Ga naar: navigatie, zoeken

The course is taught by Christian Maes. He tends to be a bit vague, however still informing. The topics are material that were mostly covered in the bachelor, nevertheless you will notice that it is needed. Among the new topics are density matrices, decoherence scattering, quantum optics, a bit of quantum information theory and foundational issues. In November there is a test which does not count for the exam.

Maes did go over some things he could ask. Here is a summary.

### 20 January

1. Problem 1: True or False
• Every quantum algorithm can be simulated on a classical computer.
• A gate set ${\displaystyle \{U_{1},\dots ,U_{m}\}}$, where each ${\displaystyle U_{i}}$ acts on a single qubit can never be a universal gate set.
• The Bell tests show that there cannot exist a hidden variable.
• The amplitude of Rabi oscillations is constant (in principle). (The point was in Rabi model it was true, but in Jaynes-Cummings not.)
• The dimension of the Hilbert space of five spin 1 particles is 243.
• The Aharonov-Bohm effect applies to neutrons and their interference pattern as well.
• ...
2. Problem 2: Show
• Show how you go from the first formula on page 42 of the lecture notes to the second one.
3. Problem 3: Calculate
• Consider a system composed out of three particles described by the state ${\displaystyle \alpha |0\rangle |0\rangle |0\rangle +\beta |0\rangle |1\rangle |0\rangle +\gamma |1\rangle |1\rangle |1\rangle }$ in the Hilbert space ${\displaystyle {\mathcal {H}}_{1}\otimes {\mathcal {H}}_{2}\otimes {\mathcal {H}}_{3}}$, where ${\displaystyle {\mathcal {H}}_{i}=\mathbb {C} ^{2}}$ with basis ${\displaystyle \{|0\rangle ,|1\rangle \}}$ and ${\displaystyle |\alpha |^{2}+|\beta |^{2}+|\gamma |^{2}=1.}$
• Calculate ${\displaystyle {\text{Tr}}\left(e^{\sigma _{x}}\sigma _{z}e^{-2\sigma _{z}}\right)}$. The Pauli matrices were given.
• Variational principle to approximate ground state of a harmonic oscillator. The Hamiltonian is ${\displaystyle H={\frac {P^{2}}{2m}}+{\frac {m\omega ^{2}}{2}}Q^{2}}$. The (still unnormalised) wavefunctions that you should use are of the form ${\displaystyle \psi _{\alpha }=e^{-\alpha |x|},\alpha >0}$. Compare with the exact solution.
4. Problem 4: Scatter:
• Differential cross section of bosons and fermions: difference with classical scattering, difference between bosons and fermions (look at 90 degrees), make a sketch of the scattering cross section and discuss briefly.
5. Problem 5: Perturb

Something similar as the following.

1. Problem 6: Fun
• How would you solve the measurement problem? Or perhaps you do not believe there is a measurement problem at all?
• What are the names of the authors of the EPR paper?

Exam 2020 English translation:

1) Oral: A. discuss about symmetries of the Schrödinger equation: when is ${\displaystyle [H,P]=0}$? What is rotation invariance? B. mixed states vs superposition of qubits: difference, give example of density matrix, give example of entangled state, calculate its density matrix, is the light in this room a mixture? C. scattering of identical bosons/fermions: difference with classical scattering, difference between bosons and fermions (look at 90 degrees), make a sketch of the scattering cross section.

2) The quantum circuit of the Toffoli gate. What does this gate do?

3) Subaddiditivity: prove that the photon number of coherent states is Poisson distributed.

4) A. Show how you go from the first formula on page 42 of the lecture notes to the second one. (Rewrite the Hamiltonian using an identity of Pauli matrices, and in terms of the magnetic field). B. Calculate ${\displaystyle {\text{Tr}}\left(e^{-\sigma _{x}}\sigma _{z}e^{2\sigma _{x}}\right)}$.