Cours : Quantum Information

Section outline

Select section Généralités

Généralités

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- Select activity Older web page of the course
  
  Older web page of the course URL
  
  Contains some additional material.
Select section Interpretation of Quantum Mechanics

Interpretation of Quantum Mechanics
- Select activity EPR paper "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?"
  
  EPR paper "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" Fichier
  
  The paper of Einstein, Podolski and Rosen pointing out that quantum mechanics is at odds with the views of classical physic about physical reality. The paper gave rise to the term "EPR pair".
- Select activity Bohr's reply "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?"
  
  Bohr's reply "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" Fichier
  
  Bohr replies that the concept of "physical reality" is inadequate, and that we have to accept new point of view that he calls "complementarity".
- Select activity Some quotes concerning the question of interpretation
  
  Some quotes concerning the question of interpretation Fichier
  
  It turns out that even if there is no real issue with the interpretation of quantum mechanics, there is at least a controversy concerning whether there is such an issue.
- Select activity An overview of existing interpretations (as of 2016)
  
  An overview of existing interpretations (as of 2016) URL
  
  An attempt to classify existing interpretations (by someone who maintains that the dispute should be solved).
- Select activity A Philosopher Looks at Quantum Mechanics (Again) by Hilary Putnam
  
  A Philosopher Looks at Quantum Mechanics (Again) by Hilary Putnam Fichier
  
  A systematic reflection on the status of the quantum mechanics by a prominent mathematician and logician (and philosopher).
Select section Mach-Zehnder interferometer

Mach-Zehnder interferometer
- Select activity Machův-Zehnderův interferometr
  
  Machův-Zehnderův interferometr Fichier
- Select activity Mach-Zehnder Interferometer
  
  Mach-Zehnder Interferometer Fichier
  
  The revised translation.
Select section Basics of Complex Linear Algebra

Basics of Complex Linear Algebra
- Select activity Základy komplexní lineární algebry
  
  Základy komplexní lineární algebry Fichier
- Select activity Basics of Complex Linear Algebra
  
  Basics of Complex Linear Algebra Fichier
  
  The revised translation.
- Select activity K Diracově značení
  
  K Diracově značení Fichier
- Select activity Comments on the Dirac notation
  
  Comments on the Dirac notation Fichier
  
  Reflection on the Dirac notation for a mathematical purist.
Select section Postulates of Quantum Mechanics

Postulates of Quantum Mechanics
- Select activity Postuláty kvantové mechaniky
  
  Postuláty kvantové mechaniky Fichier
- Select activity Postulates of Quantum Mechanics
  
  Postulates of Quantum Mechanics Fichier
  
  A revised translation
Select section Introduction into tensor products

Introduction into tensor products
- Select activity Introduction to tensor products
  
  Introduction to tensor products Fichier
  
  Corrected English version (tensors as multilinear forms)
Select section Deutsch-Jozsa algorithm

Deutsch-Jozsa algorithm
- Select activity Deutschův-Jozsův algoritmus
  
  Deutschův-Jozsův algoritmus Fichier
  
  Kapitola skript
- Select activity Deutsch-Jozsa algoritm
  
  Deutsch-Jozsa algoritm Fichier
  
  A revised translation
Select section Reversible computation

Reversible computation
- Select activity Reversible computation
  
  Reversible computation Fichier
Select section Universal set of gates

Universal set of gates
- Select activity Universal set of gates
  
  Universal set of gates Fichier
- Select activity Lecture notes implicitly invite the following task...
  
  Lecture notes implicitly invite the following tasks:
  
  Verify the function of the circuit ABC for the one-qubit controlled operator
  
  Verify the equivalence of two distinct ways of multiplication by e^{i\alpha}
  
  Verify the function of two-controlled operator
  
  Verify the equivalence of the two-controlled gate and the one-controlled gate with the CNOT
  
  Verify the ``control by zero'' with help of two X-gates
  
  Try all square roots of negation
  
  (You can also try other functions applied to an operator. For instance, what e^{i A}, where A is a normal operator looks like?)
  
  Verify the function of the two-level operator U and its equivalence with the uncontrolled operator M
  
  Make sure you understand the transformation of the complicated two-level operator to the simple one
  
  Make sure you understand the ``unitary Gaussian elimination''.
Select section Shor's algorithm

Shor's algorithm
- Select activity At the moment we can theoretically build a circuit...
  
  At the moment we can theoretically build a circuit composed of basic small gates (single cubic and CNOT) for any matrix. (Except for the as yet unproven theorem ABC necessary for the construction of a single-cubic controlled operator.) We can therefore return to the question of for which matrices there is a "small" circuit.
  
  To obtain sufficient motivation for some of the following theoretical passages (especially for the quantum Fourier transform), we will now look at the most important algorithm: Shor's algorithm for factorization.
- Select activity Shor's factorization algorithm
  
  Shor's factorization algorithm Fichier
- Select activity A note on Fermat's factorization Try to decom...
  
  A note on Fermat's factorization
  
  Try to decompose some composite number by Fermat's test. E.g. 21. We look for two squares equivalent modulo 21. We can succesively add squares to 21 and see when we get another square. We have 21 + 2² = 5². Hence 21 = (5 + 2) (5-2).
  
  Order of 2 mod 21 is 6. Thus (2³ + 1) (2³-1) = 9x7 = 63 = 0 mod 21. Nine is not a divisor, but it is a factor of the factorized number. The divisor 3 is thus found by Euclid's algorithm.
- Select activity Think carefully about what the operator W looks li...
  
  Think carefully about what the operator W looks like and how its construction is related to the reversible implementation of a classical algorithm.
  
  Is it really a permutation of basis elements?
  
  How big is the circuit for W (what number of elementary gates do we need)? Is it (measured as a function of m and n) polynomial? Quadratic? Or even linear?
- Select activity You can skip success estimates when reading the le...
  
  You can skip success estimates when reading the lecture notes. Try to imagine an algorithm for factorization of 15. The group of invertible elements modulo 15 has order 8, so all elements have the order of the power of two. For M = 16, therefore, there is an equality in the approximation (*) in this case.
- Select activity Continued fractions
  
  Continued fractions Fichier
  
  in Czech
  
  from the lecture Number Theory and RSA
  
  the relevant example is translated in the file above
- Select activity Discussion
  
  Discussion Forum
Select section Discrete Fourier transform and its quantum decomposition

Discrete Fourier transform and its quantum decomposition
- Select activity The complexity of Shor's algorithm crucially depen...
  
  The complexity of Shor's algorithm crucially depends on the circuit for the Fourier transform.
  
  We could accept the matrix of the Fourier transform as given, but we will use the opportunity to explain the principle of the discrete Fourier transform as a decomposition of a complex function (mapping) over a group into the base of periodic functions (or the base of characters, or the Fourier base).
- Select activity The Discrete Fourier Transform
  
  The Discrete Fourier Transform Fichier
  
  Linear algebraic definition of the basis of characters, or the Fourier basis.
- Select activity Quantum decomposition of the DFT
  
  Quantum decomposition of the DFT Fichier
- Select activity Examples of DFT
  
  Examples of DFT Fichier
- Select activity Discussion
  
  Discussion Forum
- Select activity DFT as phase estimation
  
  DFT as phase estimation Fichier
  
  An alternative point of view of Shor's algorithm
Select section Complex projective line representations

Complex projective line representations
- Select activity Complex projective line
  
  Complex projective line Fichier
- Select activity Optional additional material on the Hopf fibration...
  
  Optional additional material on the Hopf fibration, which is way of representation of the three dimensional real sphere 𝕊³ (that is, the surface of the four-dimensional ball) in three dimensions.
- Select activity The Hopf fibration
  
  The Hopf fibration Fichier
- Select activity An impressive visualization of the Hopf fibration ...
  
  An impressive visualization of the Hopf fibration
Select section Unitary operators as rotations

Unitary operators as rotations
- Select activity Action of unitary operators on qubits can be under...
  
  Action of unitary operators on qubits can be understood as rotations of the Bloch sphere representing qubits.
- Select activity Geometry of unitary operators
  
  Geometry of unitary operators Fichier
- Select activity A basic tool in the above representation are prope...
  
  A basic tool in the above representation are properties of quaternions.
- Select activity Quaternions
  
  Quaternions Fichier
- Select activity The inscription on the Broom Bridge in Dublin wher...
  
  The inscription on the Broom Bridge in Dublin where quaternions were born.
- Select activity Niles Johnson's lecture on the stereographic proje...
  
  Niles Johnson's lecture on the stereographic projection (slightly different from ours), Hopf fibration and quaternions.
- Select activity Remarkable interactive visualization of quaternion...
  
  Remarkable interactive visualization of quaternions and the stereographic projection
  
  https://eater.net/quaternions
Select section Extended Euler's formula and the AXBXC decomposition

Extended Euler's formula and the AXBXC decomposition
- Select activity The theorem about the AXBXC decomposition complete...
  
  The theorem about the AXBXC decomposition completes the construction of a circuit for a general unitary matrix using only CNOT and single qubit operators.
- Select activity Extended Euler's formula and the decomposition AXBXC
  
  Extended Euler's formula and the decomposition AXBXC Fichier
Select section Quantum Key Sharing principle

Quantum Key Sharing principle
- Select activity We conclude the course with a description of the f...
  
  We conclude the course with a description of the famous protocol BB884 (named after a paper by Charlese Bennetta and Gilles Brassarda from 1984). The protocol allows quantum key sharing whose security is based solely on principles of quantum mechanics, unlike classical protocols which depend on complexity assumptions like P ≠ NP.
  
  Full proof of security requires the concept of quantum entropy and more advanced parts of the quantum information theory.
- Select activity Stopa a pozitivní operátory
  
  Stopa a pozitivní operátory Fichier
- Select activity Description of the BB84 protocol
  
  Description of the BB84 protocol Fichier
- Select activity Mixed states
  
  Mixed states Fichier
- Select activity Quantum entropy
  
  Quantum entropy Fichier
- Select activity Holevo bound
  
  Holevo bound Fichier
Select section Tutorial 2023 examples

Tutorial 2023 examples
- Select activity Tutorial 1
  
  Tutorial 1 Fichier
- Select activity Tutorial 2
  
  Tutorial 2 Fichier
- Select activity Tutorial 3
  
  Tutorial 3 Fichier
- Select activity Tutorial 4
  
  Tutorial 4 Fichier
- Select activity Tutorial 5
  
  Tutorial 5 Fichier
- Select activity Tutorial 6
  
  Tutorial 6 Fichier
- Select activity Tutorial 7
  
  Tutorial 7 Fichier
Select section Tutorial 2022 examples

Tutorial 2022 examples
- Select activity Here, the examples from the tutorials will be publ...
  
  Here, the examples from the tutorials will be published.
  
  If you would like to play with the quantum circuits a bit more, you may find the following links useful:
  
  https://algassert.com/quirk - a quantum circuit simulator, you may play with some predefined examples, and some very nontrivial operations are implemented there as basic gates
  
  https://quantum-computing.ibm.com/ - after registering, you may use IBM's quantum computers (real quantum hardware and there is a free tier), as such you can see the problems of contemporary quantum computers, not as powerful as quirk, however, may still be useful
  
  there are many more (both simulators and quantum computers) e.g. http://www.quantumplayground.net/#/home https://www.rigetti.com/ however, I do not have much experience with those
- Select activity cvičení 1
  
  cvičení 1 Fichier
- Select activity cvičení 2
  
  cvičení 2 Fichier
- Select activity cvičení 3
  
  cvičení 3 Fichier
- Select activity cvičení 4 (obvody)
  
  cvičení 4 (obvody) Fichier
- Select activity Tutorial 2
  
  Tutorial 2 Fichier
- Select activity Tutorial 3
  
  Tutorial 3 Fichier
- Select activity Tutorial 4
  
  Tutorial 4 Fichier
- Select activity Tutorial 5
  
  Tutorial 5 Fichier
- Select activity Tutorial 6
  
  Tutorial 6 Fichier
- Select activity Tutorial 7
  
  Tutorial 7 Fichier
- Select activity Diskuze ke cvičení
  
  Diskuze ke cvičení Forum
Select section Tasks for credit

Tasks for credit
Select section List of topics

List of topics

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