere<\/p>\n\n\n\n
we represent information by means of a physical system, and perform ma-<\/p>\n\n\n\n
nipulations on that system to process the information in many desired ways,<\/p>\n\n\n\n
including communication. The stress here is on the realization that the basis<\/p>\n\n\n\n
of information is a physical system. The more advanced that system is and<\/p>\n\n\n\n
the set of rules it functions on, the more capable our means of information<\/p>\n\n\n\n
processing and communication. When the underlying physical system used for<\/p>\n\n\n\n
encoding and processing information is a quantum system, we have quantum<\/p>\n\n\n\n
information processing.<\/p>\n\n\n\n
While technological advances have made it possible to reach astounding<\/p>\n\n\n\n
speeds and processing power, the basic paradigm of current day information<\/p>\n\n\n\n
processing is binary logic with currents or voltages in the semiconductor cir-<\/p>\n\n\n\n
cuitry at the heart of the modern computer processors. However, it is impor-<\/p>\n\n\n\n
tant to realize that the behavior of these high and low states of the circuit<\/p>\n\n\n\n
is based on laws of classical physics. We know now that at the most funda-<\/p>\n\n\n\n
mental level, physical systems obey the laws of quantum mechanics. These<\/p>\n\n\n\n
laws are fundamentally di\ufb00erent in many ways from classical laws of physics.<\/p>\n\n\n\n
Therefore, the basic paradigm of information processing is di\ufb00erent when we<\/p>\n\n\n\n
come to quantum information processing. Not only are the algorithms and<\/p>\n\n\n\n
the processing mechanism di\ufb00erent, but there are distinct advantages of the<\/p>\n\n\n\n
quantum over the classical.<\/p>\n\n\n\n
According to recent data, the fastest current day supercomputer is capable<\/p>\n\n\n\n
of performing at a speed of hundreds of G\ufb02op\/s.<\/p>\n\n\n\n
1<\/p>\n\n\n\n
The quantum paradigm<\/p>\n\n\n\n
a\ufb00ords a speedup to many algorithms that are very slow to perform even on<\/p>\n\n\n\n
this computer! Much of modern-day information security, for instance secure<\/p>\n\n\n\n
online cash transactions, is based on classical cryptography. This has been<\/p>\n\n\n\n
proved to be vulnerable if a quantum computer is used to crack the code!<\/p>\n\n\n\n
I\u2019ve been trying to motivate the need to study quantum information. But<\/p>\n\n\n\n
1<\/p>\n\n\n\n
Giga\ufb02op per second, Giga = 10<\/p>\n\n\n\n
12<\/p>\n\n\n\n
, FLOP = \ufb02oating point operation.<\/p>\n\n\n\n
3<\/p>\n\n\n\n
4 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
the inquisitive scienti\ufb01c mind will no doubt want to grasp the basics of the<\/p>\n\n\n\n
physical laws that make complex information processing possible: the laws of<\/p>\n\n\n\n
quantum mechanics. Professor Richard Feynman of Caltech was supposed to<\/p>\n\n\n\n
have famously said that no one understands quantum mechanics! How then are<\/p>\n\n\n\n
we basing modern technology on it? With this book I\u2019d like to show you that<\/p>\n\n\n\n
despite its \u201cweirdness\u201d, by which I mean its distance from our common sense<\/p>\n\n\n\n
understanding which appears wired to classical physics, the laws of quantum<\/p>\n\n\n\n
mechanics can be apprehended by an undergraduate student, to be used as a<\/p>\n\n\n\n
set of rules by which the game is played. The more philosophically inclined will<\/p>\n\n\n\n
be drawn to ponder meaning and interpretation of these rules. And this latter<\/p>\n\n\n\n
exercise is also rewarding, in bringing out fascinating new facets of quantum<\/p>\n\n\n\n
theory, to be exploited in our ever-expanding game of information processing.<\/p>\n\n\n\n
Several considerations make the transition to the quantum inevitable while<\/p>\n\n\n\n
exploring e\ufb03cient information processing. One is from the perspective of hard-<\/p>\n\n\n\n
ware engineering, where miniaturization and the need to pack more structure<\/p>\n\n\n\n
in less space must eventually lead to the limit set by the structure of matter:<\/p>\n\n\n\n
the atomic or even electronic level. At this level, classical laws of physics are<\/p>\n\n\n\n
no longer valid and we have to consider the essentially quantum nature of the<\/p>\n\n\n\n
physical system used to store and manipulate information.<\/p>\n\n\n\n
However, from the angle of the basic physical laws of quantum mechanics,<\/p>\n\n\n\n
more complex ways of processing information should be possible. The manner<\/p>\n\n\n\n
in which a quantum system evolves, transforms information and conveys it<\/p>\n\n\n\n
in an experiment via a measurement, is fundamentally di\ufb00erent from classi-<\/p>\n\n\n\n
cal information. This was realized \ufb01rst by Feynman [33] in the 1980s when<\/p>\n\n\n\n
he pointed out that a quantum process cannot be e\ufb03ciently simulated on a<\/p>\n\n\n\n
classical computer. He showed, however, that such a system may be e\ufb03ciently<\/p>\n\n\n\n
simulated on a quantum computer.<\/p>\n\n\n\n
In the process of studying how this is possible, we are led into a deeper<\/p>\n\n\n\n
probing of the foundations of quantum physics. In implementing a quantum<\/p>\n\n\n\n
computer, physicists need to access and control individual quantum states,<\/p>\n\n\n\n
prepare them, manipulate them and \ufb01nally, measure them. The question also<\/p>\n\n\n\n
arises of how to deal with practical systems that are not ideal and isolated from<\/p>\n\n\n\n
their environments, but are subject to noise or errors due to inadvertent en-<\/p>\n\n\n\n
vironmental e\ufb00ects. These considerations lead us into an experimental regime<\/p>\n\n\n\n
of testing our ideas of quantum reality, and into discovering new quantum<\/p>\n\n\n\n
phenomena.<\/p>\n\n\n\n
The third perspective is from the theory of computation. The foundations<\/p>\n\n\n\n
of modern computer science may be said to have been laid by the work of Alan<\/p>\n\n\n\n
Turing in the 1930s [69], on abstract models of computing embodied in what<\/p>\n\n\n\n
is now known as the Turing Machine. The Universal Turing Machine (UTM)<\/p>\n\n\n\n
is an idealization of a model of computation that can execute any computable<\/p>\n\n\n\n
algorithm, in short, any task that can be run on a modern programmable<\/p>\n\n\n\n
computer. Notions of computability of problems and e\ufb03ciency of algorithms<\/p>\n\n\n\n
were developed. In rough terms, an algorithm is said to be e\ufb03cient if it takes<\/p>\n\n\n\n
polynomial time for execution. This means that the time required to run it<\/p>\n\n\n\n
<\/p>\n\n\n\n
Introduction 5<\/p>\n\n\n\n
grows as a polynomial in the number n of input bits, i.e., at most as a power<\/p>\n\n\n\n
of n. A computationally hard problem is one for which the best algorithm is<\/p>\n\n\n\n
exponential in the number of input bits, i.e., grows as a<\/p>\n\n\n\n
n<\/p>\n\n\n\n
where a is some<\/p>\n\n\n\n
constant.<\/p>\n\n\n\n
As Feynman pointed out, the evolution of a quantum system is one prime<\/p>\n\n\n\n
example of a problem that cannot be simulated e\ufb03ciently on a Turing machine,<\/p>\n\n\n\n
while a quantum computer could quite naturally do so! This was a challenge<\/p>\n\n\n\n
to the Church\u2013Turing thesis that any computation can be e\ufb03ciently simulated<\/p>\n\n\n\n
on a Turing Machine. This thesis had been modi\ufb01ed to include probabilistic<\/p>\n\n\n\n
machines (based on fuzzy logic) but now has been extended to a quantum<\/p>\n\n\n\n
version.<\/p>\n\n\n\n
Problems in computational complexity have now been extended to include<\/p>\n\n\n\n
the quantum Turing machine and the possibilities are exciting. While the<\/p>\n\n\n\n
basic notion of computability of a given problem does not change when quan-<\/p>\n\n\n\n
tum machines are included, problems that were hard classically may become<\/p>\n\n\n\n
easy. Quantum computation may also resolve other questions in the area of<\/p>\n\n\n\n
computational complexity.<\/p>\n\n\n\n
FIGURE 1.1: Three approaches to the quantum.<\/p>\n\n\n\n
These three approaches (Figure (1.1) have historically motivated the study<\/p>\n\n\n\n
of this \ufb01eld, which at present however, is rapidly blooming in several di\ufb00erent<\/p>\n\n\n\n
directions unforeseen in the last century.<\/p>\n\n\n\n
6 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
1.1 Bits and Qubits<\/p>\n\n\n\n
So what is the fundamental di\ufb00erence between classical and quantum com-<\/p>\n\n\n\n
puting?<\/p>\n\n\n\n
Computation and information processing as we know it today is built upon<\/p>\n\n\n\n
Boolean logic and algebra. Boolean algebra is binary, requiring two logical<\/p>\n\n\n\n
units called bits. You can think of them as the possible answers to a decision<\/p>\n\n\n\n
question: yes or no. The idea is that almost any problem can be reformulated<\/p>\n\n\n\n
as a series of decision questions, and therefore can be encoded in bits. A<\/p>\n\n\n\n
bit, or binary digit, is a physical system that can take on two logical states<\/p>\n\n\n\n
represented by 0 and 1. In a typical digital computer these states are the low<\/p>\n\n\n\n
and high voltage states in the microcircuitry.<\/p>\n\n\n\n
To extend the capabilities of the computational system, probabilistic al-<\/p>\n\n\n\n
gorithms are based on the notion of fuzzy bits, that can take the value 0 with<\/p>\n\n\n\n
a probability p or 1 with probability 1 \u2212 p. This is the basis of probabilistic<\/p>\n\n\n\n
computation, or so-called fuzzy logic.<\/p>\n\n\n\n
When the logic is extended to binary states of a quantum system, we ar-<\/p>\n\n\n\n
rive at the qubit, a quantum bit. A qubit can take values 0 or 1 but with<\/p>\n\n\n\n
probabilities given by the mod-squared of complex numbers! A physical qubit<\/p>\n\n\n\n
is a quantum system that will represent our Boolean units. The system can<\/p>\n\n\n\n
therefore take on two quantum states that we will now represent in the nota-<\/p>\n\n\n\n
tion |0i and |1i, to distinguish them from the states 0 and 1 of the classical<\/p>\n\n\n\n
bit. This angular bracket notation is due to physicist Paul Dirac [29]. This<\/p>\n\n\n\n
notation is very versatile and by itself a useful calculational tool.<\/p>\n\n\n\n
A qubit is generically represented as a linear superposition of the basis<\/p>\n\n\n\n
states:<\/p>\n\n\n\n
|\u03c8i = \u03b1|0i + \u03b2|1i. (1.1)<\/p>\n\n\n\n
The coe\ufb03cients \u03b1 and \u03b2 are called probability amplitudes, and satisfy such<\/p>\n\n\n\n
that |\u03b1|<\/p>\n\n\n\n
2<\/p>\n\n\n\n
+ |\u03b2|<\/p>\n\n\n\n
2<\/p>\n\n\n\n
= 1.<\/p>\n\n\n\n
The way to understand this statement is that upon measurement, the<\/p>\n\n\n\n
generic qubit takes on one of the de\ufb01nite states |0i or |1i with a probability |\u03b1|<\/p>\n\n\n\n
2<\/p>\n\n\n\n
or |\u03b2|<\/p>\n\n\n\n
2<\/p>\n\n\n\n
. In this sense, a qubit is similar to a classical bit in that measurement<\/p>\n\n\n\n
only gives one of two values. It is sometimes useful to think of these values,<\/p>\n\n\n\n
or the basis states of the qubit, as classical bits.<\/p>\n\n\n\n
Though the qubit is probabilistic, it di\ufb00ers from the fuzzy bit because<\/p>\n\n\n\n
of the possibility of interference. This is characteristic that is captured by<\/p>\n\n\n\n
the complex amplitudes. A complex number has a magnitude and well as a<\/p>\n\n\n\n
phase. While composing two or more such numbers, the phases could result<\/p>\n\n\n\n
in reinforcement or reduction in the strength of the resultant. The physical<\/p>\n\n\n\n
implications of this is familiar to us through the phenomenon of interference<\/p>\n\n\n\n
in optics. When two beams of light, described by electric \ufb01elds having de\ufb01-<\/p>\n\n\n\n
nite phase relationships to each other, are combined, then there are regions<\/p>\n\n\n\n Information as accessed by the human mind, and the myriad ways of process- ing it, is what has set humankind apart in the animal world. The ability to use the physical systems around us to encode and then to process information has been evolving in leaps and bounds ever since the dawn of our race! […]<\/p>\n","protected":false},"author":1,"featured_media":3939,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[486],"tags":[],"class_list":["post-3952","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-1-introduction"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/quantum-computing.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3952","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/comments?post=3952"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3952\/revisions"}],"predecessor-version":[{"id":3953,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3952\/revisions\/3953"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/3939"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=3952"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=3952"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=3952"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9781482238129\/files\/bg20.png\")
<\/figure>\n","protected":false},"excerpt":{"rendered":"