Introduction to quantum computation. Here the very basic maths described needed for quantum information theory as well as computation. Postulates of quantum mechanics and the Hisenberg`s Uncertainty principle. Basic operator theories are described here.
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
Quantum computing is a type of computation that harnesses the collective properties of quantum states, such as superposition, interference, and entanglement, to perform calculations.
This presentation is designed to elucidate about the Quantum Computing - History - Principles - QUBITS - Quantum Computing Models - Applications - Advantages and Disadvantages.
-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
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.
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
Quantum computing is a type of computation that harnesses the collective properties of quantum states, such as superposition, interference, and entanglement, to perform calculations.
This presentation is designed to elucidate about the Quantum Computing - History - Principles - QUBITS - Quantum Computing Models - Applications - Advantages and Disadvantages.
-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
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.
An overview of quantum computing, with its features, capabilities and types of problems it can solve. Also covers some current and future implementations of quantum computing, and a view of the patent landscape.
This is a seminar on Quantum Computing given on 9th march 2017 at CIME, Bhubaneswar by me(2nd year MCA).
Video at - https://youtu.be/vguxg0RYg7M
PDF at - http://www.slideshare.net/deepankarsandhibigraha/quantum-computing-73031375
No Cloning Theorem with essential Mathematics and PhysicsRitajit Majumdar
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.
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 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"
An overview of quantum computing, with its features, capabilities and types of problems it can solve. Also covers some current and future implementations of quantum computing, and a view of the patent landscape.
This is a seminar on Quantum Computing given on 9th march 2017 at CIME, Bhubaneswar by me(2nd year MCA).
Video at - https://youtu.be/vguxg0RYg7M
PDF at - http://www.slideshare.net/deepankarsandhibigraha/quantum-computing-73031375
No Cloning Theorem with essential Mathematics and PhysicsRitajit Majumdar
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.
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 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"
Would you bet your job on your A/B test results?Qubit
It is possible that 72% of your successful A/B tests may not be driving any business benefit, or may actually be harming your bottom line. This is because bad methodology is preventing you from detecting the truly successful ideas.
In this webinar, Qubit explains how you can avoid this problem. With special guests from Forrester who explain how A/B testing fits into the digital landscape, and Staples, who give practical advice for how to set up an A/B testing campaign, this webinar will change the way you think about A/B testing forever
This is the presentation used by Dr Charles Bennet ,Fellow IBM during his Video Conferencing Lecture on Quantum Information for students at NIT Warangal
This talk mainly focused on the protocol of quantum secret sharing(QSS). First the (k,n) threshold scheme was discussed here which was introduced by Adi Shamir and then migrated to the idea of quantum secret sharing scheme. The QSS scheme was first introduced by Hillery et al. in 1999.
Quantum Information with Continuous Variable systemskarl3s
This book deals with the study of quantum communication protocols with Continuous Variable (CV) systems. Continuous Variable systems are those described by canonical conjugated coordinates x and p endowed with infinite dimensional Hilbert spaces, thus involving a complex mathematical structure. A special class of CV states, are the so-called Gaussian states. With them, it has been possible to implement certain quantum tasks as quantum teleportation, quantum cryptography and quantum computation with fantastic experimental success. The importance of Gaussian states is two- fold; firstly, its structural mathematical description makes them much more amenable than any other CV system. Secondly, its production, manipulation and detection with current optical technology can be done with a very high degree of accuracy and control. Nevertheless, it is known that in spite of their exceptional role within the space of all Continuous Variable states, in fact, Gaussian states are not always the best candidates to perform quantum information tasks. Thus non-Gaussian states emerge as potentially good candidates for communication and computation purposes.
Quantum Computers New Generation of Computers part 6 by Prof Lili SaghafiProfessor Lili Saghafi
Qubits Out Of Diamonds
Quantum Entanglement
What Future Leads
Blind Quantum Computing
Teleportation For Error Correction
Could The Universe Be A Giant Quantum Computer?
Gamma-ray Shaping Could Lead To 'Nuclear' Quantum Computers
Research Areas
ALGORITHMS
Quantum Metrology
Quantum Noise
Potential Applications & Nasa
What questions we should ask from Quantum Computers ???
This research paper gives an overview of quantum computers – description of their operation, differences between quantum and silicon computers, major construction problems of a quantum computer and many other basic aspects. No special scientific knowledge is necessary for the reader.
This presentation was created for a first year physics project at Imperial.
A presentation describing some of the applications of quantum entanglement, for example: quantum clocks, quantum computing, teleportation and quantum cryptography. Refers to specific experiment of teleportation carried out by NIST using time-bin encoding.
Quantum teleportation is the process in which, the quantum state of a
particle is transferred to another without direct interaction. This is one of the most important consequence of quantum entanglement.
Here I have tried to explain the theory behind the teleportation and its experimental verification.
Pulse Compression Sequence (PCS) are widely used in radar to increase the range resolution. Binary sequence has the limitation that the compression ratio is small. Ternary code is suggested as an alternative. The design of ternary sequence with good Discriminating Factor (DF) and merit factor can be considered as a nonlinear multivariable optimization problem which is difficult to solve. In this paper, we proposed a new method for designing ternary sequence by using Modified Simulated Annealing Algorithm (MSAA). The general features such as global convergence and robustness of the statistical algorithm are revealed.
Quantum Computer is a machine that is used for Quantum Computation with the help of using Quantum Physics properties. Where classical computers encode information in binary “bits” that can either 0s or 1s but quantum computer use Qubits. Like the classical computer, the Quantum computer also uses 0 and 1, but qubits have a third state that allows them to represent one or zero at the same time and it’s called “Superposition”. This research paper has presented the Basics of Quantum Computer and The Future of Quantum Computer. So why Quantum Computer can be Future Computer, Because Quantum Computer is faster than any other computer, as an example, IBM’s Computer Deep Blue examined 200 million possible chess moves each second. Quantum Computer would be able to examine 1 trillion possible chess moves per second. It can be 100 million times faster than a classical computer. The computer makes human life easier and also focuses on increasing performance to make technology better. One such way is to reduce the size of the transistor and another way is to use Quantum Computer. The main aim of this paper is to know that how Quantum Computers can become the future computer.
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.
What Quantum Computing is and is not? - Manuel Rudolph, Physicist.Ari Massoudi
This is the slides showed by Manuel Rudolph during his interview-webinar on Episteme Entrepreneur: https://www.episteme-entrepreneur.com/blog/manuel-rudolph-physics-quantum-computing-machine-learning-algorithms
What Quantum Computing is and is not?
A friendly introduction to quantum computing and quantum machine learning algorithms.
Curious about how a quantum computer actually computes and how scientists write algorithms for them?
Quantum computing has been touted with great potential in various fields. Some claims may be overly ambitious, and opinions on the feasibility of building practical machines may differ. Still, there is much to be optimistic about and to be excited for. In this session, Manuel will share his experience as a quantum algorithms researcher with a diverse scientific background-- from university to a quantum computing startup and back. He will begin with a friendly introduction to the foundations of quantum computing, and then talk about his research in quantum machine learning and generative modelling. Manuel's goal is to help develop the first practical algorithms capable of running on the earliest viable quantum computers.
Manuel Rudolph is PhD Candidate in Physics at EPFL (École Polytechnique Fédérale de Lausanne) | Working on Quantum Machine Learning Algorithms | Laboratory of Quantum Information and Computation led by Prof. Zoë Holmes.
This is a seminar on Quantum Computing given on 9th march 2017 at CIME, Bhubaneswar by me(2nd year MCA).
Video at - https://youtu.be/vguxg0RYg7M
ppt at - http://www.slideshare.net/deepankarsandhibigraha/quantum-computing-73031661
On the atomic scale matter obeys the rules of quantum mechanics, which are quite different from the classical rules that determine the properties of conventional logic gates. So if computers are to become smaller in the future, new, quantum technology must replace or supplement for this.
Quantum communication and quantum computingIOSR Journals
Abstract: The subject of quantum computing brings together ideas from classical information theory, computer
science, and quantum physics. This review aims to summarize not just quantum computing, but the whole
subject of quantum information theory. Information can be identified as the most general thing which must
propagate from a cause to an effect. It therefore has a fundamentally important role in the science of physics.
However, the mathematical treatment of information, especially information processing, is quite recent, dating
from the mid-20th century. This has meant that the full significance of information as a basic concept in physics
is only now being discovered. This is especially true in quantum mechanics. The theory of quantum information
and computing puts this significance on a firm footing, and has led to some profound and exciting new insights
into the natural world. Among these are the use of quantum states to permit the secure transmission of classical
information (quantum cryptography), the use of quantum entanglement to permit reliable transmission of
quantum states (teleportation), the possibility of preserving quantum coherence in the presence of irreversible
noise processes (quantum error correction), and the use of controlled quantum evolution for efficient
computation (quantum computation). The common theme of all these insights is the use of quantum
entanglement as a computational resource.
Keywords: quantum bits, quantum registers, quantum gates and quantum networks
Quantum computing is the research area centered on creating computer technology that uses quantum theory concepts that explain the nature and conduct of energy and matter at the level of the quantum (atomic and subatomic). The development of a practical quantum computer would mark a step forward in computing capacity far greater than that of a modern supercomputer, with considerable increases in efficiency. According to the rules of quantum physics, a quantum computer could achieve enormous processing power through multi-state capacity and execute functions simultaneously using all possible permutations. This paper briefly discusses the basic elements of quantum computing and further explores the potential of quantum computing to improve analytical and computing capabilities in solving power system problems.
Manchester & Differential Manchester encoding schemeArunabha Saha
The two main variants of biphase encoding techniques are discussed here. Manchester and Differential Manchester encoding scheme are explained with examples. Comparison between several classes of polar encoding techniques are done along with the exposure about the advantages and disadvantages of both schemes.
Polar-NRZ and Polar-RZ scheme discussed with examples. The comparison with the unipolar encoding techniques are drawn here. the advantages and disadvantages to both the schemes are discussed.
Introduction to Channel Capacity | DCNIT-LDTalks-1Arunabha Saha
DCNIT-LDTalks-1
Here I have discussed the channel capacity for noiseless and noisy channels. How Nyquist capacity and Shannon capacity play a key role in the noiseless and noisy channels are discussed in detail. We will see the several expressions of SNR_dB in terms of power and amplitude and try to understand how both the capacity are different from each other. For extreme values of SNR, we will deduce the Shannon capacity formula to understand the bandwidth-limited region and power-limited region. Here I have used a few numerical examples to understand the concept clearly. In the last section of the talk, I have deduced the Shannon capacity formula from scratch to get better exposure and will understand how these ideas contribute to its mathematical framework.
Course: CMS-A-CC-4-8
youtube: https://www.youtube.com/watch?v=1OjlMqHWq6o
Data Communication, Networking & Internet Technology Lecture Series(DCNIT-LDT...Arunabha Saha
DCNIT-LDTalks
During the lockdown, I have introduced a talk series focused on the data communication and networking coursework for UG students under University of Calcutta. This is the course orientation talk.
Course: CMS-A-CC-4-8
youtube link:
https://www.youtube.com/watch?v=pLWzudPJEgk
Introduction to Bayesian classifier. It describes the basic algorithm and applications of Bayesian classification. Explained with the help of numerical problems.
Slack (or Teams) Automation for Bonterra Impact Management (fka Social Soluti...Jeffrey Haguewood
Sidekick Solutions uses Bonterra Impact Management (fka Social Solutions Apricot) and automation solutions to integrate data for business workflows.
We believe integration and automation are essential to user experience and the promise of efficient work through technology. Automation is the critical ingredient to realizing that full vision. We develop integration products and services for Bonterra Case Management software to support the deployment of automations for a variety of use cases.
This video focuses on the notifications, alerts, and approval requests using Slack for Bonterra Impact Management. The solutions covered in this webinar can also be deployed for Microsoft Teams.
Interested in deploying notification automations for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.
Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...UiPathCommunity
💥 Speed, accuracy, and scaling – discover the superpowers of GenAI in action with UiPath Document Understanding and Communications Mining™:
See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
This is a hands-on session specifically designed for automation developers and AI enthusiasts seeking to enhance their knowledge in leveraging the latest intelligent document processing capabilities offered by UiPath.
Speakers:
👨🏫 Andras Palfi, Senior Product Manager, UiPath
👩🏫 Lenka Dulovicova, Product Program Manager, UiPath
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part “Essentials of Automation” series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Here’s what you’ll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
- Practical Applications in FME Form: Delve into key user parameter types including choice, connections, and file URLs. Allow users to control how a workflow runs, making your workflows more reusable. Learn to import values and deliver the best user experience for your workflows while enhancing accuracy.
- Optimization Strategies in FME Flow: Explore the creation and strategic deployment of parameters in FME Flow, including the use of deployment and geometry parameters, to maximize workflow efficiency.
- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
Key Trends Shaping the Future of Infrastructure.pdfCheryl Hung
Keynote at DIGIT West Expo, Glasgow on 29 May 2024.
Cheryl Hung, ochery.com
Sr Director, Infrastructure Ecosystem, Arm.
The key trends across hardware, cloud and open-source; exploring how these areas are likely to mature and develop over the short and long-term, and then considering how organisations can position themselves to adapt and thrive.
Neuro-symbolic is not enough, we need neuro-*semantic*Frank van Harmelen
Neuro-symbolic (NeSy) AI is on the rise. However, simply machine learning on just any symbolic structure is not sufficient to really harvest the gains of NeSy. These will only be gained when the symbolic structures have an actual semantics. I give an operational definition of semantics as “predictable inference”.
All of this illustrated with link prediction over knowledge graphs, but the argument is general.
Accelerate your Kubernetes clusters with Varnish CachingThijs Feryn
A presentation about the usage and availability of Varnish on Kubernetes. This talk explores the capabilities of Varnish caching and shows how to use the Varnish Helm chart to deploy it to Kubernetes.
This presentation was delivered at K8SUG Singapore. See https://feryn.eu/presentations/accelerate-your-kubernetes-clusters-with-varnish-caching-k8sug-singapore-28-2024 for more details.
State of ICS and IoT Cyber Threat Landscape Report 2024 previewPrayukth K V
The IoT and OT threat landscape report has been prepared by the Threat Research Team at Sectrio using data from Sectrio, cyber threat intelligence farming facilities spread across over 85 cities around the world. In addition, Sectrio also runs AI-based advanced threat and payload engagement facilities that serve as sinks to attract and engage sophisticated threat actors, and newer malware including new variants and latent threats that are at an earlier stage of development.
The latest edition of the OT/ICS and IoT security Threat Landscape Report 2024 also covers:
State of global ICS asset and network exposure
Sectoral targets and attacks as well as the cost of ransom
Global APT activity, AI usage, actor and tactic profiles, and implications
Rise in volumes of AI-powered cyberattacks
Major cyber events in 2024
Malware and malicious payload trends
Cyberattack types and targets
Vulnerability exploit attempts on CVEs
Attacks on counties – USA
Expansion of bot farms – how, where, and why
In-depth analysis of the cyber threat landscape across North America, South America, Europe, APAC, and the Middle East
Why are attacks on smart factories rising?
Cyber risk predictions
Axis of attacks – Europe
Systemic attacks in the Middle East
Download the full report from here:
https://sectrio.com/resources/ot-threat-landscape-reports/sectrio-releases-ot-ics-and-iot-security-threat-landscape-report-2024/
Securing your Kubernetes cluster_ a step-by-step guide to success !KatiaHIMEUR1
Today, after several years of existence, an extremely active community and an ultra-dynamic ecosystem, Kubernetes has established itself as the de facto standard in container orchestration. Thanks to a wide range of managed services, it has never been so easy to set up a ready-to-use Kubernetes cluster.
However, this ease of use means that the subject of security in Kubernetes is often left for later, or even neglected. This exposes companies to significant risks.
In this talk, I'll show you step-by-step how to secure your Kubernetes cluster for greater peace of mind and reliability.
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024Albert Hoitingh
In this session I delve into the encryption technology used in Microsoft 365 and Microsoft Purview. Including the concepts of Customer Key and Double Key Encryption.
Software Delivery At the Speed of AI: Inflectra Invests In AI-Powered QualityInflectra
In this insightful webinar, Inflectra explores how artificial intelligence (AI) is transforming software development and testing. Discover how AI-powered tools are revolutionizing every stage of the software development lifecycle (SDLC), from design and prototyping to testing, deployment, and monitoring.
Learn about:
• The Future of Testing: How AI is shifting testing towards verification, analysis, and higher-level skills, while reducing repetitive tasks.
• Test Automation: How AI-powered test case generation, optimization, and self-healing tests are making testing more efficient and effective.
• Visual Testing: Explore the emerging capabilities of AI in visual testing and how it's set to revolutionize UI verification.
• Inflectra's AI Solutions: See demonstrations of Inflectra's cutting-edge AI tools like the ChatGPT plugin and Azure Open AI platform, designed to streamline your testing process.
Whether you're a developer, tester, or QA professional, this webinar will give you valuable insights into how AI is shaping the future of software delivery.
UiPath Test Automation using UiPath Test Suite series, part 3DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 3. In this session, we will cover desktop automation along with UI automation.
Topics covered:
UI automation Introduction,
UI automation Sample
Desktop automation flow
Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
UiPath Test Automation using UiPath Test Suite series, part 3
Introduction to Quantum Computation. Part - 1
1. Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Introduction to Quantum Computation
Part - I
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Ritajit Majumdar, Arunabha Saha
Postulates of Quantum
Mechanics
Next Presentation
Reference
University of Calcutta
September 9, 2013
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
1 / 70
2. Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
1
Introduction
2
Motivations for Quantum Computation
Outline
Introduction
Motivations for Quantum
Computation
3
Qubit
Qubit
Linear Algebra
4
5
Uncertainty Principle
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
6
Postulates of Quantum Mechanics
7
Next Presentation
8
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
2 / 70
3. Classical Computation vs Quantum
Computation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
It may be tempting to say that a quantum computer is
one whose operation is governed by the laws of quantum
mechanics. But since the laws of quantum mechanics
govern the behaviour of all physical phenomena, this
temptation must be resisted.
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
3 / 70
4. Classical Computation vs Quantum
Computation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
It may be tempting to say that a quantum computer is
one whose operation is governed by the laws of quantum
mechanics. But since the laws of quantum mechanics
govern the behaviour of all physical phenomena, this
temptation must be resisted.
Moore’s law roughly stated that computer power will
double for constant cost approximately once every two
years. This worked well for a long time. However, at
present, quantum effects are beginning to interfere in the
functioning of electronic devices as they are made smaller
and smaller.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
September 9, 2013
3 / 70
5. Classical Computation vs Quantum
Computation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
It may be tempting to say that a quantum computer is
one whose operation is governed by the laws of quantum
mechanics. But since the laws of quantum mechanics
govern the behaviour of all physical phenomena, this
temptation must be resisted.
Moore’s law roughly stated that computer power will
double for constant cost approximately once every two
years. This worked well for a long time. However, at
present, quantum effects are beginning to interfere in the
functioning of electronic devices as they are made smaller
and smaller.
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
One possible solution is to move to a different computing
paradigm. One such paradigm is provided by the theory of
quantum computation, which is based on the idea of using
quantum mechanics to perform computations, instead of
classical physics.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
3 / 70
6. Classical Computation vs Quantum
Computation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Quantum systems are exponentially powerful. A system of
500 particles has 2500 ”computing power”. Quantum
Computers provide a neat shortcut for solving a range of
mathematical tasks known as NP-complete problems.
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
4 / 70
7. Classical Computation vs Quantum
Computation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Quantum systems are exponentially powerful. A system of
500 particles has 2500 ”computing power”. Quantum
Computers provide a neat shortcut for solving a range of
mathematical tasks known as NP-complete problems.
For example, factorisation is an exponential time task for
classical computers. But Shor’s quantum algorithm for
factorisation is a polynomial time algorithm. It has
successfully broken RSA cryptosystem.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
September 9, 2013
4 / 70
8. Motivations for Quantum Computation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Faster than light (?) communication.
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
5 / 70
9. Motivations for Quantum Computation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Faster than light (?) communication.
Uncertainty Principle
Highly parallel and efficient quantum algorithms.
Postulates of Quantum
Mechanics
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
5 / 70
10. Motivations for Quantum Computation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Faster than light (?) communication.
Uncertainty Principle
Highly parallel and efficient quantum algorithms.
Postulates of Quantum
Mechanics
Next Presentation
Quantum Cryptography.
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
5 / 70
11. Motivations for Quantum Computation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Faster than light (?) communication.
Uncertainty Principle
Highly parallel and efficient quantum algorithms.
Postulates of Quantum
Mechanics
Next Presentation
Quantum Cryptography.
Reference
and many more...
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
5 / 70
12. Qubits: The building blocks of Quantum
Computer
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Motivations for Quantum
Computation
Qubit
In classical computer, bits of digital information are either 0 or
1. In a quantum computer, these bits are replaced by a
”superposition” of both 0 and 1.
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
1 or
their linear combination.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
6 / 70
13. Qubits: The building blocks of Quantum
Computer
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Motivations for Quantum
Computation
Qubit
In classical computer, bits of digital information are either 0 or
1. In a quantum computer, these bits are replaced by a
”superposition” of both 0 and 1.
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
1
Qubits are represented as |0 and |1 . Qubits have been
created in the laboratory using photons, ions and certain sorts
of atomic nuclei.
1 or
their linear combination.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
6 / 70
14. Superposition Principle
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Suppose we have a k-level system. So there are k
distinguishable or classical states for the system.
The possible classical states for the system: 0, 1, ..., k − 1.
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Superposition Principle
Linear Algebra
If a quantum system can be in one of k states, it can also be in
any linear superposition of those k states.
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
7 / 70
15. Superpostition Principle
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
|0 , |1 , ..., |k − 1 are called the basis states. The
superposition is denoted as a linear combination of these basis.
Outline
Introduction
Motivations for Quantum
Computation
α0 |0 + α1 |1 + ... + αk−1 |k − 1
Qubit
Linear Algebra
where,
Uncertainty Principle
αi ∈ C
Postulates of Quantum
Mechanics
Next Presentation
Reference
|αi |2 = 1
i
(more on this later)
Two level systems are called qubits. (k = 2)
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
8 / 70
16. Qubit: Physical Interpretation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
We may have various interpretations of qubits.
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
9 / 70
17. Qubit: Physical Interpretation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
We may have various interpretations of qubits.
Outline
Consider a Hydrogen atom. This atom may be treated as
a qubit. To do so, we define the ground energy state of
the electron as |0 and the first energy state as |1 .
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
9 / 70
18. Qubit: Physical Interpretation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
We may have various interpretations of qubits.
Outline
Consider a Hydrogen atom. This atom may be treated as
a qubit. To do so, we define the ground energy state of
the electron as |0 and the first energy state as |1 .
The electron dwells in some linear superposition of these
two energy levels. But during measurement, we shall find
the electron in any one of the energy states.
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
9 / 70
19. Other examples of Qubits
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Figure: Photon Polarization: The orientation of electrical field
oscillation is either horizontal or vertical.
Postulates of Quantum
Mechanics
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
10 / 70
20. Other examples of Qubits
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Figure: Photon Polarization: The orientation of electrical field
oscillation is either horizontal or vertical.
Postulates of Quantum
Mechanics
Next Presentation
Reference
Figure: Electron spin: The spin is either up or down
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
10 / 70
21. Qubit: Mathematical model
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Mathematically, a quantum state (which, as we shall see later,
is a vector) is represented by a column matrix. The two
fundamental states that we introduced before, |0 and |1 form
an orthonormal basis. We shall see more of orthonormality
when we see inner products.
The matrix representation of |0 and |1 :
|0 =
Ritajit Majumdar, Arunabha Saha (CU)
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
1
0
|1 =
Motivations for Quantum
Computation
0
1
Introduction to Quantum Computation
September 9, 2013
11 / 70
22. Qubit: Mathematical model
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
So a general quantum state |ψ = α |0 + β |1 is represented
as
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
12 / 70
23. Qubit: Mathematical model
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
So a general quantum state |ψ = α |0 + β |1 is represented
as
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
The matrix notation will be
|ψ =
=
Ritajit Majumdar, Arunabha Saha (CU)
α+0
0+β
α
β
Introduction to Quantum Computation
September 9, 2013
12 / 70
24. Introduction to Quantum
Computation
Qubit: Sign Basis
|0 and |1 are called bit basis since they can be thought of as
the quantum counter-parts of classical bits 0 and 1
respectively. However, they are not the only possible basis. We
may have infinitely many orthonormal basis for a given space.
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Another basis, called the sign basis, is denoted as |+ and |− .
|+ =
|− =
1
√
2
1
√
2
|0 +
|0 −
1
√
2
1
√
2
|1
|1
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Figure: Geometrical model of bit basis and sign basis
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
13 / 70
25. Qubit: Change of Basis
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Measure |ψ =
1
2
√
|0 +
3
2
Introduction
|1 in |+ /|− basis.
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
14 / 70
26. Introduction to Quantum
Computation
Qubit: Change of Basis
Ritajit Majumdar, Arunabha
Saha
Outline
Measure |ψ =
1
2
√
|0 +
3
2
Introduction
|1 in |+ /|− basis.
Motivations for Quantum
Computation
Qubit
Linear Algebra
It can be checked that:
|0 =
|1 =
Ritajit Majumdar, Arunabha Saha (CU)
Uncertainty Principle
1
√
2
1
√
2
|+ +
|+ −
1
√
2
1
√
2
|−
|−
Introduction to Quantum Computation
Postulates of Quantum
Mechanics
Next Presentation
Reference
September 9, 2013
14 / 70
27. Introduction to Quantum
Computation
Qubit: Change of Basis
Ritajit Majumdar, Arunabha
Saha
Outline
Measure |ψ =
1
2
√
|0 +
3
2
Introduction
|1 in |+ /|− basis.
Motivations for Quantum
Computation
Qubit
Linear Algebra
It can be checked that:
|0 =
|1 =
|ψ =
1
2
Uncertainty Principle
1
√
2
1
√
2
|+ +
|+ −
1
√
2
1
√
2
|−
|−
Postulates of Quantum
Mechanics
Next Presentation
Reference
√
|0 +
3
2
|1
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
14 / 70
28. Introduction to Quantum
Computation
Qubit: Change of Basis
Ritajit Majumdar, Arunabha
Saha
Outline
Measure |ψ =
1
2
√
|0 +
3
2
Introduction
|1 in |+ /|− basis.
Motivations for Quantum
Computation
Qubit
Linear Algebra
It can be checked that:
|0 =
|1 =
Uncertainty Principle
1
√
2
1
√
2
|+ +
|+ −
1
√
2
1
√
2
Postulates of Quantum
Mechanics
|−
|−
Next Presentation
Reference
√
|ψ = 1 |0 + 23 |1
2
1 1
1
= 2 ( √2 |+ + √2 |− ) +
Ritajit Majumdar, Arunabha Saha (CU)
√
3 √
1
2 ( 2
|+ +
1
√
2
|− )
Introduction to Quantum Computation
September 9, 2013
14 / 70
30. Introduction to Quantum
Computation
Linear Algebra
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Linear Algebra
a very short introduction
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
15 / 70
31. Introduction to Quantum
Computation
Vector Space
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Motivations for Quantum
Computation
A vector space consists of vectors(|α , |β , |γ ), together
with a set of scalars(a, b, c,....)2 , which is closed under two
operations:
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Vector addition
Reference
Scalar multiplication
2 the
scalars can be complex numbers
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
16 / 70
32. Introduction to Quantum
Computation
Vector Addition
The sum of any two vectors is another vector
Ritajit Majumdar, Arunabha
Saha
Outline
|α + |β = |γ
Introduction
Motivations for Quantum
Computation
Vector addition is commutative
Qubit
|α + |β = |β + |α
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
it is associative also
Next Presentation
|α + (|β + |γ ) = (|α + |β ) + |γ
Reference
There exists a zero(or null) vector3 with the property
|α + |0 = |α ,
∀ |α
For every vector |α there is an associative inverse
vector(|−α ) such that
|α + |−α = |0
3 |0
and 0 are different
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
17 / 70
33. Introduction to Quantum
Computation
Scalar Multiplication
Ritajit Majumdar, Arunabha
Saha
The product of any scalar with any vector is another vector
Outline
Introduction
a |α = |γ
Scalar multiplication is distributive w.r.t vector addition
Motivations for Quantum
Computation
Qubit
Linear Algebra
a(|α + |β ) = a |α + a |β
and with respect to scalar addition also
(a + b) |α = a |α + b |α
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
It is also associative w.r.t ordinary scalar multiplication
a(b |α ) = (ab) |α
Multiplication by scalars 0 and 1 has the effect
0 |α = |0 ;
Ritajit Majumdar, Arunabha Saha (CU)
1 |α = |α
Introduction to Quantum Computation
September 9, 2013
18 / 70
34. Introduction to Quantum
Computation
Basis Vectors
Ritajit Majumdar, Arunabha
Saha
Linear combination of vectors |α , |β , |γ ,... is of the form
Outline
Introduction
|α + |β + |γ + . . .
Motivations for Quantum
Computation
Qubit
A vector |λ is said to be linearly independent of the set
of vectors |α , |β , |γ ,. . . ,if it cannot be written as a
linear combination of them.
A set of vectors is linearly independent if each one is
independent of all the rest.
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
If every vector can be written as a linear combination of
members of this set then the collection of vectors said to
span the space.
A set of linearly independent vectors that spans the space
is called a basis.
The number of vectors in any basis is called the
dimension of space.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
19 / 70
35. Introduction to Quantum
Computation
Inner product
Ritajit Majumdar, Arunabha
Saha
Outline
An inner product is a function which takes two vectors as input
an gives a complex number as output.
The dual(or complex conjugate) of any vector |α is α|
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
|α
∗
= α|
Uncertainty Principle
The inner product of two vectors (|α , |β ) written as α|β
which has the properties:
α|β = β|α
α|α
4
Postulates of Quantum
Mechanics
Next Presentation
Reference
∗
0, and α|α = 0 ⇔ |α = |0
α| (b |β + c |γ ) = b α|β + c α|γ
4 This
is a complex number; α|β ∈ C
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
20 / 70
36. Introduction to Quantum
Computation
Inner Product Space
Ritajit Majumdar, Arunabha
Saha
A vector space with an inner product is called inner product
space.
i.e. the above conditions satisfied for any vectors
|α , |β , |γ ∈ V and for any scalar c
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
n
Uncertainty Principle
e.g. C has an inner product defined by
Postulates of Quantum
Mechanics
Next Presentation
b1
∗
∗ .
ai∗ bi = [a1 . . . an ] .
.
α|β ≡
i
Reference
bn
a1
b1
.
.
where |α = . and |β = .
.
.
an
Ritajit Majumdar, Arunabha Saha (CU)
bn
Introduction to Quantum Computation
September 9, 2013
21 / 70
37. Introduction to Quantum
Computation
Orthonormal Set
Inner product of any vector with itself gives a non-negative
number — its square-root of is real which is called norm
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Motivations for Quantum
Computation
α =
Qubit
α|α
Linear Algebra
It also termed as length of the vector.
A unit vector one whose norm is 1 is said to be normalized5 .
Uncertainty Principle
Two vectors whose inner product is zero is said to be
orthogonal
Next Presentation
Postulates of Quantum
Mechanics
Reference
α|α = 0
A mutually collection of orthogonal normalized vectors is
called an orthonormal set
αi |αj = δij , where δij
5 Normalization:
|ek =
Ritajit Majumdar, Arunabha Saha (CU)
= 0, i = j
= 0, i = j
|k
k
Introduction to Quantum Computation
September 9, 2013
22 / 70
38. Introduction to Quantum
Computation
Orthonormal Set(Contd.)
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
If an orthonormal basis is chosen then the inner product of
two vectors can be written as
Motivations for Quantum
Computation
Qubit
Linear Algebra
α|β =
∗
a1 b1
+
∗
a2 b2
+ ... +
∗
an bn
Uncertainty Principle
Postulates of Quantum
Mechanics
hence the norm(squared)
Next Presentation
2
2
α|α = |a1 | + |a2 | + . . . + |an |
2
Reference
each components are
aj = ej |α ,
Ritajit Majumdar, Arunabha Saha (CU)
where ej ‘s are basis vector
Introduction to Quantum Computation
September 9, 2013
23 / 70
39. Introduction to Quantum
Computation
Linear Operator and Matrices
Ritajit Majumdar, Arunabha
Saha
A linear operator between vector spaces V and W is
defined to be any function A : V → W which is linear in
its inputs
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
ai |υi
A
i
=
ai A(|υi )
i
Postulates of Quantum
Mechanics
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
24 / 70
40. Introduction to Quantum
Computation
Linear Operator and Matrices
Ritajit Majumdar, Arunabha
Saha
A linear operator between vector spaces V and W is
defined to be any function A : V → W which is linear in
its inputs
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
ai |υi
A
i
=
ai A(|υi )
i
another linear operator on any vector space V is Identity
operator, Iv defined as Iv |υ ≡ |υ .
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
Postulates of Quantum
Mechanics
Next Presentation
Reference
September 9, 2013
24 / 70
41. Introduction to Quantum
Computation
Linear Operator and Matrices
Ritajit Majumdar, Arunabha
Saha
A linear operator between vector spaces V and W is
defined to be any function A : V → W which is linear in
its inputs
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
ai |υi
A
i
=
ai A(|υi )
i
another linear operator on any vector space V is Identity
operator, Iv defined as Iv |υ ≡ |υ .
Postulates of Quantum
Mechanics
Next Presentation
Reference
Zero operator which maps all vectors to zero vector,
0 |υ ≡ |0 .
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
24 / 70
42. Introduction to Quantum
Computation
Linear Operator and Matrices
Ritajit Majumdar, Arunabha
Saha
A linear operator between vector spaces V and W is
defined to be any function A : V → W which is linear in
its inputs
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
ai |υi
A
i
=
ai A(|υi )
i
another linear operator on any vector space V is Identity
operator, Iv defined as Iv |υ ≡ |υ .
Postulates of Quantum
Mechanics
Next Presentation
Reference
Zero operator which maps all vectors to zero vector,
0 |υ ≡ |0 .
Let V, W, X are vector spaces, and A : V → W and
B : W → X are linear operators. Then the composition of
operators B and A denoted by BA and defined by
(BA)(|υ ) ≡ B(A(|υ ))
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
24 / 70
43. Matrix Representation of Linear Operators
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
already we have seen that matrices can be regarded as
linear operators..!!
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
25 / 70
44. Matrix Representation of Linear Operators
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
already we have seen that matrices can be regarded as
linear operators..!!
does linear operators has a matrix representation..??!
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
25 / 70
45. Matrix Representation of Linear Operators
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
already we have seen that matrices can be regarded as
linear operators..!!
does linear operators has a matrix representation..??!
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
yes it has..!!
Postulates of Quantum
Mechanics
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
25 / 70
46. Matrix Representation of Linear Operators
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
already we have seen that matrices can be regarded as
linear operators..!!
does linear operators has a matrix representation..??!
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
yes it has..!!
say, A : V → W is a linear operator between vector spaces
V and W. Let the basis set for V and W are
(|υ1 , . . . , |υm ) and (|ω1 , . . . , |ωn ) respectively. Then
we can say For each k in 1,2,....,m, there exist complex
numbers A1k through Ank such that
A |υk =
Postulates of Quantum
Mechanics
Next Presentation
Reference
Aik |ωi
i
The matrix whose entries are Aik is the matrix
representation of the linear operator.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
25 / 70
47. Introduction to Quantum
Computation
Identity Matrix
Ritajit Majumdar, Arunabha
Saha
Outline
The exotic way to express 1
Introduction
Definition
Motivations for Quantum
Computation
Identity matrix is one with the main diagonals and zeros
everywhere. it is denoted by In or I
Qubit
Matrix representation of Identity operator
1 0 ... 0
0 1 . . . 0
. . .. .
.. . .
..
.
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
0 0 ... 1
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
26 / 70
48. Introduction to Quantum
Computation
Identity Matrix
Ritajit Majumdar, Arunabha
Saha
Outline
The exotic way to express 1
Introduction
Definition
Motivations for Quantum
Computation
Identity matrix is one with the main diagonals and zeros
everywhere. it is denoted by In or I
Qubit
Matrix representation of Identity operator
1 0 ... 0
0 1 . . . 0
. . .. .
.. . .
..
.
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
0 0 ... 1
It satisfies the property In A = AIn = A
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
26 / 70
49. Introduction to Quantum
Computation
Identity Matrix
Ritajit Majumdar, Arunabha
Saha
Outline
The exotic way to express 1
Introduction
Definition
Motivations for Quantum
Computation
Identity matrix is one with the main diagonals and zeros
everywhere. it is denoted by In or I
Qubit
Matrix representation of Identity operator
1 0 ... 0
0 1 . . . 0
. . .. .
.. . .
..
.
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
0 0 ... 1
It satisfies the property In A = AIn = A
compact notation: (In )ij = δij
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50. Introduction to Quantum
Computation
Identity Matrix
Ritajit Majumdar, Arunabha
Saha
Outline
The exotic way to express 1
Introduction
Definition
Motivations for Quantum
Computation
Identity matrix is one with the main diagonals and zeros
everywhere. it is denoted by In or I
Qubit
Matrix representation of Identity operator
1 0 ... 0
0 1 . . . 0
. . .. .
.. . .
..
.
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
0 0 ... 1
It satisfies the property In A = AIn = A
compact notation: (In )ij = δij
It satisfies Idempotent law, I.I = I
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51. Introduction to Quantum
Computation
Pauli Matrices
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Pauli matrices are named after the physicist Wolfgang
Pauli.
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
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52. Introduction to Quantum
Computation
Pauli Matrices
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Pauli matrices are named after the physicist Wolfgang
Pauli.
Motivations for Quantum
Computation
Qubit
Linear Algebra
These are a set of 2 x 2 complex matrices which are
Hermitian and unitary.
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
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53. Introduction to Quantum
Computation
Pauli Matrices
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Pauli matrices are named after the physicist Wolfgang
Pauli.
Motivations for Quantum
Computation
Qubit
Linear Algebra
These are a set of 2 x 2 complex matrices which are
Hermitian and unitary.
They look like
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
σ0 ≡ I ≡
10
,
01
σ2 ≡ σy ≡ Y ≡
Ritajit Majumdar, Arunabha Saha (CU)
σ1 ≡ σx ≡ X ≡
0 −i
,
i 0
01
10
σ3 ≡ σz ≡ Z ≡
1 0
0 −1
Introduction to Quantum Computation
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54. Introduction to Quantum
Computation
Pauli Matrices(properties)
Ritajit Majumdar, Arunabha
Saha
Outline
Some properties of Pauli matrices
2
σ1
=
2
σ2
=
2
σ3
= −iσ1 σ2 σ3 =
Introduction
1 0
0 1
=I
det(σi ) = −1
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Tr (σi ) = 0
Each Pauli matrices has two eigenvalues +1 and -1.
Postulates of Quantum
Mechanics
Next Presentation
Normalized eigenvectors are
Reference
ψx+ =
1
√
2
1
,
1
ψx− =
1
√
2
1
−1
ψy + =
1
√
2
1
,
i
ψy − =
1
√
2
1
−i
ψz+ =
1
,
0
Ritajit Majumdar, Arunabha Saha (CU)
ψz− =
0
1
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55. Pauli Matrices and Quantum Computation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Pauli matrices are used here as rotation6 operators.
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
On the basis of Pauli matrices the X, Y, Z quantum gates7 are
designed.
The Pauli-X gate is the quantum equivalent of NOT
gate. It maps |0 to |1 and |1 to |0 .
Postulates of Quantum
Mechanics
Next Presentation
Reference
6 rotation
7 all
of Bloch sphere
acts on single qubit
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56. Pauli Matrices and Quantum Computation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Pauli-Y gate maps |0 to ı |1 and |1 to −˙ |0 .
˙
ı
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
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57. Pauli Matrices and Quantum Computation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Pauli-Y gate maps |0 to ı |1 and |1 to −˙ |0 .
˙
ı
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Pauli-Z gate leaves the basis state |0 unchanged and
maps |1 to -|1 .
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
Reference
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58. Introduction to Quantum
Computation
Hilbert Space
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Hilbert Space
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Wave functions live in Hilbert space
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59. Hilbert Space: Few Basics
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
The mathematical concept of Hilbert space named after
David Hilbert, but this term coined by John von Neumann.
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
8 any Cauchy sequence of functions in Hilbert space converges to a
function that is also in the space.
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60. Hilbert Space: Few Basics
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
The mathematical concept of Hilbert space named after
David Hilbert, but this term coined by John von Neumann.
Motivations for Quantum
Computation
Basically this is the generalization of the notion of
Euclidean space i.e. it extends the methods of algebra
and calculus of 2D Euclidean plane and 3D space to space
of any finite or infinite dimensions.
Linear Algebra
Qubit
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
8 any Cauchy sequence of functions in Hilbert space converges to a
function that is also in the space.
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Introduction to Quantum Computation
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61. Hilbert Space: Few Basics
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
The mathematical concept of Hilbert space named after
David Hilbert, but this term coined by John von Neumann.
Motivations for Quantum
Computation
Basically this is the generalization of the notion of
Euclidean space i.e. it extends the methods of algebra
and calculus of 2D Euclidean plane and 3D space to space
of any finite or infinite dimensions.
Linear Algebra
Hilbert space is an abstract vector space with inner
product defined in it, which allows length and angle to be
measured.
Qubit
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
8 any Cauchy sequence of functions in Hilbert space converges to a
function that is also in the space.
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Introduction to Quantum Computation
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62. Hilbert Space: Few Basics
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
The mathematical concept of Hilbert space named after
David Hilbert, but this term coined by John von Neumann.
Motivations for Quantum
Computation
Basically this is the generalization of the notion of
Euclidean space i.e. it extends the methods of algebra
and calculus of 2D Euclidean plane and 3D space to space
of any finite or infinite dimensions.
Linear Algebra
Hilbert space is an abstract vector space with inner
product defined in it, which allows length and angle to be
measured.
Qubit
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Hilbert space must be complete.8
8 any Cauchy sequence of functions in Hilbert space converges to a
function that is also in the space.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
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63. Hilbert Space: Formal Approach
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
The set of all functions of x constitute a vector space. To
represent a possible physical state,the wave function needed to
be normalized
|ψ|2 dx ≡ ψ|ψ = 1
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
The set of all square-integrable functions on a specified
interval,9
b
a
f (x) such that
|f (x)|2 dx < ∞,
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
constitutes a smaller vector space. It is known to
mathematician as L2 (a, b); physicists call it Hilbert space.
Definition
A Euclidean space Rn is a vector space endowed with the inner
product x|y = y |x ∗ norm x =
x|x and associated metric
x − y , such that every Cauchy sequence takes a limit in Rn .
This makes Rn a Hilbert space.
9 The
limits(a and b) can be ±∞
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64. Introduction to Quantum
Computation
Observables
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Observables
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
woow..! It looks good..!!
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65. Introduction to Quantum
Computation
Observables
Ritajit Majumdar, Arunabha
Saha
Outline
A system observable is a measurable operator, where the
property of the system state can be determined by some
sequence of physical operations.
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
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66. Introduction to Quantum
Computation
Observables
Ritajit Majumdar, Arunabha
Saha
Outline
A system observable is a measurable operator, where the
property of the system state can be determined by some
sequence of physical operations.
Introduction
In quantum mechanics the measurement process affects
the state in a non-deterministic, but in a statistically
predictable way. In particular, after a measurement is
applied, the state description by a single vector may be
destroyed, being replaced by a statistical ensemble.
Linear Algebra
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
Motivations for Quantum
Computation
Qubit
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
September 9, 2013
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67. Introduction to Quantum
Computation
Observables
Ritajit Majumdar, Arunabha
Saha
Outline
A system observable is a measurable operator, where the
property of the system state can be determined by some
sequence of physical operations.
Introduction
In quantum mechanics the measurement process affects
the state in a non-deterministic, but in a statistically
predictable way. In particular, after a measurement is
applied, the state description by a single vector may be
destroyed, being replaced by a statistical ensemble.
Linear Algebra
Motivations for Quantum
Computation
Qubit
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
In quantum mechanics each dynamical variable (e.g.
position, translational momentum, orbital angular
momentum, spin, total angular momentum, energy, etc.)
is associated with a Hermitian operator that acts on the
state of the quantum system and whose eigenvalues
correspond to the possible values of the dynamical
variable.
Ritajit Majumdar, Arunabha Saha (CU)
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68. Introduction to Quantum
Computation
Observables
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
e.g. let |α is an eigenvector of the observable A, with
eigenvalue a and exits in a d-dimensional Hilbert space,
then
A |α = a |α
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
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69. Introduction to Quantum
Computation
Observables
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
e.g. let |α is an eigenvector of the observable A, with
eigenvalue a and exits in a d-dimensional Hilbert space,
then
A |α = a |α
This equation states that if a measurement of the
observable A is made while the system of interest is in
state |α , then the observed value of the particular
measurement must return the eigenvalue a with certainty.
If the system is in the general state |φ ∈ H then the
eigenvalue a return with probability | α|φ |2 (Born rule).
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
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70. Introduction to Quantum
Computation
Observables
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
e.g. let |α is an eigenvector of the observable A, with
eigenvalue a and exits in a d-dimensional Hilbert space,
then
A |α = a |α
This equation states that if a measurement of the
observable A is made while the system of interest is in
state |α , then the observed value of the particular
measurement must return the eigenvalue a with certainty.
If the system is in the general state |φ ∈ H then the
eigenvalue a return with probability | α|φ |2 (Born rule).
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
More precisely, the observables are Hermitian operator
so its represented by Hermitian matrix.
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71. Introduction to Quantum
Computation
Hermitian Operator
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Hermitian
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
hmmm...sounds like me!! is it a new breed ??!!
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72. Introduction to Quantum
Computation
Hermitian Operator
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
No, its nothing new. Its the self-adjoint operator.
Motivations for Quantum
Computation
Qubit
Definition
Linear Algebra
Hermitian matrix is a square matrix with complex entries that
is equal to its own conjugate transpose i.e. the element in
the i-th row and j-th column is equal to the complex conjugate
of the element in the j-th row and i-th column, for all indices i
and j.
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
∗
mathematically aij = aji or in matrix notation, A = (AT )∗
In compact notation, A = A†
Ritajit Majumdar, Arunabha Saha (CU)
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73. Hermitian Operators: Expectation Value
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
As previously said that the measurements are
non-deterministic, so we can get a probabilistic measure of any
observable, that is known as expectation value
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
ˆ
Q =
ˆ
ˆ
ψ ∗ Qψ = ψ Qψ
Operators representing observables have the very special
property that,
ˆ
ˆ
∀f (x)
f Qf = Qf f
Postulates of Quantum
Mechanics
Next Presentation
Reference
More strong condition for hermiticity,
ˆ
ˆ
f Qg = Qf g
∀f (x) and g (x)
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74. Hermitian Operators:Properties
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Eigenvalues of Hermitian operators are real.
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
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75. Hermitian Operators:Properties
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Eigenvalues of Hermitian operators are real.
Eigenfunctions belonging to distinct eigenvalues are
orthogonal.
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
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Introduction to Quantum Computation
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76. Hermitian Operators:Properties
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Eigenvalues of Hermitian operators are real.
Eigenfunctions belonging to distinct eigenvalues are
orthogonal.
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
The eigenfunctions of a Hermitian operator is complete.
Any function(in Hilbert space) can be expressed as the
linear combination of them.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
Reference
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77. Introduction to Quantum
Computation
Uncertainty Principle
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
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78. Introduction to Quantum
Computation
Wave Particle Duality
Ritajit Majumdar, Arunabha
Saha
According to de Broglie hypothesis:
Outline
Introduction
p=
h
λ
=
2π
λ
Motivations for Quantum
Computation
Qubit
where p is the momentum, λ is the wavelength and h is called
Plank’s constant. It has a value of 6.63 × 10−34 Joule-sec.
h
= 2π is called the reduced Plank constant.
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
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79. Introduction to Quantum
Computation
Wave Particle Duality
Ritajit Majumdar, Arunabha
Saha
According to de Broglie hypothesis:
Outline
Introduction
p=
h
λ
=
2π
λ
Motivations for Quantum
Computation
Qubit
where p is the momentum, λ is the wavelength and h is called
Plank’s constant. It has a value of 6.63 × 10−34 Joule-sec.
h
= 2π is called the reduced Plank constant.
This formula essentially states that every particle has a wave
nature and vice versa.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
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80. Heisenberg Uncertainty Principle
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
The Uncertainty Principle is a direct consequence of the wave
particle duality. The wavelength of a wave is well defined, while
asking for its position is absurd. Vice versa is the case for a
particle. And from de Broglie hypothesis, we get that
momentum is inversely proportional to wavelength. Since every
substance has both wave and particle nature -
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Uncertainty Principle
One can never know with perfect accuracy both the position
and the momentum of a particle.
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81. Uncertainty Principle: Mathematical Notation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
According to Heisenberg, the uncertainty in the position and
momentum of a substance must be at least as big as 2 . So we
can write the mathematical notation of the uncertainty
principle:
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
δx.δp ≥
2
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
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82. Introduction to Quantum Mechanics
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
’Not only is the Universe stranger than we
think, it is stranger than we can think’
- Warner Heisenberg
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
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Introduction to Quantum Computation
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83. Introduction to Quantum
Computation
Quantum Mechanics
Ritajit Majumdar, Arunabha
Saha
By the late nineteenth century the laws of physics were
based on Mechanics and laws of Gravitation from Newton,
Maxwell’s equations describing Electricity and Magnetism
and on Statistical Mechanics describing the state of large
collection of matter.
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
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Introduction to Quantum Computation
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84. Introduction to Quantum
Computation
Quantum Mechanics
Ritajit Majumdar, Arunabha
Saha
By the late nineteenth century the laws of physics were
based on Mechanics and laws of Gravitation from Newton,
Maxwell’s equations describing Electricity and Magnetism
and on Statistical Mechanics describing the state of large
collection of matter.
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
These laws of physics described nature very well under
most conditions. However, some experiments of the late
19th and early 20th century could not be explained.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
Postulates of Quantum
Mechanics
Next Presentation
Reference
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85. Introduction to Quantum
Computation
Quantum Mechanics
Ritajit Majumdar, Arunabha
Saha
By the late nineteenth century the laws of physics were
based on Mechanics and laws of Gravitation from Newton,
Maxwell’s equations describing Electricity and Magnetism
and on Statistical Mechanics describing the state of large
collection of matter.
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
These laws of physics described nature very well under
most conditions. However, some experiments of the late
19th and early 20th century could not be explained.
Postulates of Quantum
Mechanics
Next Presentation
Reference
The problems with classical physics led to the development
of Quantum Mechanics and Special Relativity.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
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86. Introduction to Quantum
Computation
Quantum Mechanics
Ritajit Majumdar, Arunabha
Saha
By the late nineteenth century the laws of physics were
based on Mechanics and laws of Gravitation from Newton,
Maxwell’s equations describing Electricity and Magnetism
and on Statistical Mechanics describing the state of large
collection of matter.
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
These laws of physics described nature very well under
most conditions. However, some experiments of the late
19th and early 20th century could not be explained.
Postulates of Quantum
Mechanics
Next Presentation
Reference
The problems with classical physics led to the development
of Quantum Mechanics and Special Relativity.
Some of the problems leading to the development of
Quantum Mechanics are
Black Body Radiation
Photoelectric Effect
Double Slit Experiment
Compton Scattering
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87. Double Slit Experiment: Setup
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
This experiment shows an aberrant result which cannot be
explained using classical laws of physics.
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
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88. Double Slit Experiment: Setup
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
This experiment shows an aberrant result which cannot be
explained using classical laws of physics.
The experiment setup consists of a monochromatic source of
light and two extremely small slits, big enough for only one
photon particle to pass through it.
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
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Introduction to Quantum Computation
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89. Double Slit Experiment: One Slit Open
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
If initially only one slit is open, we get a probability distribution
as shown in figure.
So if the two slits are opened individually, the two distinct
probability distributations are obtained in the screen.
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
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Introduction to Quantum Computation
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90. Double Slit Experiment: The Classical
Expectation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Since the opening of two slits individually are independent
events, classically we expect that if the two slits are opened
together, the two probability distributions should add up giving
the new probability distribution.
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
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Introduction to Quantum Computation
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91. Double Slit Experiment: The Anomaly
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
What we observe in reality when both slits are opened
together, is an interference pattern.
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
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Introduction to Quantum Computation
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92. Double Slit Experiment: Quantum Explanation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
There is no classical explanation to this observation.
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
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Introduction to Quantum Computation
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93. Double Slit Experiment: Quantum Explanation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
There is no classical explanation to this observation.
However, using quantum mechanics, it can be explained.
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
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94. Double Slit Experiment: Quantum Explanation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
There is no classical explanation to this observation.
However, using quantum mechanics, it can be explained.
It is the wave particle duality of light that is responsible
for such aberrant observation. The wave nature of light is
responsible for the interference pattern observed.
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
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Introduction to Quantum Computation
September 9, 2013
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95. Double Slit Experiment: Complete Picture
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Figure: Double Slit Experiment showing the anomaly - deviation of
the observed result from the one predicted by Classical Physics.
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Introduction to Quantum Computation
September 9, 2013
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96. Nobody understands Quantum Mechanics
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Quantum mechanics is a very counter-intuitive theory.
The results of quantum mechanics is nothing like what we
experience in everyday life. It is just that nature behaves
very strangely at the level of elementary particles. And
this strange way is described by quantum mechanics.
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
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Introduction to Quantum Computation
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97. Nobody understands Quantum Mechanics
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Quantum mechanics is a very counter-intuitive theory.
The results of quantum mechanics is nothing like what we
experience in everyday life. It is just that nature behaves
very strangely at the level of elementary particles. And
this strange way is described by quantum mechanics.
In fact, so strange is the theory that Richard Feynman
once said
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
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Introduction to Quantum Computation
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98. 1st Postulate: State Space
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
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.
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
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Introduction to Quantum Computation
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99. State Vector: Mathematical Realisation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Let us consider a quantum state |ψ = α |0 + β |1
Motivations for Quantum
Computation
Since a state is a unit vector, the norm of the vector must be
unity, or in mathematical notation,
Qubit
ψ|ψ = 1
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Hence, the condition that |ψ is a unit vector is equivalent to
Reference
|α|2 + |β|2 = 1
This condition is called the normalization condition of the state
vector.
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Introduction to Quantum Computation
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100. State: Why unit vector?
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Let |ψ = α0 |0 + α1 |1 + . . . + αk−1 |k − 1 be a quantum
state. As we shall see later, the superposition is not observable.
When a state is observed, it collapses into one of the basis
states.
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
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101. State: Why unit vector?
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Motivations for Quantum
Computation
The square of the amplitude |αi |2 gives the probability that the
system collapses to the state |i .
Qubit
Linear Algebra
Uncertainty Principle
Since the total probability is always 1, we must have:
Postulates of Quantum
Mechanics
Next Presentation
|αi |2 = 1
Reference
This condition is satisfied only when the state vector is a unit
vector.
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Introduction to Quantum Computation
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102. 2nd Postulate: Evolution
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Postulate 2
Outline
The evolution of a closed quantum system is described by a
unitary transformation.
That is, if |ψ1 is the state of the system at time t1 and |ψ2 at
time t2 , then:
|ψ2 = U(t1 , t2 ) |ψ1
where U(t1 , t2 ) is a unitary operator.
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
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Introduction to Quantum Computation
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103. 2nd Postulate: Evolution
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Postulate 2
Outline
The evolution of a closed quantum system is described by a
unitary transformation.
That is, if |ψ1 is the state of the system at time t1 and |ψ2 at
time t2 , then:
|ψ2 = U(t1 , t2 ) |ψ1
where U(t1 , t2 ) is a unitary operator.
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Figure: Quantum systems evolve by the rotation of the Hilbert
Space
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Introduction to Quantum Computation
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104. Time Evolution: Schrodinger Equation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
The time evolution of a closed quantum system is given by the
Schrodinger Equation:
Introduction
Motivations for Quantum
Computation
Qubit
i
d|ψ
dt
= H |ψ
Linear Algebra
Uncertainty Principle
where H is the hamiltonian and it is defined as the total energy
(kinetic + potential) of the system.
Postulates of Quantum
Mechanics
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
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105. Time Evolution: Schrodinger Equation
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
The time evolution of a closed quantum system is given by the
Schrodinger Equation:
Introduction
Motivations for Quantum
Computation
Qubit
i
d|ψ
dt
= H |ψ
Linear Algebra
Uncertainty Principle
where H is the hamiltonian and it is defined as the total energy
(kinetic + potential) of the system.
Postulates of Quantum
Mechanics
Next Presentation
Reference
The connection between the hamitonian picture and the
unitary operator picture is given by:
|ψ2 = exp −iH(t2 −t1 ) |ψ1 = U(t1 , t2 ) |ψ1
where we define, U(t1 , t2 ) ≡ exp −iH(t2 −t1 )
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
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106. Attempt at 3rd Postulate
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
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.
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
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Introduction to Quantum Computation
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107. Attempt at 3rd Postulate
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
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.
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
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 .
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
Postulates of Quantum
Mechanics
Next Presentation
Reference
September 9, 2013
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108. Attempt at 3rd Postulate
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
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.
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
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 .
Postulates of Quantum
Mechanics
Next Presentation
Reference
After measurement, the state of the system collapses to
either |0 or |1 with the said probability.
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Introduction to Quantum Computation
September 9, 2013
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109. Attempt at 3rd Postulate
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
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.
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
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 .
Postulates of Quantum
Mechanics
Next Presentation
Reference
After measurement, the state of the system collapses to
either |0 or |1 with the said probability.
However, after measurement if the new state of the
system is |0 (say), then further measurements in the
same basis gives outcome |0 with probability 1.
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Introduction to Quantum Computation
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110. 3rd Postulate: Measurement
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Postulate 3
Motivations for Quantum
Computation
Quantum measurements are described by a collection {Mm } of
measurement operators. The index m refers to the
measurement outcomes that may occur in the experiment. If
the state of the quantum system is |ψ before experiment, then
the probability that result m occurs is given by,
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
p(m) = ψ| Mm † .Mm |ψ
And the state of the system after measurement is
√
Ritajit Majumdar, Arunabha Saha (CU)
Mm |ψ
ψ|Mm †.Mm |ψ
Introduction to Quantum Computation
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111. So what is the Big Deal?
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
The postulate states that a quantum system stays in a
superposition when it is not observed. When a measurement is
done, it immediately collapses to one of its eigenstates. Hence
we can never observe what the original superposition of the
system was. We merely observe the state after it collapses.
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
This inherent ambiguity provides an excellent security in
Quantum Cryptography.
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
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112. Introduction to Quantum
Computation
Schrodinger’s Cat
Ritajit Majumdar, Arunabha
Saha
This is a thought experiment proposed by Erwin
Schrodinger.
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
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113. Introduction to Quantum
Computation
Schrodinger’s Cat
Ritajit Majumdar, Arunabha
Saha
This is a thought experiment proposed by Erwin
Schrodinger.
Place a cat in a steel chamber with a device containing a
vial of hydrocyanic acid and a radioactive substance. If
even a single atom of the substance decays, it will trip a
hammer and break the vial which in turn kills the cat.
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
63 / 70
114. Introduction to Quantum
Computation
Schrodinger’s Cat
Ritajit Majumdar, Arunabha
Saha
This is a thought experiment proposed by Erwin
Schrodinger.
Place a cat in a steel chamber with a device containing a
vial of hydrocyanic acid and a radioactive substance. If
even a single atom of the substance decays, it will trip a
hammer and break the vial which in turn kills the cat.
Without opening the box, an observer cannot know
whether the cat is alive or dead. So the cat may be said
to be in a superposition of the two states.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
September 9, 2013
63 / 70
115. Introduction to Quantum
Computation
Schrodinger’s Cat
Ritajit Majumdar, Arunabha
Saha
This is a thought experiment proposed by Erwin
Schrodinger.
Place a cat in a steel chamber with a device containing a
vial of hydrocyanic acid and a radioactive substance. If
even a single atom of the substance decays, it will trip a
hammer and break the vial which in turn kills the cat.
Without opening the box, an observer cannot know
whether the cat is alive or dead. So the cat may be said
to be in a superposition of the two states.
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
However, when the box is opened, we observe
deterministically that the cat is either dead or alive. We
can, by no means, observe the superposition.
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
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116. 4th Postulate: Composite System
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
What we have seen so far was a single qubit system. What
happens when there are multiple qubits? This is given by the
last postulate:
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Postulate 4
Next Presentation
The state space of a composite physical system is the tensor
product of the state spaces of the component physical systems.
Moreover, if we have n systems, and the system number i is
prepared in state |ψi , then the joint state of the total system is
Reference
|ψ1 ⊗ |ψ2 ⊗ ... ⊗ |ψn
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Introduction to Quantum Computation
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117. Introduction to Quantum
Computation
Two Qubit System
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Let us consider a two qubit system. Classically, with two bits,
we can have 4 states - 00, 01, 10, 11. A quantum system is a
linear superposition of all these four states.
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
So, a general two qubit quantum state can be represented as
|ψ = α00 |00 + α01 |01 + α10 |10 + α11 |11
10
Postulates of Quantum
Mechanics
Next Presentation
Reference
where,
|α00 |2 + |α01 |2 + |α10 |2 + |α11 |2 = 1
10 |0
⊗ |0 ≡ |0 |0 ≡ |00
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
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118. Measurement in Two Qubit System
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Motivations for Quantum
Computation
Measurement is similar to single qubit system. When we
measure the two qubit system we get outcome j with
probability |αj |2 and the new state will be |j .
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
That is for the system
Next Presentation
Reference
|ψ = α00 |00 + α01 |01 + α10 |10 + α11 |11 ,
we get outcome |00 with probability |α00 |2 and the new state
of the system will be |00 .
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
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119. Introduction to Quantum
Computation
Partial Measurement
Ritajit Majumdar, Arunabha
Saha
Outline
So what if we want to measure only the first qubit? Or maybe
only the second one?
Introduction
Motivations for Quantum
Computation
Qubit
We take the same two qubit system,
|ψ = α00 |00 + α01 |01 + α10 |10 + α11 |11 ,
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
If only the first qubit is measured then we get the outcome |0
for the first qubit with probability |α00 |2 + |α01 |2
Reference
and the state collapses to
√
|φ = α00 |00
+α01 |01
|α00 |2 +|α01 |2
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Introduction to Quantum Computation
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120. Coming up in next talk...
Introduction to Quantum
Computation
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Motivations for Quantum
Computation
Einstein-Polosky-Rosen (EPR) Paradox
Qubit
Bell State
Linear Algebra
Uncertainty Principle
Quantum Entanglement
Density Matrix Notation of Quantum Mechanics
Quantum Gates
Postulates of Quantum
Mechanics
Next Presentation
Reference
and many more...
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
September 9, 2013
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121. Introduction to Quantum
Computation
Reference
Ritajit Majumdar, Arunabha
Saha
Michael A. Nielsen, Isaac Chuang
Quantum Computation and Quantum Information
Cambridge University Press
David J. Griffiths
Introduction to Quantum Mechanics
Prentice Hall, 2nd Edition
Umesh Vazirani, University of California Berkeley
Quantum Mechanics and Quantum Computation
https://class.coursera.org/qcomp-2012-001/
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
Postulates of Quantum
Mechanics
Next Presentation
Reference
Michael A. Nielsen, University of Queensland
Quantum Computing for the determined
http://michaelnielsen.org/blog/
quantum-computing-for-the-determined/
James Branson, University of California San Diego
Quantum Physics (UCSD Physics 130)
http://quantummechanics.ucsd.edu/ph130a/130_
notes/130_notes.html
Ritajit Majumdar, Arunabha Saha (CU)
Introduction to Quantum Computation
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122. Introduction to Quantum
Computation
Reference
Ritajit Majumdar, Arunabha
Saha
Outline
Introduction
Motivations for Quantum
Computation
Qubit
Linear Algebra
Uncertainty Principle
N. David Mermin, Cornell University
Lecture Notes on Quantum Computation
Postulates of Quantum
Mechanics
Next Presentation
http://www.wikipedia.org
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Introduction to Quantum Computation
Reference
September 9, 2013
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