This document provides an overview of concepts from linear algebra that are necessary for understanding quantum mechanics. It reviews vectors, vector spaces, linear independence, bases, linear operators, and complex numbers. It then introduces key concepts for quantum mechanics, including Dirac notation, inner products, outer products, eigenvalues and eigenvectors, unitary and Hermitian operators, and tensor products. The goal is to cover the necessary mathematical foundations and notations systematically to enable the study of quantum mechanics postulates.
This document discusses the history and future of quantum computing. It explains how quantum computers work using principles of quantum mechanics like superposition and entanglement. Quantum computers can perform multiple computations simultaneously by exploiting the ability of qubits to exist in superposition. Current research involves building larger quantum registers with more qubits and performing calculations with 2 qubits. The future of quantum computing may enable solving certain problems much faster than classical computers, with desktop quantum computers potentially arriving within 10 years.
1. Planning involves finding a sequence of actions that achieves a goal starting from an initial state. It uses a set of operators that define the possible actions and their effects.
2. A plan is a sequence of operator instances that transforms the initial state into a goal state. Classical planning assumes fully observable, deterministic environments.
3. Planning problems can be represented using a logical language that describes states, goals, actions and their preconditions and effects. This representation allows planning algorithms to operate over problems.
Quantum computing uses quantum mechanics phenomena like superposition and entanglement to perform calculations exponentially faster than classical computers for certain problems. While quantum computers have shown promise in areas like optimization, simulation, and encryption cracking, significant challenges remain in scaling up quantum bits and reducing noise and errors. Current research aims to build larger quantum registers of 50+ qubits to demonstrate quantum advantage and explore practical applications, with the future potential to revolutionize fields like artificial intelligence, materials design, and drug discovery if full-scale quantum computers can be realized.
1) Quantum entanglement is a property where quantum states of objects cannot be described independently, even if separated spatially. A practical example involves two cups of hot chocolate where tasting one instantly reveals the other's state.
2) Bra-ket notation is used to describe quantum states as vectors or functionals in a Hilbert space. Operators act on these states to model physical quantities.
3) A qubit is the quantum analogue of a classical bit, existing in superposition of states |0> and |1>. Quantum computers use entanglement between qubits to perform computations in parallel.
Quantum computing uses quantum bits (qubits) that can exist in superpositions of states rather than just 1s and 0s. This allows quantum computers to perform exponentially more calculations in parallel than classical computers. Some of the main challenges to building quantum computers are preventing qubit decoherence from environmental interference, developing effective error correction methods, and observing outputs without corrupting data. Quantum computers may one day be able to break current encryption methods and solve optimization problems much faster than classical computers.
Quantum computing is a new method of computing based on quantum mechanics that offers greater computational power than classical computers. Quantum computers use quantum bits or qubits that can exist in superpositions of states allowing massive parallelism. Several approaches like ion traps, quantum dots and NMR have demonstrated quantum computing. However, challenges remain around errors from decoherence and a lack of reliable reading mechanisms. If these obstacles can be overcome, quantum computers may solve problems in artificial intelligence, cybersecurity, drug design and more exponentially faster than classical computers.
Quantum computing and quantum communications utilize principles of quantum mechanics such as superposition and entanglement to process and transmit information in novel ways. Current research is exploring how to build reliable quantum computers and networks using technologies like ion traps, quantum dots, and optical methods. While still in early stages, quantum information science shows promise for solving computationally difficult problems in fields such as artificial intelligence, cybersecurity, and drug discovery. Pioneering work by groups like D-Wave, IBM, and China are helping advance our understanding of how to harness quantum effects for powerful new computing and communication applications.
This document discusses the history and future of quantum computing. It explains how quantum computers work using principles of quantum mechanics like superposition and entanglement. Quantum computers can perform multiple computations simultaneously by exploiting the ability of qubits to exist in superposition. Current research involves building larger quantum registers with more qubits and performing calculations with 2 qubits. The future of quantum computing may enable solving certain problems much faster than classical computers, with desktop quantum computers potentially arriving within 10 years.
1. Planning involves finding a sequence of actions that achieves a goal starting from an initial state. It uses a set of operators that define the possible actions and their effects.
2. A plan is a sequence of operator instances that transforms the initial state into a goal state. Classical planning assumes fully observable, deterministic environments.
3. Planning problems can be represented using a logical language that describes states, goals, actions and their preconditions and effects. This representation allows planning algorithms to operate over problems.
Quantum computing uses quantum mechanics phenomena like superposition and entanglement to perform calculations exponentially faster than classical computers for certain problems. While quantum computers have shown promise in areas like optimization, simulation, and encryption cracking, significant challenges remain in scaling up quantum bits and reducing noise and errors. Current research aims to build larger quantum registers of 50+ qubits to demonstrate quantum advantage and explore practical applications, with the future potential to revolutionize fields like artificial intelligence, materials design, and drug discovery if full-scale quantum computers can be realized.
1) Quantum entanglement is a property where quantum states of objects cannot be described independently, even if separated spatially. A practical example involves two cups of hot chocolate where tasting one instantly reveals the other's state.
2) Bra-ket notation is used to describe quantum states as vectors or functionals in a Hilbert space. Operators act on these states to model physical quantities.
3) A qubit is the quantum analogue of a classical bit, existing in superposition of states |0> and |1>. Quantum computers use entanglement between qubits to perform computations in parallel.
Quantum computing uses quantum bits (qubits) that can exist in superpositions of states rather than just 1s and 0s. This allows quantum computers to perform exponentially more calculations in parallel than classical computers. Some of the main challenges to building quantum computers are preventing qubit decoherence from environmental interference, developing effective error correction methods, and observing outputs without corrupting data. Quantum computers may one day be able to break current encryption methods and solve optimization problems much faster than classical computers.
Quantum computing is a new method of computing based on quantum mechanics that offers greater computational power than classical computers. Quantum computers use quantum bits or qubits that can exist in superpositions of states allowing massive parallelism. Several approaches like ion traps, quantum dots and NMR have demonstrated quantum computing. However, challenges remain around errors from decoherence and a lack of reliable reading mechanisms. If these obstacles can be overcome, quantum computers may solve problems in artificial intelligence, cybersecurity, drug design and more exponentially faster than classical computers.
Quantum computing and quantum communications utilize principles of quantum mechanics such as superposition and entanglement to process and transmit information in novel ways. Current research is exploring how to build reliable quantum computers and networks using technologies like ion traps, quantum dots, and optical methods. While still in early stages, quantum information science shows promise for solving computationally difficult problems in fields such as artificial intelligence, cybersecurity, and drug discovery. Pioneering work by groups like D-Wave, IBM, and China are helping advance our understanding of how to harness quantum effects for powerful new computing and communication applications.
Grover's algorithm is a quantum algorithm that allows finding an item in an unstructured database with fewer queries than classical algorithms. It works by amplifying the probability of measuring the target item. The key steps are: (1) initialize a superposition of all possible inputs, (2) apply an oracle to mark the target item, (3) apply Grover's diffusion operator to amplify the target amplitude, and (4) repeat steps 2-3 until the target can be measured with high probability. An example is using Grover's algorithm to find a phone number in an unsorted database with only sqrt(N) queries, compared to N/2 queries classically.
Quantum computers use principles of quantum mechanics rather than classical binary logic. They have qubits that can represent superpositions of 0 and 1, allowing massive parallelism. Key effects like superposition, entanglement, and tunneling give them advantages over classical computers for problems like factoring and searching. Early quantum computers have been built with up to a few hundred qubits, and algorithms like Shor's show promise for cryptography applications. However, challenges remain around error correction and controlling quantum states as quantum computers scale up. D-Wave has produced commercial quantum annealing systems with over 1000 qubits, but debate continues on whether these demonstrate quantum advantage. Overall, quantum computing could transform fields like AI, simulation, and optimization if challenges around building reliable large-scale quantum
A quantum computer performs calculations using quantum mechanics and quantum properties like superposition and entanglement. It uses quantum bits (qubits) that can exist in superpositions of states unlike classical computer bits. A quantum computer could solve some problems, like factoring large numbers, much faster than classical computers. The document discusses the history of computing generations and quantum computing, how quantum computers work using qubits, superpositions and entanglement, and potential applications like encryption cracking and simulation.
This document discusses density operators and their use in quantum information and computing. It begins by introducing density operators and how they can be used to describe quantum systems whose state is not precisely known or composite systems. The key properties of density operators are that they must have a trace of 1 and be positive operators. The document then covers reduced density operators which describe subsystems by taking the partial trace. Finally, it discusses how the reduced density operator gives the correct measurement statistics for observations on a subsystem.
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.
Fundamentals of quantum computing part i revPRADOSH K. ROY
This document provides an introduction to the fundamentals of quantum computing. It discusses computational complexity classes such as P and NP and essential matrix algebra concepts like Hermitian, unitary, and normal matrices. It also contrasts the classical and quantum worlds. In the quantum world, quantum systems can exist in superposition states and qubits can represent more than just binary 0s and 1s. The document introduces the concept of a qubit register and how multiple qubits can be represented using tensor products. It discusses characteristics of quantum systems like superposition, Born's rule for probabilities, and the measurement postulate which causes wavefunction collapse.
Multi Layer Perceptron & Back PropagationSung-ju Kim
This document discusses multi-layer perceptrons (MLPs), including their advantages over single-layer perceptrons. MLPs can classify problems that single-layer perceptrons cannot by using multiple hidden layers between the input and output layers. MLPs are trained using an error-based learning method called backpropagation, which calculates errors between the target and actual output values and adjusts weights in the network accordingly starting from the output layer and propagating backwards. MLPs are well-suited for parallel processing architectures.
This presentation provides a basic introduction to quantum computers architecture including basic concepts related to the theory, quantum vs classical mechanics, qubits, quantum gates and some related algorithms.
This document discusses quantum error correction. It explains that while quantum states and operators are theoretically perfect, in reality approximations must be made which can cause errors. Quantum error correction deals with these imperfections. It describes different types of quantum errors and discusses barriers to quantum error correction, such as the no-cloning theorem. The document introduces classical error correction techniques and explains how similar techniques can be applied to encode quantum states to correct bit flip and phase flip errors by measuring the parity of qubits without collapsing their superpositions. Specific quantum error correcting codes are presented, including Shor's code which can correct both types of errors.
Matrices are used extensively in computer applications related to graphics and image processing. Matrices represent images as a collection of coordinate points, and changing the values in the matrix allows images to be transformed through operations like scaling, rotation, and distortion. Matrices are also used to encrypt and decrypt codes and messages. Overall, matrices play a vital role in computer applications by enabling graphical representations and transformations that would otherwise be very complicated to achieve.
This document discusses kernel methods and radial basis function (RBF) networks. It begins with an introduction and overview of Cover's theory of separability of patterns. It then revisits the XOR problem and shows how it can be solved using Gaussian hidden functions. The interpolation problem is explained and how RBF networks can perform strict interpolation through a set of training data points. Radial basis functions that satisfy Micchelli's theorem allowing for a nonsingular interpolation matrix are presented. Finally, the structure and training of RBF networks using k-means clustering and recursive least squares estimation is covered.
The document discusses flip-flops, which are basic electronic circuits that have two stable states and can serve as one bit of digital memory. It defines what a flip-flop is and describes several common types of flip-flops, including SR, JK, T, D, and master-slave edge-triggered flip-flops. The document provides brief explanations of how each flip-flop type works and is implemented using logic gates.
Artificial Intelligence: Introduction, Typical Applications. State Space Search: Depth Bounded
DFS, Depth First Iterative Deepening. Heuristic Search: Heuristic Functions, Best First Search,
Hill Climbing, Variable Neighborhood Descent, Beam Search, Tabu Search. Optimal Search: A
*
algorithm, Iterative Deepening A*
, Recursive Best First Search, Pruning the CLOSED and OPEN
Lists
This document provides an overview of quantum computing, including:
- The current state of quantum computing technology, which involves noisy intermediate-scale quantum computers with 10s to 100s of qubits and moderate error rates.
- The difference between quantum and classical information, noting that quantum information uses superposition and entanglement, exponentially increasing computational power.
- An example quantum algorithm, Bernstein-Vazirani, which can solve a problem in one query that classical computers require n queries to solve, demonstrating quantum computing's potential computational advantages.
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
Convolutional neural networks (CNNs) learn multi-level features and perform classification jointly and better than traditional approaches for image classification and segmentation problems. CNNs have four main components: convolution, nonlinearity, pooling, and fully connected layers. Convolution extracts features from the input image using filters. Nonlinearity introduces nonlinearity. Pooling reduces dimensionality while retaining important information. The fully connected layer uses high-level features for classification. CNNs are trained end-to-end using backpropagation to minimize output errors by updating weights.
Group members for the project are Falah Hassan, Maidah Malik, and Maria Khan. The document discusses half adders and full adders. A half adder adds two binary digits and produces a sum and carry output. It is built from two logic gates. A full adder accepts two input bits and a carry input, and produces a sum and carry output. It is implemented using two half adders joined by an OR gate. The main difference between a half adder and full adder is that a full adder has three inputs and two outputs, allowing multiple adders to be chained to add more bits.
This document discusses several numerical analysis methods for finding roots of equations or solving systems of equations. It describes the bisection method for finding roots of continuous functions, the method of false positions for approximating roots between two values with opposite signs of a function, Gauss elimination for transforming a system of equations into triangular form, Gauss-Jordan method which further eliminates variables in equations below, and iterative methods which find solutions through successive approximations rather than direct computation.
This document summarizes quantum computing. It begins with an introduction explaining the differences between classical and quantum bits, with qubits being able to exist in superpositions of states. The history of quantum computing is discussed, including early explorations in the 1970s-80s and Peter Shor's breakthrough in 1994. D-Wave Systems is mentioned as the first company to develop a quantum computer in 2011. The scope, architecture, working principles, advantages and applications of quantum computing are then outlined at a high level. The document concludes by discussing the growing field of quantum computing research and applications.
Data science-retreat-how it works plus advice for upcoming data scientistsJose Quesada
The document describes a data science retreat program aimed at helping junior data scientists transition to senior roles. It discusses the challenges companies face in finding qualified data scientists and proposes that the retreat, which involves portfolio projects, mentoring, and pair programming, can help address this skills gap. Companies can sponsor candidates in the program, receiving discounts on their initial salaries if hired. The retreat director advocates this approach as a way for companies to develop strong relationships with candidates and assess their skills directly.
Grover's algorithm is a quantum algorithm that allows finding an item in an unstructured database with fewer queries than classical algorithms. It works by amplifying the probability of measuring the target item. The key steps are: (1) initialize a superposition of all possible inputs, (2) apply an oracle to mark the target item, (3) apply Grover's diffusion operator to amplify the target amplitude, and (4) repeat steps 2-3 until the target can be measured with high probability. An example is using Grover's algorithm to find a phone number in an unsorted database with only sqrt(N) queries, compared to N/2 queries classically.
Quantum computers use principles of quantum mechanics rather than classical binary logic. They have qubits that can represent superpositions of 0 and 1, allowing massive parallelism. Key effects like superposition, entanglement, and tunneling give them advantages over classical computers for problems like factoring and searching. Early quantum computers have been built with up to a few hundred qubits, and algorithms like Shor's show promise for cryptography applications. However, challenges remain around error correction and controlling quantum states as quantum computers scale up. D-Wave has produced commercial quantum annealing systems with over 1000 qubits, but debate continues on whether these demonstrate quantum advantage. Overall, quantum computing could transform fields like AI, simulation, and optimization if challenges around building reliable large-scale quantum
A quantum computer performs calculations using quantum mechanics and quantum properties like superposition and entanglement. It uses quantum bits (qubits) that can exist in superpositions of states unlike classical computer bits. A quantum computer could solve some problems, like factoring large numbers, much faster than classical computers. The document discusses the history of computing generations and quantum computing, how quantum computers work using qubits, superpositions and entanglement, and potential applications like encryption cracking and simulation.
This document discusses density operators and their use in quantum information and computing. It begins by introducing density operators and how they can be used to describe quantum systems whose state is not precisely known or composite systems. The key properties of density operators are that they must have a trace of 1 and be positive operators. The document then covers reduced density operators which describe subsystems by taking the partial trace. Finally, it discusses how the reduced density operator gives the correct measurement statistics for observations on a subsystem.
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.
Fundamentals of quantum computing part i revPRADOSH K. ROY
This document provides an introduction to the fundamentals of quantum computing. It discusses computational complexity classes such as P and NP and essential matrix algebra concepts like Hermitian, unitary, and normal matrices. It also contrasts the classical and quantum worlds. In the quantum world, quantum systems can exist in superposition states and qubits can represent more than just binary 0s and 1s. The document introduces the concept of a qubit register and how multiple qubits can be represented using tensor products. It discusses characteristics of quantum systems like superposition, Born's rule for probabilities, and the measurement postulate which causes wavefunction collapse.
Multi Layer Perceptron & Back PropagationSung-ju Kim
This document discusses multi-layer perceptrons (MLPs), including their advantages over single-layer perceptrons. MLPs can classify problems that single-layer perceptrons cannot by using multiple hidden layers between the input and output layers. MLPs are trained using an error-based learning method called backpropagation, which calculates errors between the target and actual output values and adjusts weights in the network accordingly starting from the output layer and propagating backwards. MLPs are well-suited for parallel processing architectures.
This presentation provides a basic introduction to quantum computers architecture including basic concepts related to the theory, quantum vs classical mechanics, qubits, quantum gates and some related algorithms.
This document discusses quantum error correction. It explains that while quantum states and operators are theoretically perfect, in reality approximations must be made which can cause errors. Quantum error correction deals with these imperfections. It describes different types of quantum errors and discusses barriers to quantum error correction, such as the no-cloning theorem. The document introduces classical error correction techniques and explains how similar techniques can be applied to encode quantum states to correct bit flip and phase flip errors by measuring the parity of qubits without collapsing their superpositions. Specific quantum error correcting codes are presented, including Shor's code which can correct both types of errors.
Matrices are used extensively in computer applications related to graphics and image processing. Matrices represent images as a collection of coordinate points, and changing the values in the matrix allows images to be transformed through operations like scaling, rotation, and distortion. Matrices are also used to encrypt and decrypt codes and messages. Overall, matrices play a vital role in computer applications by enabling graphical representations and transformations that would otherwise be very complicated to achieve.
This document discusses kernel methods and radial basis function (RBF) networks. It begins with an introduction and overview of Cover's theory of separability of patterns. It then revisits the XOR problem and shows how it can be solved using Gaussian hidden functions. The interpolation problem is explained and how RBF networks can perform strict interpolation through a set of training data points. Radial basis functions that satisfy Micchelli's theorem allowing for a nonsingular interpolation matrix are presented. Finally, the structure and training of RBF networks using k-means clustering and recursive least squares estimation is covered.
The document discusses flip-flops, which are basic electronic circuits that have two stable states and can serve as one bit of digital memory. It defines what a flip-flop is and describes several common types of flip-flops, including SR, JK, T, D, and master-slave edge-triggered flip-flops. The document provides brief explanations of how each flip-flop type works and is implemented using logic gates.
Artificial Intelligence: Introduction, Typical Applications. State Space Search: Depth Bounded
DFS, Depth First Iterative Deepening. Heuristic Search: Heuristic Functions, Best First Search,
Hill Climbing, Variable Neighborhood Descent, Beam Search, Tabu Search. Optimal Search: A
*
algorithm, Iterative Deepening A*
, Recursive Best First Search, Pruning the CLOSED and OPEN
Lists
This document provides an overview of quantum computing, including:
- The current state of quantum computing technology, which involves noisy intermediate-scale quantum computers with 10s to 100s of qubits and moderate error rates.
- The difference between quantum and classical information, noting that quantum information uses superposition and entanglement, exponentially increasing computational power.
- An example quantum algorithm, Bernstein-Vazirani, which can solve a problem in one query that classical computers require n queries to solve, demonstrating quantum computing's potential computational advantages.
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
Convolutional neural networks (CNNs) learn multi-level features and perform classification jointly and better than traditional approaches for image classification and segmentation problems. CNNs have four main components: convolution, nonlinearity, pooling, and fully connected layers. Convolution extracts features from the input image using filters. Nonlinearity introduces nonlinearity. Pooling reduces dimensionality while retaining important information. The fully connected layer uses high-level features for classification. CNNs are trained end-to-end using backpropagation to minimize output errors by updating weights.
Group members for the project are Falah Hassan, Maidah Malik, and Maria Khan. The document discusses half adders and full adders. A half adder adds two binary digits and produces a sum and carry output. It is built from two logic gates. A full adder accepts two input bits and a carry input, and produces a sum and carry output. It is implemented using two half adders joined by an OR gate. The main difference between a half adder and full adder is that a full adder has three inputs and two outputs, allowing multiple adders to be chained to add more bits.
This document discusses several numerical analysis methods for finding roots of equations or solving systems of equations. It describes the bisection method for finding roots of continuous functions, the method of false positions for approximating roots between two values with opposite signs of a function, Gauss elimination for transforming a system of equations into triangular form, Gauss-Jordan method which further eliminates variables in equations below, and iterative methods which find solutions through successive approximations rather than direct computation.
This document summarizes quantum computing. It begins with an introduction explaining the differences between classical and quantum bits, with qubits being able to exist in superpositions of states. The history of quantum computing is discussed, including early explorations in the 1970s-80s and Peter Shor's breakthrough in 1994. D-Wave Systems is mentioned as the first company to develop a quantum computer in 2011. The scope, architecture, working principles, advantages and applications of quantum computing are then outlined at a high level. The document concludes by discussing the growing field of quantum computing research and applications.
Data science-retreat-how it works plus advice for upcoming data scientistsJose Quesada
The document describes a data science retreat program aimed at helping junior data scientists transition to senior roles. It discusses the challenges companies face in finding qualified data scientists and proposes that the retreat, which involves portfolio projects, mentoring, and pair programming, can help address this skills gap. Companies can sponsor candidates in the program, receiving discounts on their initial salaries if hired. The retreat director advocates this approach as a way for companies to develop strong relationships with candidates and assess their skills directly.
1. The document introduces Dirac notation to represent quantum states as kets (vectors) in a Hilbert space H. Kets can be added and multiplied by scalars, obeying certain rules.
2. An inner product assigns a complex number to pairs of kets, satisfying properties like linearity and producing a positive real number when a ket is taken with itself.
3. Linear functionals (bras) map kets to complex numbers in a linear way. Bras form the dual space H† of H. Operators map elements of H to other elements of H in a linear way.
This document introduces Dirac notation, which provides a concise way to represent states in a linear space without specifying a coordinate system. It summarizes:
1) Dirac notation uses "kets" to represent vectors and "bras" to represent their complex conjugates. Combining a bra and ket represents an inner product.
2) The outer product of two vectors represents a linear operator. In Dirac notation, a ket-bra combination represents an outer product, while a bra-ket combination represents an inner product.
3) Dirac notation allows representations of states, operators, and transformations between bases to be written independently of any particular coordinate system.
The document discusses systems theory and provides definitions and principles about systems. It defines a system as a collection of components bound more strongly to each other than their environment. Systems can exist because of stable components and binding forces. Complex systems can exhibit emergent behaviors from simple local rules operating at a large scale. All complex adaptive systems use some form of computation, and the theory of evolution describes how selective pressure favors replication of better adapted systems in large ecosystems of variable systems.
This lecture discusses operator methods in quantum mechanics. Some key points:
1. Operators allow quantum mechanics to be formulated without relying on a particular basis. The Hamiltonian operator H acts on state vectors.
2. Dirac notation represents state vectors as "kets" and defines inner products between states. A resolution of identity allows expanding states in a basis.
3. Hermitian operators correspond to physical observables. Their eigenfunctions form a complete basis. The time-evolution operator evolves states forward in time.
4. The uncertainty principle relates the uncertainties of non-commuting operators like position and momentum. Symmetries of the Hamiltonian are represented by unitary operators that commute with it.
“No hidden variables!”: From Neumann’s to Kochen and Specker’s theorem in qua...Vasil Penchev
The talk addresses a philosophical comparison and thus interpretation of both theorems having one and the same subject:
The absence of the “other half” of variables, called “hidden” for that, to the analogical set of variables in classical mechanics:
These theorems are:
John’s von Neumann’s (1932)
Simon Kochen and Ernst Specker’s (1968)
This document discusses key concepts in quantum mechanics including wave functions, operators, linear vector spaces, inner products, orthogonal and orthonormal bases, Hilbert spaces, and the expansion theorem. It defines wave functions and operators as the two main constructs in quantum mechanics. It also explains that the natural language of quantum mechanics is linear algebra and describes concepts like linear vector spaces, inner products, orthogonal and orthonormal bases, and Hilbert spaces in the context of quantum mechanics.
Concepts and Problems in Quantum Mechanics, Lecture-II By Manmohan DashManmohan Dash
9 problems (part-I and II) and in depth the ideas of Quantum; such as Schrodinger Equation, Philosophy of Quantum reality and Statistical interpretation, Probability Distribution, Basic Operators, Uncertainty Principle.
This document provides a summary of key developments in the foundations of quantum mechanics. It discusses Planck's discovery that led to defining Planck's constant h, which established that energy is quantized. Einstein's work on the photoelectric effect supported this and introduced the photon concept. Bohr used classical mechanics and energy quantization to develop his model of the hydrogen atom. The document outlines the revolutionary changes brought by quantum theory and its greater scope and applicability compared to classical physics. It provides context for understanding quantum mechanics from first principles.
This document discusses the construction of a parking garage and park structure. It provides details on the columns, slabs, beams and other structural elements that were post-tensioned concrete. Photos show various stages of construction including the lift shaft, pedestrian bridges and spandrel walls. The post-tensioned design allowed for increased strength with less deflection and more rapid construction.
Lecture 8: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
- Dimensionality reduction techniques assign instances to vectors in a lower-dimensional space while approximately preserving similarity relationships. Principal component analysis (PCA) is a common linear dimensionality reduction technique.
- Kernel PCA performs PCA in a higher-dimensional feature space implicitly defined by a kernel function. This allows PCA to find nonlinear structure in data. Kernel PCA computes the principal components by finding the eigenvectors of the normalized kernel matrix.
- For a new data point, its representation in the lower-dimensional space is given by projecting it onto the principal components in feature space using the kernel trick, without explicitly computing features.
- The document provides an introduction to linear algebra and MATLAB. It discusses various linear algebra concepts like vectors, matrices, tensors, and operations on them.
- It then covers key MATLAB topics - basic data types, vector and matrix operations, control flow, plotting, and writing efficient code.
- The document emphasizes how linear algebra and MATLAB are closely related and commonly used together in applications like image and signal processing.
A very very brief introduction to vectors, matrices, and their properties. I used to use this presentation to help students with no linear algebra background so they can catch up with materials taught in my complex systems courses.
The document provides an overview of basic math concepts for computer graphics, including:
- Sets, mappings, and Cartesian coordinates are introduced to represent vectors and points in 2D and 3D space.
- Linear interpolation is described as a fundamental operation in graphics used to connect data points.
- Parametric and implicit equations are discussed for representing common 2D curves and lines.
- Concepts like the dot product, cross product, and gradient are covered, which are important for calculations involving vectors.
Seismic data processing introductory lectureAmin khalil
This document provides a syllabus for a course on seismic data processing. The syllabus outlines topics that will be covered, including the mathematical foundations of Fourier transforms, sampling considerations for seismic time series, basic processing sequences, velocity analysis, filtering and migration techniques, acquisition of seismic data both on land and at sea, 3D seismic data processing, and other advanced topics such as Radon transforms and AVO analysis. References for the course include books on seismic data processing and digital signal processing. The document explains that seismic data processing is important to remove unwanted signals and noise and enhance signal-to-noise ratios, as reflection seismic signals may be obscured by other seismic arrivals like ground roll and direct waves.
This document provides an overview of fundamentals of linear algebra and vector calculus concepts that are important for machine learning. It begins with an introduction to Euclidean spaces and vectors, then covers key linear algebra topics like matrices, matrix operations, and solving systems of linear equations. It also discusses vectors norms, eigenvalues/eigenvectors, and the Cholesky decomposition. Finally, it introduces concepts in vector calculus like derivatives, gradients, Hessians, and their application to backpropagation in neural networks. The document is intended to provide machine learning practitioners and students with the essential mathematical foundations.
Aplicaciones y subespacios y subespacios vectoriales en laemojose107
se enfoca en la enseñanza del Álgebra Lineal en carreras de ingeniería. Los conceptos vinculados a esta rama de las matemáticas se estudian en los cursos básicos de los primeros años de los planes de estudio en esas carreras. Se estudian conceptos tales como vectores, matrices, sistemas de ecuaciones lineales, espacios vectoriales, transformaciones lineales, valores y. vectores propios, y diagonalización de matrices.
MATLAB is an interactive development environment and programming language used by engineers and scientists for technical computing, data analysis, and algorithm development. It allows users to access data from files, web services, applications, hardware, and databases, and perform data analysis and visualization. MATLAB can be used for applications in areas like control systems, signal processing, communications, and more.
This document provides an overview of linear models and matrix algebra concepts that are important for economics. It discusses the objectives of using mathematics for economics, including understanding problems by stating the unknown and known variables. The document then covers key topics in linear algebra like the history of matrices, what matrices are, basic matrix operations, and properties of matrix addition and multiplication. It also introduces concepts like the inverse and transpose of a matrix. Finally, it provides an example of how matrices and vectors can represent systems of linear equations used in economic models.
PCA is a dimensionality reduction technique that uses linear transformations to project high-dimensional data onto a lower-dimensional space while retaining as much information as possible. It works by identifying patterns in data and expressing the data in such a way as to highlight their similarities and differences. Specifically, PCA uses linear combinations of the original variables to extract the most important patterns from the data in the form of principal components. The first principal component accounts for as much of the variability in the data as possible, and each succeeding component accounts for as much of the remaining variability as possible.
This document provides an overview of MATLAB, including the MATLAB desktop, variables, vectors, matrices, matrix operations, array operations, built-in functions, data visualization, flow control using if and for statements, and user-defined functions. It introduces key MATLAB concepts like the command window, workspace, and editor. It also demonstrates how to create and manipulate variables, vectors, matrices, and plots in MATLAB.
Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions.
Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linear map that best approximates the function near that point.
1. The document provides an introduction to linear algebra concepts including matrix arithmetic, properties, eigenvectors and eigenvalues.
2. Key concepts are explained such as adding and multiplying matrices, and how matrices can transform vectors through scaling, rotation, and other transformations.
3. Special matrices like diagonal, identity, and normal matrices are discussed. Finding eigenvectors and eigenvalues is described as a way to understand how matrices transform vectors.
Support Vector Machines aim to find an optimal decision boundary that maximizes the margin between different classes of data points. This is achieved by formulating the problem as a constrained optimization problem that seeks to minimize training error while maximizing the margin. The dual formulation results in a quadratic programming problem that can be solved using algorithms like sequential minimal optimization. Kernels allow the data to be implicitly mapped to a higher dimensional feature space, enabling non-linear decision boundaries to be learned. This "kernel trick" avoids explicitly computing coordinates in the higher dimensional space.
Here are the steps to plot the given functions using MATLAB:
1. Plot y = 0.4x + 1.8 for 0 ≤ x ≤ 35 and 0 ≤ y ≤ 3.5:
x = 0:35;
y = 0.4.*x + 1.8;
plot(x,y)
xlim([0 35])
ylim([0 3.5])
2. Plot imaginary vs real parts of 0.2 + 0.8i*n for 0 ≤ n ≤ 20:
n = 0:20;
z = 0.2 + 0.8i*n;
plot(real(z),imag(z))
xlabel('Real Part')
The document discusses vector spaces and their properties. It defines a vector space as a collection of vectors that can be added and scaled by real numbers, while satisfying certain properties like closure and distributivity. Examples of vector spaces include Rn, the space of matrices, and function spaces. A subspace is a subset of a vector space that is also a vector space. The column space of a matrix contains all linear combinations of its columns and is an important subspace.
This document introduces the topic of convex optimization. It discusses mathematical optimization problems and notes that certain classes of problems, such as least-squares problems, linear programs, and convex optimization problems, can be solved efficiently and reliably. The document then provides examples of convex optimization problems and discusses how they can be solved. It outlines the goals and topics that will be covered in the course, including convex sets and functions, algorithms, and applications.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Convex Optimization Modelling with CVXOPTandrewmart11
An introduction to convex optimization modelling using cvxopt in an IPython environment. The facility location problem is used as an example to demonstrate modelling in cvxopt.
➽=ALL False flag-War Machine-War profiteering-Energy (oil/Gas) Iraq, Iran,…oil and gas
USA invades other countries just to own their natural resources and to place them in the hands of American corporations. Facebook doesn’t call that terrorism. They call it democracy. BBC, CNN, FOX NEWS, FR 24, ITV/CH 4, SKY, EURO NEWS, ITV trash Sun paper,… Facebook all are protector and preserver of the propaganda classifying IR Iran as a dangerous terrorist organization. But FB, BBC, CNN, FOX NEWS, FR 24, ITV/CH 4-SKY, EURO NEWS, ITV do know well, that USA is the biggest terrorist country in the world.
‘terrorism’ the unlawful use of violence and intimidation, especially against civilians, in the pursuit of political aims.
"the fight against terrorism" is the fight against the unlawful use of violence and intimidation and carpet bombing.
Ever since the beginning of the 19th century, the West has been sucking on the jugular vein of the Moslem body politic like a veritable vampire whose thirst for Moslem blood is never sated and who refused to let go. Since 1979, Iran, which has always played the role of the intellectual leader of the Islamic world, has risen up to put a stop to this outrage against God’s law and will, and against all decency.
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PART 1 (IN TOTAL 12 PARTS)
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NEWS YOU WON’T FIND ON BBC-CNN-FOX NEWS, FRNACE 24, EURO NEWS
Articles for Political Science, Mathematics and Productivity the Student Room BSc, MSc & PhD Project Mangers etc
PPTs in SLIDESHARE International Studies Research Degrees (MPhilPhD)
The document discusses construction productivity in the UK and other countries. It provides factors that impact productivity such as site/project management, resource management, labor characteristics, and motivation. Productivity in the UK construction industry has improved over the past decade but still lags countries like Germany and France. Reasons given for relatively lower UK productivity include issues with subcontracting, materials handling, training, and technology adoption. Improving areas like planning, prefabrication, and reducing waste could further increase construction productivity.
Redefining brain tumor segmentation: a cutting-edge convolutional neural netw...IJECEIAES
Medical image analysis has witnessed significant advancements with deep learning techniques. In the domain of brain tumor segmentation, the ability to
precisely delineate tumor boundaries from magnetic resonance imaging (MRI)
scans holds profound implications for diagnosis. This study presents an ensemble convolutional neural network (CNN) with transfer learning, integrating
the state-of-the-art Deeplabv3+ architecture with the ResNet18 backbone. The
model is rigorously trained and evaluated, exhibiting remarkable performance
metrics, including an impressive global accuracy of 99.286%, a high-class accuracy of 82.191%, a mean intersection over union (IoU) of 79.900%, a weighted
IoU of 98.620%, and a Boundary F1 (BF) score of 83.303%. Notably, a detailed comparative analysis with existing methods showcases the superiority of
our proposed model. These findings underscore the model’s competence in precise brain tumor localization, underscoring its potential to revolutionize medical
image analysis and enhance healthcare outcomes. This research paves the way
for future exploration and optimization of advanced CNN models in medical
imaging, emphasizing addressing false positives and resource efficiency.
artificial intelligence and data science contents.pptxGauravCar
What is artificial intelligence? Artificial intelligence is the ability of a computer or computer-controlled robot to perform tasks that are commonly associated with the intellectual processes characteristic of humans, such as the ability to reason.
› ...
Artificial intelligence (AI) | Definitio
Batteries -Introduction – Types of Batteries – discharging and charging of battery - characteristics of battery –battery rating- various tests on battery- – Primary battery: silver button cell- Secondary battery :Ni-Cd battery-modern battery: lithium ion battery-maintenance of batteries-choices of batteries for electric vehicle applications.
Fuel Cells: Introduction- importance and classification of fuel cells - description, principle, components, applications of fuel cells: H2-O2 fuel cell, alkaline fuel cell, molten carbonate fuel cell and direct methanol fuel cells.
Comparative analysis between traditional aquaponics and reconstructed aquapon...bijceesjournal
The aquaponic system of planting is a method that does not require soil usage. It is a method that only needs water, fish, lava rocks (a substitute for soil), and plants. Aquaponic systems are sustainable and environmentally friendly. Its use not only helps to plant in small spaces but also helps reduce artificial chemical use and minimizes excess water use, as aquaponics consumes 90% less water than soil-based gardening. The study applied a descriptive and experimental design to assess and compare conventional and reconstructed aquaponic methods for reproducing tomatoes. The researchers created an observation checklist to determine the significant factors of the study. The study aims to determine the significant difference between traditional aquaponics and reconstructed aquaponics systems propagating tomatoes in terms of height, weight, girth, and number of fruits. The reconstructed aquaponics system’s higher growth yield results in a much more nourished crop than the traditional aquaponics system. It is superior in its number of fruits, height, weight, and girth measurement. Moreover, the reconstructed aquaponics system is proven to eliminate all the hindrances present in the traditional aquaponics system, which are overcrowding of fish, algae growth, pest problems, contaminated water, and dead fish.
2. Goals:Goals:
• Review circuit fundamentals
• Learn more formalisms and different notations.
• Cover necessary math more systematically
• Show all formal rules and equations
3. Introduction to Quantum MechanicsIntroduction to Quantum Mechanics
• This can be found inThis can be found in MarinescuMarinescu and inand in Chuang andChuang and
NielsenNielsen
• Objective
– To introduce all of the fundamental principles of Quantum
mechanics
• Quantum mechanics
– The most realistic known description of the world
– The basis for quantum computing and quantum information
• Why Linear Algebra?
– LA is the prerequisite for understanding Quantum Mechanics
• What is Linear Algebra?
– … is the study of vector spaces… and of
– linear operations on those vector spaces
4. Linear algebra -Linear algebra -Lecture objectivesLecture objectives
• Review basic concepts from Linear Algebra:
– Complex numbers
– Vector Spaces and Vector Subspaces
– Linear Independence and Bases Vectors
– Linear Operators
– Pauli matrices
– Inner (dot) product, outer product, tensor product
– Eigenvalues, eigenvectors, Singular Value Decomposition (SVD)
• Describe the standard notations (the Dirac notations)
adopted for these concepts in the study of Quantum
mechanics
• … which, in the next lecture, will allow us to study the
main topic of the Chapter: the postulates of quantum
mechanics
5. Review: Complex numbersReview: Complex numbers
• A complex number is of the form
where and i2
=-1
• Polar representation
• With the modulus or magnitude
• And the phase
• Complex conjugateconjugate
nnn
ibaz +=
R,ba nn
∈Czn
∈
where, R,θueuz nn
i
nn
n
∈= θ
22
nn
n bau +=
=
n
n
n
a
barctanθ
( )nnnn
iuz θθ sincos += ( ) nnnnn
ibaibaz −=+=
∗∗
6. Review: The Complex Number SystemReview: The Complex Number System
• Another definitions and Notations:
• It is the extension of the real number system via closure
under exponentiation.
• (Complex) conjugate:
c* = (a + bi)* ≡ (a − bi)
• Magnitude or absolute value:
|c|2
= c*c = a2
+b2
1-≡i )( RC ∈∈ a,b,c
bc
ac
bac
≡
≡
+=
][
][
Im
Re
i
“Real” axis
+
+i
−
−i
“Imaginary”
axis
The “imaginary”
unit
a
b c
22*
))(( bababaccc +=+−=≡ ii
7. Review: ComplexReview: Complex
ExponentiationExponentiation
• Powers of i are complex
units:
• Note:
eπi/2
= i
eπi
= −1
e3π i/2
= − i
e2π i
= e0
= 1
θθ sincos iiθ
+≡e
+1
+i
−1
−i
eθi
θ
Z1=2 eZ1=2 e πi
Z1Z122
= (2 e= (2 e πi
)2
= 2 2
(eeπi
)2
= 4= 4 (ee πi
)2
==
4 e4 e 22πi
44
22
8. Recall:Recall: What is a qubit?What is a qubit?
• A bit has two possible states
• Unlike bits, a qubit can be in a state other than
• We can form linear combinations of states
• A qubit state is a unit vector in a two-dimensional
complex vector space
0 or 1
0 or 1
0 1ψ α β= +
9. Properties of QubitsProperties of Qubits
• Qubits are computational basis states
- orthonormal basis
- we cannot examine a qubit to determine its quantum state
- A measurement yields
0 for
1 for
ij ij
i j
i j
i j
δ δ
≠
= =
=
2
0 with probability α
2
1 with probability β
2 2
where 1α β+ =
10. (Abstract) Vector Spaces(Abstract) Vector Spaces
• A concept from linear algebra.
• A vector space, in the abstract, is any set of objects that can be
combined like vectors, i.e.:
– you can add them
• addition is associative & commutative
• identity law holds for addition to zero vector 0
– you can multiply them by scalars (incl. −1)
• associative, commutative, and distributive laws hold
• Note: There is no inherent basis (set of axes)
– the vectors themselves are the fundamental objects
– rather than being just lists of coordinates
11. VectorsVectors
• Characteristics:
– Modulus (or magnitude)
– Orientation
• Matrix representation of a vector
[ ] vector)(row,,
dualitsandcolumn),(a
1
1
∗∗
==
=
n
n
zz
z
z
vv
v
τ
This is adjoint, transpose andThis is adjoint, transpose and
next conjugatenext conjugate
Operations
on vectors
12. Vector Space, definition:Vector Space, definition:
• A vector space (of dimension n) is a set of n vectors
satisfying the following axioms (rules):
– Addition: add any two vectors and pertaining to a
vector space, say Cn
, obtain a vector,
the sum, with the properties :
• Commutative:
• Associative:
• Any has a zero vector (called the origin):
• To every in Cn
corresponds a unique vector - v such as
– Scalar multiplication: next slide
v '
v
+
+
=+
'
'
11
'
nn zz
zz
vv
vvvv +=+ ''
( ) ( )'''''' vvvvvv ++=++
v0v =+v
v
( ) 0vv =−+
Operations
on vectors
13. Vector Space (cont)Vector Space (cont)
ScalarScalar multiplication:multiplication: foranyscalarforanyscalar
Multiplication by scalars is Associative:Multiplication by scalars is Associative:
distributive with respect to vectoraddition:distributive with respect to vectoraddition:
Multiplication by vectors isMultiplication by vectors is
distributive with respect to scalaraddition:distributive with respect to scalaraddition:
A VectorsubspaceA Vectorsubspace in anin an n-dimensional vectorn-dimensional vector spacespace isis
a non-empty subset of vectors satisfying the same axiomsa non-empty subset of vectors satisfying the same axioms
hatsuch way tinproduct,scalarthe,
vectoraistherevvectorand
1
=
∈∈
n
n
zz
zz
z
CCz
v
( ) ( ) vv '' zzzz =
vv =1
( ) '' vvvv zzz +=+
( ) vvv '' zzzz +=+
in such way thatin such way that
Operations
on vectors
17. Spanning Set and Basis vectorsSpanning Set and Basis vectors
OrOr SPANNING SET forCSPANNING SET forCnn
: any set of: any set of nn vectors such thatvectors such that any
vector in the vectorspacein the vectorspace CCnn
can be written using thecan be written using the nn basebase
vectorsvectors
Example forC2
(n=2):
which is a linearcombination of the 2-dimensional basis
vectors and 0 1
Spanning setSpanning set
is a set of allis a set of all
such vectorssuch vectors
for any alphafor any alpha
and betaand beta
18. Bases and LinearBases and Linear
IndependenceIndependence
Always exists!Always exists!
in the spacein the space
Red and blueRed and blue
vectors add to 0,vectors add to 0,
are not linearlyare not linearly
independentindependent
LinearlyLinearly
independentindependent
vectorsvectors
21. So far we talked only about vectors and operations onSo far we talked only about vectors and operations on
them. Now we introduce matricesthem. Now we introduce matrices
Linear OperatorsLinear Operators
A is linearA is linear
operatoroperator
22. Hilbert spacesHilbert spaces
• A Hilbert space is a vector space in which the
scalars are complex numbers, with an inner
product (dot product) operation • : H×H → C
– Definition of inner product:
x•y = (y•x)* (* = complex conjugate)
x•x ≥ 0
x•x = 0 if and only if x = 0
x•y is linear, under scalar multiplication
and vector addition within both x and y
x
y
x•y/|x|
yxyx ≡•
Another notation often used:
“Component”
picture:
“bracket”
Black dot is anBlack dot is an
inner productinner product
23. Vector Representation of StatesVector Representation of States
• Let S={s0, s1, …} be a maximal set of
distinguishable states, indexed by i.
• The basis vector vi identified with the ith
such state
can be represented as a list of numbers:
s0 s1 s2 si-1 si si+1
vi = (0, 0, 0, …, 0, 1, 0, … )
• Arbitrary vectors v in the Hilbert space can then be
defined by linear combinations of the vi:
),,( 10 ccc
i
ii == ∑ vv
∑=
i
iyxi
*
yx
24. Dirac’s Ket NotationDirac’s Ket Notation
• Note: The inner product
definition is the same as the
matrix product of x, as a
conjugated row vector, times
y, as a normal column vector.
• This leads to the definition, for state s, of:
– The “bra” 〈s| means the row matrix [c0* c1* …]
– The “ket” |s〉 means the column matrix →
• The adjoint operator †
takes any matrix M
to its conjugate transpose M†
≡ MT
*, so
† †
∑=
i
iyxi
*
yx
[ ]
=
2
1
*
2
*
1 y
y
xx
“Bracket”
2
1
c
c
You have to be familiarYou have to be familiar
with these three notationswith these three notations
26. Properties: Unitary
and Hermitian
( ) kkk ∀= ,Iσσ
τ
( ) kk σσ =
τ
Pauli Matrices =Pauli Matrices = examplesexamples
X is like inverterX is like inverter
This is adjointThis is adjoint
29. This is new, we
did not use inner products yet
Inner Products of vectorsInner Products of vectors
We already talked
about this when we
defined Hilbert spacespace
Be able to prove these properties from definitions
Complex numbersComplex numbers
30. Slightly other formalism for InnerSlightly other formalism for Inner
ProductsProducts
Be familiar withBe familiar with
various formalismsvarious formalisms
33. Outer Products ofOuter Products of
vectorsvectors
This is KroneckerThis is Kronecker
operationoperation
34. We will illustrate how this
can be used formally to
create unitary and other
matrices
Outer Products of vectorsOuter Products of vectors
|u> <v||u> <v| is an outeris an outer
productproduct of |u> andof |u> and
|v>|v>
|u> is from U, ||u> is from U, |
v> is from V.v> is from V.
|u><v||u><v| is a mapis a map
VV UU
35. Eigenvalues of
matrices are used in
analysis and synthesis
Eigenvectors of linear operatorsEigenvectors of linear operators andand
their Eigenvaluestheir Eigenvalues
36. Eigenvalues andEigenvalues and EigenvectorsEigenvectors
versus diagonalizable matricesversus diagonalizable matrices
Eigenvector ofEigenvector of
Operator AOperator A
37. Diagonal Representations of matricesDiagonal Representations of matrices
Diagonal matrixDiagonal matrix
From last slideFrom last slide
41. Unitary and Positive Operators:Unitary and Positive Operators:
some propertiessome properties and a new notationand a new notation
Other notation for adjointOther notation for adjoint
(Dagger is also used(Dagger is also used
Positive operatorPositive operator
Positive definitePositive definite
operatoroperator
42. Hermitian Operators: someHermitian Operators: some
propertiesproperties in different notationin different notation
These are important and
useful properties of our
matrices of circuits
43. Tensor ProductsTensor Products of Vectorof Vector
SpacesSpaces
Note variousNote various
notationsnotations
Notation for vectors inNotation for vectors in
space Vspace V
45. Tensor Products of vectors andTensor Products of vectors and
Tensor Products of OperatorsTensor Products of Operators
Properties of tensor
products for vectors
Tensor productTensor product
for operatorsfor operators
46. Properties of Tensor ProductsProperties of Tensor Products
of vectorsof vectors and operatorsand operators
We repeat them inWe repeat them in
different notationdifferent notation
herehere
These can be vectors of any sizeThese can be vectors of any size
48. For Normal OperatorsFor Normal Operators
there is also Spectralthere is also Spectral
DecompositionDecomposition
If A is represented like this Then f(A) can be represented like
this
49. Trace of a matrix andTrace of a matrix and aa
Commutator of matricesCommutator of matrices
50. Quantum NotationQuantum Notation
(Sometimes denoted by bold fonts)(Sometimes denoted by bold fonts)
(Sometimes called Kronecker(Sometimes called Kronecker
multiplication)multiplication)
Review to rememberReview to remember
51. Exam ProblemsExam Problems
Review systematically from basicReview systematically from basic
Dirac elementsDirac elements
.|a|a〉〉|a|a〉〉
|a|a〉〉|a|a〉〉 xx
|a|a〉〉〈〈a|a| xx
vectorvector
numbernumber
matrixmatrix
numbernumber
|a|a〉〉 〈〈a|a|xx
The most important new idea that we introduced inThe most important new idea that we introduced in
this lecture is inner products, outer products,this lecture is inner products, outer products,
eigenvectors and eigenvalues.eigenvectors and eigenvalues.
52. Exam ProblemsExam Problems
• Diagonalization of unitary matrices
• Inner and outer products
• Use of complex numbers in quantum theory
• Visualization of complex numbers and Bloch Sphere.
• Definition and Properties of Hilbert Space.
• Tensor Products of vectors and operators – properties and proofs.
• Dirac Notation – all operations and formalisms
• Functions of operators
• Trace of a matrix
• Commutator of a matrix
• Postulates of Quantum Mechanics.
• Diagonalization
• Adjoint, hermitian and normal operators
• Eigenvalues and Eigenvectors
53. Bibliography & acknowledgementsBibliography & acknowledgements
• Michael Nielsen and Isaac Chuang, Quantum
Computation and Quantum Information, Cambridge
University Press, Cambridge, UK, 2002
• R. Mann,M.Mosca, Introduction to Quantum
Computation, Lecture series, Univ. Waterloo, 2000
http://cacr.math.uwaterloo.ca/~mmosca/quantumcourse
• Paul Halmos, Finite-Dimensional Vector Spaces,
Springer Verlag, New York, 1974
54. • Covered in 2003, 2004, 2005, 2007
• All this material is illustrated with examples in
next lectures.