Difference between revisions of "MAT4XXX"

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=Mathematical Physics - MAT4XXX/5XXX=
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=Introduction to Quantum Information Science and Engineering - MAT4XXX/5XXX=
  
 
==Course description==
 
==Course description==
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The course intends to be a basic introduction to the mathematical
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This course will be an introduction accessible and welcoming to all STEM students. No prior quantum mechanics courses are expected since all the principles and techniques of quantum information will be taught during the course. The focus will be on qubits, entanglement, and decoherence, three key building blocks of quantum computing.  
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and computational techniques in applied mathematics, computational science & engineer�ing, and data science & machine learning. This course will stress then how the methods of
 
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mathematical modeling in the STEM disciplines have transitioned from the analytical (as in
 
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Theoretical Physics) to the numerical (as in traditional methods in Computational Science
 
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and Engineering) and more recently to Data-based methods (as in current developments in
 
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Data Science and Machine Learning). The student will acquire the basic skills needed broadly
 
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in Computational Science and Engineering, of which Computational Physics, Data Science,
 
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Machine Learning, and Numerical Modeling in the Mathematical Sciences are a subset.
 
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==Catalog entry==
 
==Catalog entry==
  
 
''Prerequisite'':  
 
''Prerequisite'':  
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Calculus III [[MAT2214]] and Differential Equations I [[MAT3613]] with a letter grade of C- or better, or successful completion of at least three credits of equivalent courses.
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Linear Algebra [[MAT2233]] or Applied Linear Algebra [[MAT2253]], or equivalent (can be waived with approval of instructor), with a letter grade of C- or better, or successful completion of at least three credits of equivalent courses.  
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''Content'':  
 
''Content'':  
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1. Computational Science, Engineering, and Mathematics
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Foundations of quantum mechanics such as unitary time evolution, entanglement, and the EPR paradox approached from the information perspective, and quantum entropy. Information and its encoding into physical systems such as photons, atoms, and superconducting circuits.
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(a) Linear Algebra and Computational Science & Engineering
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Quantum control using quantum logic gates, providing a foundation for quantum programming.
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(b) Applied Math and Computational Science & Engineering
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Applications: quantum teleportation, quantum cryptography, quantum computing.
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(c) Fourier Series and Integrals
 
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(d) Laplace Transform and Spectral Methods
 
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(e) Initial Value Problems
 
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(f) Conjugate Gradients and Krylov Subspaces
 
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(g) Minimum Principles
 
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2. Data Science and Machine Learning: a Mathematical Perspective
 
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(a) Principal Components and the Best Low Rank Matrix
 
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(b) Randomized Linear Algebra
 
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(c) Low Rank and Compressed Sensing
 
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(d) Markov Chains
 
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(e) Stochastic Gradient Descent and ADAM
 
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(f) Introduction to Machine Learning: Neural Networks
 
  
  
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'''Textbooks:'''
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'''Textbook:'''
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* Strang, G. Computational Science & Engineering. USA, Wellesley-Cambridge, 2007.
 
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* Strang, G. Linear Algebra and Learning from Data. Wellesley-Cambridge Press, 2019.
 
  
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* Nielsen, M. and Chuang, I. Quantum Computation and Quantum Information. UK, Cambridge University Press, 2012.
  
  
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Strang's 4 special matrices
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An overview of quantum computing and information.
 
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Differences, Derivatives, BC. Gradient, Divergence. Laplace equation.
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Classical Information Theory. Connection between information and thermodynamics
 
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Inverses. Positive Definite Matrices
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Communications Theory. Physical qubits: spinning particles and photons
 
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Stiffness Matrices. Oscillations & Newton's Laws.
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Operators in Quantum Mechanics. Classical cryptography
 
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Graph Models. Networks. Clustering and k-means.
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Quantum cryptography. Entanglement.
 
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Fourier Series. Chebyshev, Legendre, and Bessel
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Mixed states and the density operator. Local measurements & open quantum systems.
 
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Fast Fourier Transform (FFT). Convolution and Signal Processing.
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Quantum non-locality and the Einstein-Podolsky-Rosen paradox
 
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Fourier Integrals. Deconvolution, Integral Equations. Wavelets, Signal Processing.
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Bell’s inequality. Quantum dense coding. Quantum teleportation
 
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Computational implementation of Laplace and z- Transforms. Spectral Methods.
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Quantum non-locality and the Einstein-Podolsky-Rosen paradox
 
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Finite Difference for ODEs. Accuracy & Stability. Conservation Laws, diffusion, fluids
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Quantum computation.
 
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Elimination with reordering, multigrid methods, conjugate gradients, Krylov subspaces
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Von Neumann measurements
 
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Regular. least sq. Linear programming. Adjoint. Stoch. Gradient Descent. ADAM.
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Many-worlds interpretation of quantum mechanics.
 
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Matrix-matrix Multiplication. 4 Fundamental Subspaces. Orthogonal Matrices. Best low rank matrix. Rayleigh quotients. Factoring matrices and tensors.
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Selected topic 1.
 
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Randomized Linear Algebra. Low rank signals. Singular values. Compressed sensing. Covariance Matrices. Multivariate Gaussian. Weighted least squares. Markov chains. Neural Networks. Backpropagation. Machine Learning.
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Presentations by students
 
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Latest revision as of 15:50, 24 January 2025

Introduction to Quantum Information Science and Engineering - MAT4XXX/5XXX

Course description

This course will be an introduction accessible and welcoming to all STEM students. No prior quantum mechanics courses are expected since all the principles and techniques of quantum information will be taught during the course. The focus will be on qubits, entanglement, and decoherence, three key building blocks of quantum computing.

Catalog entry

Prerequisite: Linear Algebra MAT2233 or Applied Linear Algebra MAT2253, or equivalent (can be waived with approval of instructor), with a letter grade of C- or better, or successful completion of at least three credits of equivalent courses.


Content: Foundations of quantum mechanics such as unitary time evolution, entanglement, and the EPR paradox approached from the information perspective, and quantum entropy. Information and its encoding into physical systems such as photons, atoms, and superconducting circuits. Quantum control using quantum logic gates, providing a foundation for quantum programming. Applications: quantum teleportation, quantum cryptography, quantum computing.


Textbook:

  • Nielsen, M. and Chuang, I. Quantum Computation and Quantum Information. UK, Cambridge University Press, 2012.


Topics List

Date Sections Topics Prerequisite Skills Student Learning Outcomes
Week 1

An overview of quantum computing and information.

Week 2

Classical Information Theory. Connection between information and thermodynamics

Week 3

Communications Theory. Physical qubits: spinning particles and photons

Week 4

Operators in Quantum Mechanics. Classical cryptography

Week 5

Quantum cryptography. Entanglement.

Week 6

Mixed states and the density operator. Local measurements & open quantum systems.

Week 7

Quantum non-locality and the Einstein-Podolsky-Rosen paradox

Week 8

Bell’s inequality. Quantum dense coding. Quantum teleportation

Week 9

Quantum non-locality and the Einstein-Podolsky-Rosen paradox

Week 10

Quantum computation.

Week 11

Von Neumann measurements

Week 12

Many-worlds interpretation of quantum mechanics.

Week 13

Selected topic 1.

Week 14

Presentations by students

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