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<\/p>\n\n\n\nIntroduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
(a) Experimental setup: 1 slit closed<\/p>\n\n\n\n
(b) Classically expected result (c) Actual result: interference<\/p>\n\n\n\n
Introduction 9<\/p>\n\n\n\n
1.2 Properties of Qubits<\/p>\n\n\n\n
Quantum systems have certain properties that are counter-intuitive and<\/p>\n\n\n\n
completely outside our range of experience in the classical world. These<\/p>\n\n\n\n
\u201cweird\u201d properties are best understood as inevitable consequences of axioms<\/p>\n\n\n\n
on which quantum laws are based. These axioms have been arrived at af-<\/p>\n\n\n\n
ter considerable e\ufb00ort and study of experimental phenomena, and are now<\/p>\n\n\n\n
accepted among physicists as the complete theory which describes the real<\/p>\n\n\n\n
world at the fundamental level.<\/p>\n\n\n\n
The basic mathematical properties of a qubit can be analyzed and studied<\/p>\n\n\n\n
independent of the physical system that realizes it. By treating the qubit as<\/p>\n\n\n\n
an abstract mathematical entity, we can develop a general theory of quan-<\/p>\n\n\n\n
tum information processing. Some of the strange new properties that become<\/p>\n\n\n\n
relevant are now discussed.<\/p>\n\n\n\n
Superposition and quantum parallelism: The main implication of<\/p>\n\n\n\n
states like that of Equation 1.1 is that a single state contains the potential<\/p>\n\n\n\n
for the system to be in either basis state. In some sense the system, say an<\/p>\n\n\n\n
electron characterized by its spin value, simultaneously exists in both states<\/p>\n\n\n\n
until measured. Physically this does not seem to make sense to our classical<\/p>\n\n\n\n
minds unless we say that the electron has not decided which of the two possible<\/p>\n\n\n\n
states it should be in, until forced into one of them by a measurement.<\/p>\n\n\n\n
This feature is exploited in quantum computation to implement what is<\/p>\n\n\n\n
called quantum parallelism: an operation that acts on a bit can now simul-<\/p>\n\n\n\n
taneously act on both possible values of the bit if the input is a qubit in a<\/p>\n\n\n\n
quantum superposition.<\/p>\n\n\n\n
Size of computational space: If we want to do an n-bit computation,<\/p>\n\n\n\n
Classically the \u201cspace\u201d available for computation is of size n. In terms of a<\/p>\n\n\n\n
quantum system of n qubits, the number of possible basis states is 2<\/p>\n\n\n\n
n<\/p>\n\n\n\n
, and<\/p>\n\n\n\n
this is the size of the space available for computation. The size of the space of<\/p>\n\n\n\n
states available for computation grows exponentially with the number of bits<\/p>\n\n\n\n
(Figure 1.5). This is the power we wish to exploit in quantum computation.<\/p>\n\n\n\n
Entanglement and quantum correlations: Multiple qubit systems can<\/p>\n\n\n\n
exist in superposition states that are known as entangled states. These states<\/p>\n\n\n\n
possess intrinsic correlations between the component systems that are dif-<\/p>\n\n\n\n
ferent from classical correlations. These correlations can survive even if the<\/p>\n\n\n\n
component systems are taken physically far apart from each other. For exam-<\/p>\n\n\n\n
ple, 2-qubit states are in general linear superpositions of |00i, |01i, |10i, and<\/p>\n\n\n\n
|11i. Look at the state<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(|00i + |11i). In such a state, the \ufb01rst and second<\/p>\n\n\n\n
systems are correlated quantum mechanically: the value of the second qubit<\/p>\n\n\n\n
is always equal to that of the \ufb01rst qubit, irrespective of what measurement<\/p>\n\n\n\n
we make on which bit and when. Such a state is called \u201centangled\u201d because<\/p>\n\n\n\n
of this correlation.<\/p>\n\n\n\n
Quantum correlations can be exploited to generate new methods of pro-<\/p>\n\n\n\n
<\/p>\n\n\n\n
10 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
FIGURE 1.5: Computational power: quantum vs classical.<\/p>\n\n\n\n
cessing, increasing the e\ufb03ciency by allowing controlled operations to be per-<\/p>\n\n\n\n
formed. These correlations are an invaluable resource in quantum information<\/p>\n\n\n\n
theory and we will see their basic applications in quantum state teleportation<\/p>\n\n\n\n
and secure information transfer over a distance.<\/p>\n\n\n\n
Measurement and state collapse: Though a qubit could exist in a su-<\/p>\n\n\n\n
perposition of basis states, a measurement of the qubit would give one of the<\/p>\n\n\n\n
two basis states alone. Measurement of a quantum system causes it to collapse<\/p>\n\n\n\n
into one of the basis states, which destroys the superposition, including any<\/p>\n\n\n\n
information that may be encoded in the probability amplitudes. Some au-<\/p>\n\n\n\n
thors express this property as a qubit existing in a superposition not having<\/p>\n\n\n\n
a de\ufb01nite state. Measurement results can be predicted with 100% certainty in<\/p>\n\n\n\n
\u201cde\ufb01nite\u201d states, and the system exists in a basis state. When a system is not<\/p>\n\n\n\n
in a de\ufb01nite state, measurement disturbs the system and one can never know<\/p>\n\n\n\n
the original state exactly. It is a quantitative and in-depth study of quantum<\/p>\n\n\n\n
measurements that has uncovered new laws of quantum information.<\/p>\n\n\n\n
Unitary evolution and reversibility: Quantum dynamical laws gov-<\/p>\n\n\n\n
erning the evolution of an isolated quantum system are what are known as<\/p>\n\n\n\n
unitary evolutions. Thus the functioning of a quantum computer is necessarily<\/p>\n\n\n\n
via unitary transformations of the initial quantum state. Unitary operations<\/p>\n\n\n\n
are fully reversible and, from a large body of study on the energetics of com-<\/p>\n\n\n\n
putation, are said to lead to greater energy e\ufb03ciency.<\/p>\n\n\n\n
No cloning: This is another peculiar property of generic quantum states:<\/p>\n\n\n\n
quantum states that are not basis states cannot be perfectly cloned or copied.<\/p>\n\n\n\n
The fact that classical states can be copied and kept aside for further process-<\/p>\n\n\n\n
ing is often taken for granted. When implementing a function in a classical<\/p>\n\n\n\n
circuit, we often send copies of a particular input to di\ufb00erent parts of the<\/p>\n\n\n\n
circuit. Such an operation is no longer possible in quantum computing. This<\/p>\n\n\n\n
changes the way we look at a quantum computation. And on the upside,<\/p>\n\n\n\n
this also makes it possible to exchange information in a secure manner since<\/p>\n\n\n\n
tapping a quantum line disturbs the system irrevocably!<\/p>\n\n\n\n
Introduction 11<\/p>\n\n\n\n
These properties lead us naturally to a model of computation often called<\/p>\n\n\n\n
the \u201ccircuit model,\u201d based on classical logic-gate circuits, of quantum compu-<\/p>\n\n\n\n
tation, which is what we will primarily study in this book. However, several<\/p>\n\n\n\n
other models of quantum information processing have also evolved, such as<\/p>\n\n\n\n
measurement-based computation, continuous-variable computation and adia-<\/p>\n\n\n\n
batic evolution. The interested reader may refer to the literature for these.<\/p>\n\n\n\n
1.3 Practical Considerations<\/p>\n\n\n\n
Theoretically, the examination of the paradigm of quantum computation<\/p>\n\n\n\n
has been very promising and exciting. However these considerations need to be<\/p>\n\n\n\n
grounded in reality. Pure quantum systems are found at the microscopic scale<\/p>\n\n\n\n
and are di\ufb03cult to access except by special technological means. To initialize<\/p>\n\n\n\n
any information process, we must have the means to assign any desired state<\/p>\n\n\n\n
to the qubit. Manipulation of the states of an individual qubit requires a high<\/p>\n\n\n\n
level of technological ingenuity. We need not just one qubit but large qubit<\/p>\n\n\n\n
registers. These may be built out of a collection of non-interacting qubits but<\/p>\n\n\n\n
whether such a register can be built for the system at hand brings in questions<\/p>\n\n\n\n
of scalability.<\/p>\n\n\n\n
In implementing a quantum gate, we would be required to assemble some<\/p>\n\n\n\n
means to applying forces on the system in a precise and accurate manner.<\/p>\n\n\n\n
These operations would have to be impervious to error. The major prob-<\/p>\n\n\n\n
lem in practice with quantum superposition states is that they are extremely<\/p>\n\n\n\n
fragile. The slightest interactions would cause a disturbance by which the co-<\/p>\n\n\n\n
herence is lost and the prepared system ends up in one of the basis states!<\/p>\n\n\n\n
This phenomenon, known in literature as decoherence, is also crucial in un-<\/p>\n\n\n\n
derstanding how the classical world emerges from the quantum substrate.<\/p>\n\n\n\n
However, the discovery of quantum error correction and the subsequent con-<\/p>\n\n\n\n
struction of fault-tolerant computing has infused con\ufb01dence in the success of<\/p>\n\n\n\n
the paradigm despite this issue.<\/p>\n\n\n\n
The \ufb01nal big challenge is in interpreting the results of a measurement on<\/p>\n\n\n\n
the system. The whole computational process must be set up such that the<\/p>\n\n\n\n
end result is one of the basis states so that measurements give de\ufb01nite and<\/p>\n\n\n\n
not probabilistic outcomes.<\/p>\n\n\n\n
It may indeed be justi\ufb01able to ask if quantum computation is just in theory,<\/p>\n\n\n\n
a matter of fanciful speculation, or possible in concrete implementation. While<\/p>\n\n\n\n
there are technical challenges in the building of a feasible quantum computer,<\/p>\n\n\n\n
the actual implementation is not only possible but also a reality. Various<\/p>\n\n\n\n
ingenious techniques in quantum physics have been implemented, and newer<\/p>\n\n\n\n
ones are being rapidly developed.<\/p>\n\n\n\n
In developing a viable physical implementation, a bunch of criteria, \ufb01rst<\/p>\n\n\n\n
to be underlined by DiVincenzo [30], are to be satis\ufb01ed:<\/p>\n\n\n\n
12 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
1. A robust, error-tolerant system for qubits<\/p>\n\n\n\n
2. A method of initializing (preparing initial states)<\/p>\n\n\n\n
3. Scalability: quantum systems that must be replicated to larger numbers<\/p>\n\n\n\n
to make bigger registers<\/p>\n\n\n\n
4. Ability to manipulate individual quantum states: this is the most chal-<\/p>\n\n\n\n
lenging engineering task that is required to make the computer work<\/p>\n\n\n\n
5. Readout of output: the end result of the computation must be readable,<\/p>\n\n\n\n
that is, measurement with unambiguous results.<\/p>\n\n\n\n
Several systems have been analyzed with these criteria in mind. In a given<\/p>\n\n\n\n
system too, there could be di\ufb00erent possible realizations of a qubit. In Ta-<\/p>\n\n\n\n
ble 1.1, we list a few such systems to give you an idea of the variety in the<\/p>\n\n\n\n
physics that is involved.<\/p>\n\n\n\n
TABLE 1.1: Summary of common physical implementations of quantum com-<\/p>\n\n\n\n
puting systems.<\/p>\n\n\n\n
System Information carrier Method of control<\/p>\n\n\n\n
Quantum Optics photon polarization polarizers, half wave<\/p>\n\n\n\n
plates, quarter wave<\/p>\n\n\n\n
plates<\/p>\n\n\n\n
presence of a single photon<\/p>\n\n\n\n
in one of two modes<\/p>\n\n\n\n
beamsplitters, mirrors,<\/p>\n\n\n\n
and non-linear optical<\/p>\n\n\n\n
media<\/p>\n\n\n\n
Cavity QED two-level atom interacting<\/p>\n\n\n\n
with a single photon<\/p>\n\n\n\n
phase-shifters, beam split-<\/p>\n\n\n\n
ters, and other linear opti-<\/p>\n\n\n\n
cal elements<\/p>\n\n\n\n
Trapped Ions hyper\ufb01ne energy levels<\/p>\n\n\n\n
and the vibrational modes<\/p>\n\n\n\n
of the atom<\/p>\n\n\n\n
pulsed laser light to ma-<\/p>\n\n\n\n
nipulate the atomic state<\/p>\n\n\n\n
Nuclear Magnetic<\/p>\n\n\n\n
Resonance (NMR)<\/p>\n\n\n\n
nuclear spin states pulsed RF \ufb01elds in the<\/p>\n\n\n\n
presence of a strong exter-<\/p>\n\n\n\n
nal magnetic \ufb01eld<\/p>\n\n\n\n
Superconducting<\/p>\n\n\n\n
Circuits<\/p>\n\n\n\n
Cooper-pair box electrostatic gates and<\/p>\n\n\n\n
Josephson junctions<\/p>\n\n\n\n
\ufb02ux-coupled SQUID magnetic \ufb01elds, spin inter-<\/p>\n\n\n\n
actions<\/p>\n\n\n\n
current-biased junction pulsed microwave \ufb01elds<\/p>\n\n\n\n
Quantum Dots electron spin magnetic \ufb01elds and volt-<\/p>\n\n\n\n
age pulses<\/p>\n\n\n\n
charge state electrostatic gates and<\/p>\n","protected":false},"excerpt":{"rendered":"
Introduction to Quantum Physics and Information Processing (a) Experimental setup: 1 slit closed (b) Classically expected result (c) Actual result: interference Introduction 9 1.2 Properties of Qubits Quantum systems have certain properties that are counter-intuitive and completely outside our range of experience in the classical world. These \u201cweird\u201d properties are best understood as inevitable consequences […]<\/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-3954","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\/3954","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=3954"}],"version-history":[{"count":3,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3954\/revisions"}],"predecessor-version":[{"id":4550,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3954\/revisions\/4550"}],"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=3954"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=3954"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=3954"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}