Previous QUB teaching
Previous QUB teaching
Embedded Files
Please contact me if there is content behind a broken link that you would like to access.
MTH 4023 (Mathematical Methods for Quantum Information Processing)
MTH 4023 (Mathematical Methods for Quantum Information Processing)
General Information:
A level 4/master's course to develop theory and mathematical foundations needed for quantum information, quantum communication, and quantum computing.
Welcome slides (handwritten) and Welcome slides (blank, with clickable links). Updated 13/01.
Suggested Routine:
Tuesday/Friday. Read ahead a few pages in (last year's) notes, posted online.
Monday/Wednesday. Review* the previous lecture.
Thursday. Attend the tutorial, work on your homework.
*what does it mean to "Review a lecture"? You probably won't have understood or internalised everything that happened in a 2 hour lecture -- hence it is important to invest time after class to reviewing and understanding the days' material. This booklet from Loughborough University shows you how to train yourself to "self-explain", to check if you understand mathematics lectures. Usually this involves rereading notes, redoing examples without the answer before you, trying homework problems, and explaining ideas to your classmates.
Remember: learning = doing. In order to get your brain to learn something, you generally have to make it do something active.
These notes have been adapted from lecture material of Gabriele De Chiara, Steve Campbell, Alessandro Ferraro, Mauro Paternostro, and Hannah McAleese, with thanks. Several sections are also drawn from Nielsen & Chuang, Quantum Computation and Quantum Information (2010), and Jozsa, Quantum Information and Computation: Lecture Notes, University of Cambridge (2021).
Lectures 1 & 2. Introduction and Overview of Quantum Mechanics (optional handout). Typed notes only. Updated 13/01.
Lectures 3 & 4. Linear Algebra in Dirac Notation -- Introduction. Typed notes only. Updated 16/01.
Lectures 5 & 6. Linear Algebra in Dirac Notation -- Outer products and Hermiticity. Handwritten notes and Typed notes. Updated 20/01.
Lectures 7 & 8. Linear Algebra in Dirac Notation -- Unitary operators, tensors, and trace. Handwritten notes and Typed notes. Updated 23/01.
Lectures 9 & 10. End of "Linear Algebra in Dirac Notation", Start of "Postulates of Quantum Mechanics". Handwritten notes and Typed notes. Updated 27/01.
Lectures 11 & 12. Postulates of Quantum Mechanics: Measurement. Handwritten notes and Typed notes. Updated 30/01.
Lectures 13 & 14. Postulates of Quantum Mechanics: More on Measurement. Handwritten notes and Typed notes. Updated Typed notes 12/02 -- correction to final exercise.
Lectures 15 & 16. Quantum Communication: No-cloning & Teleportation. Handwritten notes and Typed notes. Updated 7/02. Includes S2.5: Generalised Uncertainty Principle.
Lectures 17 & 18. Quantum Communication: Superdense coding & BB84. Handwritten notes and Typed notes. Updated 10/02.
Lectures 19 & 20. Quantum Communication: Bell's inequality & E91. Handwritten notes (calculation supplement) and Typed notes. Updated 13/02.
Lectures 21 & 22. Quantum Mechanics with Density Operators: Postulates, and the Bloch Ball. Handwritten notes and Typed notes. Updated 17/02.
Lectures 23 & 24. Quantum Mechanics with Density Operators: Bloch Ball, and Partial Trace. Handwritten notes and Typed notes. Updated 20/02.
Lectures 25 & 26. End of "QM with Density Operators", Start of "Quantum Circuits and Algorithms". Handwritten notes and Typed notes. Updated 24/02.
Lectures 27 & 28. Quantum Circuits and Algorithms: Multiqubit gates. Handwritten notes and Typed notes. Updated 02/03.
Lectures 29 & 30. Quantum Circuits and Algorithms: Classical circuit simulations, oracles, and parallelism. Handwritten notes and Typed notes. Updated 03/03.
Lectures 31 & 32. Quantum Circuits and Algorithms: Deutsch-Jozsa's & Grover's algorithms. Handwritten notes and Typed notes. Updated 06/03.
Lectures 33 & 34. Quantum Maps: Principal-Environment systems, and Kraus Operators. Handwritten notes and Typed notes. Updated 10/03.
Lectures 35 & 36. Quantum Maps: Three Examples, and Error-Correction. Handwritten notes and Typed notes. Updated 13/03.
Lectures 37 & 38. Pure & Mixed State Entanglement: Measures of Entanglement, and Dilution. Handwritten notes and Typed notes. Updated 20/03 -- extra material in box on p. 86.
Lectures 39 & 40. Pure & Mixed State Entanglement: Diluation, Distillation, and PPT. Handwritten notes and Typed notes. Updated 14/04 -- extra material in boxes on pp. 87-90.
In the revision class on 30/04, we covered some questions from the 2022 paper, and a variant on a question from Homework 10. Handwritten notes. Best of luck in your exam! Updated 1/05.
The course has now finished; your exam will be in May, in the “main exam period”. Thank you for your attention! Right: Professor Bob Coecke delivering a guest lecture on Quantum Picturalism.
John Stewart Bell (1928-1990) was a Belfast born, Queen's educated physicist known for making fundamental contributions to quantum physics. His most well-known result, Bell's Theorem, demonstrates how quantum mechanics can empirically be shown as incompatible with local hidden-variable theories; the idea that quantum physics is not probabilistic in nature, but rather is determined by "hidden variables" that we cannot access. ("Local" refers to the idea that an object is only influenced by its immediate surroundings.) Bell's Theorem, proven in 1964, shows that a local hidden-variable theory cannot produce the measurement statistics that quantum mechanics (specifically, entanglement) predicts; in 2022, the Nobel Prize in Physics was awarded to Alain Aspect, John Clauser, and Anton Zeilinger for their experimental validation of Bell's theorem.
Bell, shown right in CERN in 1982, first entered QUB as a technician in the Physics Department after finishing school. Bell was able to enter the department as a student in 1945, and obtained bachelor's degrees in experimental physics (1948) and mathematical physics (1949). Bell died unexpectedly in Geneva in 1990; unknown to him, he had reportedly been nominated for a Nobel Prize that year.
MTH 2011 (Linear Algebra)
MTH 2011 (Linear Algebra)
General Information:
A level 2 course in linear algebra to develop theory and methods needed in subsequent modules.
Welcome slides (Monday) and Tutorial Information slides (Thursday).
Suggested Routine:
Before Monday's lecture. Have a read over the planned lecture notes (on Canvas).
After Monday's lecture. Review* the lecture. See can you use the notes to answer questions you have.Tuesday/Wednesday. Have a go at the homework. Attend your tutorial, work with your classmates, and ask questions. Read over the planned lecture notes (on Canvas) for Thursday.
After Thursday's lecture. Review* the lecture. Submit your homework. Read over the planned lecture notes (on Canvas) for Friday.
After Friday's lecture. Review* the lecture, try to square away what you've learned this week. Have a good weekend!
*what does it mean to "Review a lecture"? Well, you probably won't have understood everything that happened in the 50 minute period. It is very normal to feel confused and a little lost when material is first presented, which is why it is so important to invest time after class to reviewing and understanding the days' material. This booklet from Loughborough University shows you how to train yourself to "self-explain", to check if you understand mathematics lectures. Usually this involves rereading notes, redoing examples without the answer before you, trying homework problems, and explaining ideas to your classmates.
Remember: learning = doing. In order to get your brain to learn something, you generally have to make it do something active. Conversely, if you do something, you are learning! Mathematics courses are designed to keep us moving through material quickly -- even if it feels like you are just treading water, that is making progress.
Our lecture notes are heavily based on the lecture notes provided by Dr David Barnes (with thanks).
Lecture 1: Linear Equations and Gaussian Elimination. Handwritten notes (Dave) and Typed notes. Updated 18/09.
Lecture 2: Backsubstitution and Matrices. Handwritten notes and Typed notes. Updated 18/09.
Lecture 3: Matrix Equations. Handwritten notes and Typed notes. Updated 19/09.
Lecture 4: Elementary Operations. Handwritten notes and Typed notes. Updated 22/09.
Lecture 5: Calculating the inverse. Handwritten notes and Typed notes. Updated 25/09.
Lecture 6: LU factorisation. Handwritten notes and Typed notes. Updated 26/09.
Lecture 7: Determinants. Handwritten notes and Typed notes. Updated 29/09.
Lecture 8: Further Properties of Determinants. Handwritten notes (correction made in purple, in proof of Proposition 8.4) and Typed notes. Updated 02/10.
Lecture 9: Fields and Vector Spaces. Handwritten notes and Typed notes. Updated 3/10.
Lecture 10: Subspaces. Handwritten notes and Typed notes. Updated 6/10.
Lecture 11: Spanning sets and linear independence. Handwritten notes and Typed notes. Updated 9/10.
Lecture 12: Bases and dimension. Handwritten notes and Typed notes. Updated 10/10. (This lecture was delivered online -- the recording can be found on Canvas, under "Announcements".)
Lecture 13: Linear complements. Handwritten notes and Typed notes. Updated 13/10.
Lecture 14: Rank of a matrix. Handwritten notes and Typed notes. Updated 16/10.
Lecture 15: The Extended Kronecker-Capelli Theorem. Handwritten notes and Typed notes. Updated 17/10.
Lecture 16: Linear maps. Handwritten notes and Typed notes. Updated 20/10.
Lecture 17: Properties of Linear maps. Handwritten notes and Typed notes. Updated 24/10.
Lecture 18: Linear isomorphisms. Handwritten notes and Typed notes. Updated 24/10.
Lecture 19: Kernel, image, and rank of a linear map. Handwritten notes and Typed notes. Updated 27/10.
Lecture 20: On the connection between matrices and linear maps. Handwritten notes and Typed notes. Updated 30/10.
Lecture 21: Matrix multiplication and composition of linear maps. Handwritten notes and Typed notes. Updated 31/10.
Lecture 22: Eigenvalues and eigenvectors. Handwritten notes and Typed notes. Updated 3/11.
Lecture 23: Examples of Eigenvalues and Eigenvectors. Handwritten notes and Typed notes. Updated 6/11.
Lecture 24: Diagonalisable linear maps. Handwritten notes and Typed notes. Updated 7/11. See information about formulas for solving polynomial equations below, as well!
Lecture 25: Inner product spaces. Handwritten notes and Typed notes. Updated 10/11.
Lecture 26: Norms, and Cauchy-Schwarz. Handwritten notes and Typed notes. Updated 13/11.
Lecture 27: Linear Isometries. Handwritten notes and Typed notes. Updated 14/11.
Lecture 28: Orthonormal sets. Handwritten notes and Typed notes. Updated 17/11.
Lecture 29: Gram-Schmidt Orthogonalisation. Handwritten notes and Typed notes. Updated 20/11.
Lecture 30: Adjoints. Handwritten notes and Typed notes. Updated 21/11.
Lecture 31: Adjoints and matrices. Handwritten notes and Typed notes. Updated 24/11.
Lecture 32: Orthonormal Diagonalisation (Complex case). Handwritten notes and Typed notes. Updated 27/11.
Lecture 33: Orthonormal Diagonalisation (Real case). Handwritten notes and Typed notes. Updated 28/11.
Lecture 34: Orthogonal complements. Handwritten notes and Typed notes. Updated 1/12.
The course has now finished; your exam will be in April/May, in the “main exam period”. Thank you for your attention!
Non-Examinable Lectures.
Lecture 35: Jordan Form (Overview). Handwritten notes and Typed notes. Updated 4/12.
Lecture 36: Jordan Form (Continued). Handwritten notes and Typed notes. Updated 5/12.
"Gaussian Elimination" is an algorithm for solving systems of linear equations, by applying "elementary row operations" to a matrix to reduce it to a simpler form (row-echelon form -- incidentally, the word echelon comes from the French word échelon, which means the "rung" or step of a ladder. Can you guess why the name applies?).
According to Wikipedia, "The algorithm that is taught in high school was named for Gauss only in the 1950s as a result of confusion over the history of the subject" -- Carl Friedrich Gauss in 1810 developed a notation for row reduction that was subsequently widely adopted, but he was not the inventor of the method! This method appears (without proof) as a calculation tool in a Chinese mathematical text approximately 2000 years ago (see the above article from the American Mathematical Society for more information). In Europe, the method was introduced by Isaac Newton in the early 1700s, as he found there was not a systematic or complete account of solving simultaneous linear equations available in the algebra textbooks of the time. According to the AMS article, the method is taught today in universities via matrices in a presentation described by the English mathematician Alan Turing, the "father" of computer science.
Eigenvalues and Eigenvectors
Eigenvalues and Eigenvectors
A linear map can be thought of as a transformation of vectors that does some combination of rotating, stretching, and shearing. To the left is a shearing transformation applied to the Mona Lisa: if every point in the image is represented by a vector pointing from the centre of the image to that point, multiplying (on the left) by a certain 2 x 2 matrix A brings us from the left to the right image.
Vectors off of the centre horizontal line such as v are sheared by A. However vectors such as v remain pointing in the same direction. v is an eigenvector of this map: it is a nonzero vector that satisfies the equation Av = λv, where λ is a scalar (in this example, it appears λ = 1).
An eigenvector v is a nonzero vector whose direction remains unchanged by the transformation v --> Av. It may be stretched/contracted, hence the equation Av = λv.
Formulas for solving polynomial equations
Formulas for solving polynomial equations
We learn the "-b" formula in school, to determine the solutions to a quadratic polynomial equation ax^2 + bx+ c = 0. There are formulas for solving a cubic (degree 3) or quartic (degree 4) equation -- which are at the end of these notes. What about a formula -- using +, x, and taking roots -- for a quintic (degree 5) polynomial equation? It has been mathematically proven that no such formula can exist! (Even stronger than this is the following specific example: the roots of the polynomial x^5 - x - 1 = 0 cannot be expressed using addition, multiplication, and taking roots.)
This discovery marked a significant moment in the history of mathematics. The original proof of this fact was due to Paolo Ruffini and Niels Abel (the history of the result is a little roundabout). A deeper and (in a sense) more general proof, that allowed one to determine which polynomials had no such formulas, was given by the French mathematician and political activist Évariste Galois (pictured right). Galois died in the midst of producing his great mathematical works -- for reasons that remain unknown, he fought in a duel and died from his wounds at the age of 20. Convinced of his impending death, Galois wrote a lengthy letter the night before the duel, outlining and collecting his mathematical ideas. This letter has been called "the most substantial piece of writing in the whole literature of mankind" (Hermann Weyl). Before his death, Galois introduced a precise connection between the roots of a polynomial and (a group of) permutations of those roots. This allows us to characterise properties of the solutions of a polynomial equation -- e.g., whether they can be expressed using +, x, and roots -- in terms of structural properties of the group of permutations. This connection is now known as Galois theory, in his honour.
Page updated
Google Sites
Report abuse