This is the first project report at my University. This report describes No Cloning Theorem, an introductory topic of Quantum Computation and Quantum Information Theory. The report also covers the necessary mathematics and physics.
-It is a good ppt for a beginner to learn about Quantum
Computer.
-Quantum computer a solution for every present day computing
problems.
-Quantum computer a best solution for AI making
Quantum computing is a rapidly developing field of computer science that explores the application of quantum mechanics to information processing. It promises to revolutionize the way we solve complex problems that are currently beyond the capabilities of classical computers.
This PowerPoint presentation provides an introduction to the basics of quantum computing, including the principles of quantum mechanics, the properties of quantum bits or qubits, quantum entanglement, quantum superposition, and types of quantum computing .
-It is a good ppt for a beginner to learn about Quantum
Computer.
-Quantum computer a solution for every present day computing
problems.
-Quantum computer a best solution for AI making
Quantum computing is a rapidly developing field of computer science that explores the application of quantum mechanics to information processing. It promises to revolutionize the way we solve complex problems that are currently beyond the capabilities of classical computers.
This PowerPoint presentation provides an introduction to the basics of quantum computing, including the principles of quantum mechanics, the properties of quantum bits or qubits, quantum entanglement, quantum superposition, and types of quantum computing .
Quantum computation uses the quantistic physics principles to store and to process information on computational devices.
Presentation for a workshop during the event "SUPER, Salone delle Startup e Imprese Innovative"
In this deck from the Argonne Training Program on Extreme-Scale Computing 2019, Jonathan Baker from the University of Chicago presents: Quantum Computing: The Why and How.
"Jonathan Baker is a second year Ph.D student at The University of Chicago advised by Fred Chong. He is studying quantum architectures, specifically how to map quantum algorithms more efficiently to near term devices. Additionally, he is interested in multivalued logic and taking advantage of quantum computing’s natural access to higher order states and using these states to make computation more efficient. Prior to beginning his Ph.D., he studied at the University of Notre Dame where he obtained a B.S. of Engineering in computer science and a B.S. in Chemistry and Mathematics."
Watch the video: https://wp.me/p3RLHQ-l1i
Learn more: https://extremecomputingtraining.anl.gov/
Sign up for our insideHPC Newsletter: http://insidehpc.com/newsletter
The Presentation is about the quantum computers and quantum computing describing the quantum phenomena which makes the future computers 1000 times more powerful than the current computers .Also include an Artificial intelligence to tell the difference of computing power between the a conventional computer computing and a quantum computer computing.Quantum computers are still under research and development and not available for common peoples and businesses but major organization are investing highly on these future machine hardware especially U.S is spending billions of Dollars to make it happened for their future security purposes.
Quantum computing description in short. History about quantum computers. Hero's of quantum computers,. introductions abstract what are quantum computers
Quantum computing is the computing which uses the laws of quantum mechanics to process information. Quantum computer works on qubits, which stands for "Quantum Bits".
With quantum computers, factoring of prime numbers are possible.
This presentation is about quantum computing.which going to be new technological concept for computer operating system.In this subject the research is going on.
Quantum computing is an emerging new theory of computation based on the principles of quantum mechanics. It is the basis for a fundamentally new information processing model that is garnering increasing attention in the media and from commercial information technology companies. In certain computing tasks, it can theoretically arrive at a solution more efficiently than classical computers. In this session, we explore the basic principles behind quantum computing, including qubit superposition and entanglement -- the basis for quantum parallelism. We explore quantum logic gates as an abstracted representation of underlying hardware and discuss a simple quantum gate circuit that demonstrates parallelism. We also review the current state of the technology and what has been demonstrated compared to what is theoretically predicted. Current trends in the quantum computing industry will be presented along with proposed possible uses in biomedical informatics.
I will explain why quantum computing is interesting, how it works and what you actually need to build a working quantum computer. I will use the superconducting two-qubit quantum processor I built during my PhD as an example to explain its basic building blocks. I will show how we used this processor to achieve so-called quantum speed-up for a search algorithm that we ran on it. Finally, I will give a short overview of the current state of superconducting quantum computing and Google's recently announced effort to build a working quantum computer in cooperation with one of the leading research groups in this field.
With the introduction of quantum computing on the horizon, computer security organizations are stepping up research and development to defend against a new kind of computer power. Quantum computers pose a very real threat to the global information technology infrastructure of today. Many security implementations in use based on the difficulty for modern-day computers to perform large integer factorization. Utilizing a specialized algorithm such as mathematician Peter Shor’s, a quantum computer can compute large integer factoring in polynomial time versus classical computing’s sub-exponential time. This theoretical exponential increase in computing speed has prompted computer security experts around the world to begin preparing by devising new and improved cryptography methods. If the proper measures are not in place by the time full-scale quantum computers produced, the world’s governments and major enterprises could suffer from security breaches and the loss of massive amounts of encrypted data
Quantum Computers new Generation of Computers part 7 by prof lili saghafi Qua...Professor Lili Saghafi
Quantum algorithm
algorithm for factoring, the general number field sieve
Optimization algorithm
deterministic quantum algorithm Deutsch-Jozsa algorithm
Entanglement
Enigma
Quantum Teleportation
Quantum Computers New Generation of Computers PART1 by Prof Lili SaghafiProfessor Lili Saghafi
This lecture is intended to introduce the concepts and terminology used in Quantum Computing, to provide an overview of what a Quantum Computer is, and why you would want to program one.
The material here is using very high level concepts and is designed to be accessible to both technical and non-technical audiences.
Some background in physics, mathematics and programming is useful to help understand the concepts presented.
Exploits Quantum Mechanical effects
Built around “Qubits” rather than “bits”
Operates in an extreme environment
Enables quantum algorithms to solve very hard problems
Quantum computation uses the quantistic physics principles to store and to process information on computational devices.
Presentation for a workshop during the event "SUPER, Salone delle Startup e Imprese Innovative"
In this deck from the Argonne Training Program on Extreme-Scale Computing 2019, Jonathan Baker from the University of Chicago presents: Quantum Computing: The Why and How.
"Jonathan Baker is a second year Ph.D student at The University of Chicago advised by Fred Chong. He is studying quantum architectures, specifically how to map quantum algorithms more efficiently to near term devices. Additionally, he is interested in multivalued logic and taking advantage of quantum computing’s natural access to higher order states and using these states to make computation more efficient. Prior to beginning his Ph.D., he studied at the University of Notre Dame where he obtained a B.S. of Engineering in computer science and a B.S. in Chemistry and Mathematics."
Watch the video: https://wp.me/p3RLHQ-l1i
Learn more: https://extremecomputingtraining.anl.gov/
Sign up for our insideHPC Newsletter: http://insidehpc.com/newsletter
The Presentation is about the quantum computers and quantum computing describing the quantum phenomena which makes the future computers 1000 times more powerful than the current computers .Also include an Artificial intelligence to tell the difference of computing power between the a conventional computer computing and a quantum computer computing.Quantum computers are still under research and development and not available for common peoples and businesses but major organization are investing highly on these future machine hardware especially U.S is spending billions of Dollars to make it happened for their future security purposes.
Quantum computing description in short. History about quantum computers. Hero's of quantum computers,. introductions abstract what are quantum computers
Quantum computing is the computing which uses the laws of quantum mechanics to process information. Quantum computer works on qubits, which stands for "Quantum Bits".
With quantum computers, factoring of prime numbers are possible.
This presentation is about quantum computing.which going to be new technological concept for computer operating system.In this subject the research is going on.
Quantum computing is an emerging new theory of computation based on the principles of quantum mechanics. It is the basis for a fundamentally new information processing model that is garnering increasing attention in the media and from commercial information technology companies. In certain computing tasks, it can theoretically arrive at a solution more efficiently than classical computers. In this session, we explore the basic principles behind quantum computing, including qubit superposition and entanglement -- the basis for quantum parallelism. We explore quantum logic gates as an abstracted representation of underlying hardware and discuss a simple quantum gate circuit that demonstrates parallelism. We also review the current state of the technology and what has been demonstrated compared to what is theoretically predicted. Current trends in the quantum computing industry will be presented along with proposed possible uses in biomedical informatics.
I will explain why quantum computing is interesting, how it works and what you actually need to build a working quantum computer. I will use the superconducting two-qubit quantum processor I built during my PhD as an example to explain its basic building blocks. I will show how we used this processor to achieve so-called quantum speed-up for a search algorithm that we ran on it. Finally, I will give a short overview of the current state of superconducting quantum computing and Google's recently announced effort to build a working quantum computer in cooperation with one of the leading research groups in this field.
With the introduction of quantum computing on the horizon, computer security organizations are stepping up research and development to defend against a new kind of computer power. Quantum computers pose a very real threat to the global information technology infrastructure of today. Many security implementations in use based on the difficulty for modern-day computers to perform large integer factorization. Utilizing a specialized algorithm such as mathematician Peter Shor’s, a quantum computer can compute large integer factoring in polynomial time versus classical computing’s sub-exponential time. This theoretical exponential increase in computing speed has prompted computer security experts around the world to begin preparing by devising new and improved cryptography methods. If the proper measures are not in place by the time full-scale quantum computers produced, the world’s governments and major enterprises could suffer from security breaches and the loss of massive amounts of encrypted data
Quantum Computers new Generation of Computers part 7 by prof lili saghafi Qua...Professor Lili Saghafi
Quantum algorithm
algorithm for factoring, the general number field sieve
Optimization algorithm
deterministic quantum algorithm Deutsch-Jozsa algorithm
Entanglement
Enigma
Quantum Teleportation
Quantum Computers New Generation of Computers PART1 by Prof Lili SaghafiProfessor Lili Saghafi
This lecture is intended to introduce the concepts and terminology used in Quantum Computing, to provide an overview of what a Quantum Computer is, and why you would want to program one.
The material here is using very high level concepts and is designed to be accessible to both technical and non-technical audiences.
Some background in physics, mathematics and programming is useful to help understand the concepts presented.
Exploits Quantum Mechanical effects
Built around “Qubits” rather than “bits”
Operates in an extreme environment
Enables quantum algorithms to solve very hard problems
COMPUTATIONAL PERFORMANCE OF QUANTUM PHASE ESTIMATION ALGORITHMcsitconf
A quantum computation problem is discussed in this paper. Many new features that make
quantum computation superior to classical computation can be attributed to quantum coherence
effect, which depends on the phase of quantum coherent state. Quantum Fourier transform
algorithm, the most commonly used algorithm, is introduced. And one of its most important
applications, phase estimation of quantum state based on quantum Fourier transform, is
presented in details. The flow of phase estimation algorithm and the quantum circuit model are
shown. And the error of the output phase value, as well as the probability of measurement, is
analysed. The probability distribution of the measuring result of phase value is presented and
the computational efficiency is discussed.
COMPUTATIONAL PERFORMANCE OF QUANTUM PHASE ESTIMATION ALGORITHMcscpconf
A quantum computation problem is discussed in this paper. Many new features that make quantum computation superior to classical computation can be attributed to quantum coherence
effect, which depends on the phase of quantum coherent state. Quantum Fourier transform algorithm, the most commonly used algorithm, is introduced. And one of its most important
applications, phase estimation of quantum state based on quantum Fourier transform, is presented in details. The flow of phase estimation algorithm and the quantum circuit model are
shown. And the error of the output phase value, as well as the probability of measurement, is analysed. The probability distribution of the measuring result of phase value is presented and the computational efficiency is discussed.
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.
The basics of quantum computing, associated mathematics, DJ algorithms and coding details are covered.
These slides are used in my videos https://youtu.be/6o2jh25lrmI, https://youtu.be/Wj73E4pObRk, https://youtu.be/OkFkSXfGawQ and https://youtu.be/OkFkSXfGawQ
Quantum Computing 101, Part 1 - Hello Quantum WorldAaronTurner9
This is the first part of a blog series on quantum computing, broadly derived from CERN’s Practical introduction to quantum computing video series, Michael Nielson’s Quantum computing for the determined video series, and the following (widely regarded as definitive) references:
• [Hidary] Quantum Computing: An Applied Approach
• [Nielsen & Chuang] Quantum Computing and Quantum Information [a.k.a. “Mike & Ike”]
• [Yanofsky & Mannucci] Quantum Computing for Computer Scientists
My objective is to keep the mathematics to an absolute minimum (albeit not quite zero), in order to engender an intuitive understanding. You can think it as a quantum computing cheat sheet.
Similar to No Cloning Theorem with essential Mathematics and Physics (20)
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
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Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
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No Cloning Theorem with essential Mathematics and Physics
1. Calcutta University
Computer Science and Engineering
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Term Paper - I
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No Cloning Theorem
Ritajit Majumdar
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Roll No: 91/CSE/111006
Registration No: 0029169 of 2008-09
Supervisor:
Supervisor:
Guruprasad Kar
Department of Computer
Science and Engineering
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Physics and Applied
Mathematics Unit
Pritha Banerjee
Indian Statistical Institute,
Kolkata
February 12, 2014
Calcutta University
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Abstract
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In this report, I present the idea of No Cloning Theorem, which
was proposed by Wootters and Zurek. This theorem essentially
states that non-orthogonal states of a closed quantum system
cannot be reliably distinguished, and hence cannot be copied.
The linearity of quantum mechanics prohibits the presence of a
perfect cloning device. Hence, generally speaking, it is not possible to develop a universal cloning apparatus which can clone
any arbitrary quantum state.
4. 4.1.1
4.1.2
4.1.3
Photon Emission . . . . . . . . . . . . . . 15
Linearity of Quantum Mechanics . . . . . 16
A single quantum cannot be cloned . . . . 17
19
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5 Conclusion
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Chapter 1
Introduction
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The modern incarnation of computer science was announced by
the great mathematician Alan Turing. He developed a model
of computation known as Turing Machine. He claimed that if
an algorithm can be performed in a computer, then there is an
equivalent algorithm for the Universal Turing Machine.
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Furthermore, Moore’s law stated that computer power will
double for constant cost roughly once in every two years. In
spite of dramatic miniaturization in computer technology in last
few decades, our basic understanding of how a computer works is
still the same (i.e. the Turing Machine). However, conventional
approaches to the fabrication of computer technology are beginning to run up against fundamental difficulties of size. Quantum
effects are beginning to interfere in the functioning of electronic
devices as they are made smaller and smaller [1].
One possible solution to the problem posed by the eventual failure of Moore’s law is to move to a different computing paradigm. Richard Feynman first proposed the concept of
a computer which exploits quantum laws. He said “Atoms on
3
6. small scale behave like nothing on a large scale, for they satisfy
the laws of quantum mechanics. So, as we go down and fiddle
around with the atoms there, we are working with different laws,
and we can expect to do different things”
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A quantum computer is not just a computer following quantum laws. Rather it is a machine which can make explicit use
of certain quantum phenomena which are not present in the
classical realm - e.g. - Superposition [3.1.1 State Space], Entanglement [2] etc.
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Recent studies have shown that quantum algorithms are usually faster than classical ones. Quantum Algorithms are essentially parallel. Polynomial time quantum algorithms for some
NP problems (e.g. prime factorization) have been developed.
Quantum Cryptography holds the promise of better (nearly perfect) secrecy than classical cryptography. And information processing using quantum laws are also more efficient than their
classical counterparts.
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In this report, a basic introductory topic of quantum computation, namely No Cloning Theorem, has been described.
The report starts with the necessary mathematical and physical background and then enters the theorem.
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Chapter 2
2.1.1
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Hilbert Space
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Introductory Mathematics for
Quantum Computation
Linear Vector Space
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For Quantum Computation, the vector space of interest is Cn ,
which is the complex vector space of n dimension. The elements
of the space are called vectors, and are represented by the
column matrix
v1
v
2
.
.
.
vn
The vectors are written as |v , while their complex conjugate
(i.e. the row matrix) is written as v|.
Let |v , |w and |z be three vectors in a space V and α and β
are two scalars (usually complex numbers). Then V is a Linear
Vector Space if the following conditions are satisfied 5
8. 1. Closure: |v + |w ∈ V .
2. Closure: α |v ∈ V .
3. Commutative: |v + |w = |w + |v .
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4. Associative: α(β |v ) = (αβ) |v
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5. Associative: |v + (|w + |z ) = (|v + |w ) + |z .
6. There is a zero vector such that |v + 0 = |v .
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7. There is an additive inverse which maps a vector to the zero
vector |v + |−v = 0
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8. Distributive: α(|v + |w ) = α |v + α |w
9. Distributive: (α + β) |v = α |v + β |v
Inner Product Space
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2.1.2
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An inner product is a linear function which takes as input two
vectors and outputs a complex number. So mathematically inner product is a function that maps from V × V −→ C.
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Inner product between two vectors and |w is computed
|v
v1
w1
v
w
2
2
as w|v . So if |v = . and |w = .
.
.
.
.
vn
wn
then the inner product
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∗
where wi is the complex conjugate of wi .
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∗
∗
w|v = w1 w2
v1
v
∗ 2
· · · wn . .
.
.
vn
Again, let |v and |w be two vectors and λ is a scalar. A
Linear Vector Space is an Inner Product Space if the following
conditions are satisfied -
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1. v|λw = λ v|w
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2. v|w = ( w|v )∗
3. v|v ≥ 0, the value is 0 iff |v = 0
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The length of a vector is defined as v =
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v|v
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And the length is called norm. A vector is said to be unit vector or normalised if its norm is 1.
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Two vectors whose inner product is zero are said to be orthogonal. A collection of mutually orthogonal, normalised
vectors is called an orthonormal set.
αi |αj = δij ,
where δij
= 0, i = j
= 0, i = j
A linear vector space with inner product is called
a Hilbert Space.
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10. 2.2
2.2.1
Observables and Tensor Product
Observables
U †U = U U † = I
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A quantum state is defined as a vector in Hilbert Space. The operators (e.g time evolution operator, quantum gates) are called
observables. Observables in Quantum mechanics are unitary
matrices. A matrix U is said to unitary if
where U † = (U ∗ )T and I is the identity matrix.
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If a state vector is of dimension n, then the observable operating on it is of dimension n × n.
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Operators in quantum mechanics are Hermitian Matrices. A
matrix H is said to be hermitian if
H = H†
Tensor Products
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Thus, unlike classical gates, quantum gates are reversible.
Operating the same gate twice on the state returns the original
state.
Tensor product is a way of putting vector spaces together to
form a larger vector space. Let V and W be two vector spaces
of dimensions m and n respectively. Then the tensor product
V
W is a vector space of mn dimension [1].
Let A =
a11 a12
a21 a22
and B =
8
b11 b12
b21 b22
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b12
b22
b12
b22
a12 b12
a12 b22
a22 b12
a22 b12
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Then the tensor product A B is
b11 b12
b11
a11 b21 b22 a12 b21
b b
b
a21 11 12 a22 11
b21 b22
b21
a11 b11 a11 b12 a12 b11
a11 b21 a11 b22 a12 b21
=
a21 b11 a21 b12 a22 b11
a21 b21 a21 b22 a22 b11
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Chapter 3
3.1.1
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Postulates of Quantum Mechanics
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Introductory Quantum
Mechanics
State Space
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Postulate 1: Associated to any isolated system is a
Hilbert Space called the state space. The system is
completely defined by the state vector, which is a
unit vector in the state space [1].
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In classical computer, a bit can be either 0 or 1. In quantum
mechanics, the quantum bit or qubit is a vector in the state
space. And a qubit can be in any linear superposition of |0 or
|1 . In general, a qubit is mathematically represented as |ψ = α0 |0 + α1 |1
where α0 and α1 are complex numbers. α0 and α1 are called
the amplitudes of |0 and |1 respectively. The square of the
amplitude, |αi |2 , gives the probability that the system collapses
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13. to state |i (this will be further elaborated in the 3rd postulate).
Since the total probability is always 1, hence
|α0 |2 + |α1 |2 = 1
3.1.2
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This is called the normalisation condition. Hence, a qubit is
a unit vector in the state space.
Evolution
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Postulate 2: The evolution of a closed quantum system is described by a unitary transformation [1].
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If |ψ(t1 ) is the state of the quantum system at time t1 and
|ψ(t2 ) is the state of the system at time t2 , then
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|ψ(t2 ) = U (t1 , t2 ) |ψ(t1 )
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where U (t1 , t2 ) is a unitary matrix.
Schr¨dinger Equation
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The time evolution of a closed quantum system is given by the
Schr¨dinger equation o
i
d|ψ
dt
= H |ψ
where H is the total enery (kinetic + potential) of the system
and is called the Hamiltonian.
Integrating the equation within the time limits t1 and t2 , we
have |ψ2 = exp −i
11
(t2 −t1 )
H
|ψ1
14. Now taking U ≡ exp −i
picture of time evolution.
we get back the unitary matrix
Measurement
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3.1.3
(t2 −t1 )
H
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Unlike classical physics, measurement in quantum mechanics is
not deterministic. Even if we have the complete knowledge of
a system, we can at most predict the probability of a certain
outcome from a set of possible outcomes. If we have a quantum
state |ψ = α |0 + β |1 , then the probability of getting outcome
|0 is |α|2 and that of |1 is |β|2 . After measurement, the state of
the system collapses to either |0 or |1 with the said probability.
†
p(m) = ψ| Mm Mm |ψ
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Postulate 3: Quantum measurements are described
by a collection of measurement operators {Mm }.
These are operators acting on the state space of
the system being measured. The index m refers to
the measurement outcomes that may occur in the experiment. If the state of the quantum system is |ψ
immediately before the measurement then the probability that result m occurs is [1]
and the state of the system after measurement is
Mm |ψ
†
ψ| Mm Mm |ψ
The measurement operators satisfy the completeness relation
†
m Mm Mm
12
=I
15. 3.1.4
Composite System
Postulate 4: The state space of a composite physical
system is the tensor product of the state spaces of
the component physical systems [1].
|ψ2
···
|ψn
M
|ψ1
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If we have n systems, numbered 1 through n, and the ith
system is prepared in state |ψi , then the composite state of the
total system is
3.2.1
Distinguising Quantum States
Orthogonal States
PY
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3.2
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In a classical computer, if we have a RAM of size m and
add another RAM of size n, then the composite size is m + n.
However, from this postulate, it is obvious that in a quantum
computer, if we have a qubit of dimension m and another qubit
of dimension n, then the dimension of the composite system is
mn.
CO
An observable M can be written in the form |m m|. This is
called Spectral Decomposition 1 .
Consider a set of projectors Mi = |ψi ψi | for i = 1, · · · , n
and M0 = I −
1
i=0 |ψi
ψi | (from Completeness Relation)
For further information, see Nielsen, Chuang Page 72
13
16. So if an orthonormal state |ψi is prepared, then
p(i) = ψi | Mi |ψi = ψi |ψi ψi |ψi = 1
3.2.2
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Thus result i occurs with certainty and hence it is possible
to reliably distinguish orthonormal states |ψi .
Non Orthogonal States
M
Consider two quantum states
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|ψ2
|ψ1 = |0
= α |0 + β |1
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If a measurement is performed, then |ψ1 is projected to |0
with probability 1. However, |ψ2 is also projected to |0 with
probability |α|2 .
IG
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So, if the outcome is |0 , then it is not possible to say whether
the state was |ψ1 or |ψ2 .
CO
PY
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Hence, non orthogonal states cannot be reliably distinguished.
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17. AJ
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Chapter 4
4.1
No Cloning Theorem
M
No Cloning Theorem
Photon Emission
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4.1.1
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Possibly the most prominent feature that distinguishes between
classical and quantum information theory is the “no cloning theorem” which prevents in producing perfect copies of an arbitrary
quantum mechanical state [4].
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When a photon having a definite polarization encounters an excited atom, there is some probability that the atom will emit a
photon due to stimulation. If there is such an emission, then the
second photon is guaranteed to have the same polarization as
the original photon [3].
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18. R
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img : en.wikipedia.org/wiki/Stimulated emission
Linearity of Quantum Mechanics
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4.1.2
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This phenomena gave rise to the question whether using this
method (or any other method), it is possible to clone a quantum
state.
IG
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From the 2nd postulate of quantum mechanics, we have
|ψ(t2 ) = U (t1 , t2 ) |ψ(t1 )
CO
PY
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Now, from Schr¨dinger equation, it is evident that quantum
o
laws are linear. So if we have any arbitrary quantum state,
|ψ = α |ψ1 + β |ψ2 , and if we take,
U |ψ1 = |φ1 and U |ψ2 = |φ2
then operating U on the state |ψ U |ψ = U (α |ψ1 + β |ψ2 )
= αU |ψ1 + βU |ψ2
= α |φ1 + β |φ2
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19. 4.1.3
A single quantum cannot be cloned
The proof of this theorem is indirect, i.e. proof by contradiction.
Let us consider |A is a universal quantum cloner. Let |s be
the state of the incident photon. So we should have -
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|A |s = |As |ss
M
where |As is the state of the apparatus after operation (we
are not much interested in the state of the apparatus) and |ss
represents the state of the two photons, one original and the
other its copy.
T
Now, let us consider two orthogonal states |0 and |1 . So,
IT
AJ
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|A |0 = |A0 |00
|A |1 = |A1 |11
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So for an arbitrary quantum state |ψ = α |0 + β |1 , we
expect = |Aψ
|A |ψ = |Aψ |ψψ
(α |0 + β |1 )(α |0 + β |1 )
PY
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= |Aψ (α2 |00 + αβ |01 + αβ |10 + β 2 |11 )
(4.1)
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However, from linearity of quantum mechanics, it is evident
that
|A |ψ = |A (α |0 + β |1 )
= α |A |0 + β |A |1
= α |00 + β |11
17
(4.2)
20. Equations (4.1) and (4.2) will be equal only if αβ = 0. However, for αβ to be 0, either α = 0 or β = 0.
AJ
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But if α = 0, then the term |00 vanishes in equation (4.1)
and if β = 0, then the term |11 vanishes. Thus the linearity of
quantum mechanics prohibits a perfect copy. Hence, A single
quantum cannot be cloned.
T
M
However, the above argument does not prohibit the copying of orthogonal states. This is because orthogonal states are
distinguishable. Since non-orthogonal states are not reliably distinguishable, we cannot make a perfect copy of them.
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Hence, in general, the theorem states that it is not possible to
have a universal quantum cloner that can create perfect clones
of any arbitrary quantum state.
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21. AJ
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Chapter 5
Conclusion
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M
The linearity of quantum mechanics prohibits the existance of a
universal quantum copy machine. This has both advantages and
disadvantages. The disadvantages are immediately prominent.
Unlike classical gates, fan out is not possible in quantum gates.
If a quantum computer is made, copying, at least perfect copying, will not be possible so easily. However, a huge advantage
is observable in quantum cryptography. Hackers cannot make a
copy of the data being sent.
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An immediate consequence of no-cloning theorem is that information cannot be copied. So for information sending, a protocol named quantum teleportation [5] has been proposed which
states that to send an information, the information at the sender’s
end must be destroyed. Studies are being pursued on performing
partial cloning of a quantum state [4][6][7][8].
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22. AJ
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Bibliography
[1] Quantum Computation and Quantum Information
Nielsen, Chuang
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M
[2] Einstein, Podolsky, Rosen
PRL Vol 47, May 15, 1935
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[3] A single quantum cannot be cloned
Wootters, Zurek
Nature, Vol 299, 1982
IG
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[4] S. Bandyopadhyay, G. Kar
arXiv:quant-ph/9902073v3
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[5] Bennett et. al
PRL Vol 70, number 13, March 1993
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[6] R. F. Werner
PRA Vol 58, number 3, September 1998
[7] Buzek, Hillery
PRA Vol 54, number 3, September 1996
[8] Buzek et. al
PRA vol 55, number 5, May 1997
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