Problem set 2

Problem set 2#

Short guidelines#

Description#

Today you will be working on a specific challenge related to quantum physics or quantum computing. In groups, your task is to develop a solution using the concepts and tools you’ve learned throughout the course. You can use any modules or libraries you find useful, such as NumPy, SciPy or QuTiP.

We will be using AI copilots to assist you in coding and problem-solving. The focus will be on collaboration, creativity, and critical thinking.

The final part of the session will involve a short presentation (15’), where each group will share their approach, solution, and reflections on the use of AI tools in their process.

Objectives#

  1. Understand the problem statement and requirements.

  2. Collaborate with your team to brainstorm ideas and approaches.

  3. Develop a solution using Python or Mathematica. You can use AI tools to assist you in coding and problem-solving.

  4. Prepare a presentation to share your findings and reflect on the use of AI in your process.

Resources you can use#

  • Access to AI copilots/tools for coding assistance. Some examples include GenAI, ChatGPT, or other similar tools.

  • Documentation and resources from previous weeks.

Short presentation outline (15’)#

  1. Background:

    • What was the question?

    • How did you approach it?

  2. Solution development and implementation

  3. Implementation (mathematical and coding)

  4. Critical reflection on the use of AI copilots. Eg:

    • Which AI tool/s did you use?

    • Was it helpful, misleading, or counterproductive?

    • What resources did you need other than AI?

    • How confident are you in your solution?

    • Which parts are less clear or hard to verify?

  5. Coding skills evaluation

    • To what extent do you understand the code generated with AI assistance?

    • Were there functions or code constructs used that you didn’t know before? Do you understand them now?

Problem 2#

Minimal driving time for a single control without constraint#

Let the dynamics of a two-level system be described by the Hamiltonian

\[ H = \Gamma(t) \sigma_{3} + \omega \sigma_{1} \]

The goal is to find an optimal driving \(\Gamma(t)\) such that the corresponding time-development operator \(U_{H}(t,0)\) evolves from an initial state \(|\psi_{in}\rangle\) at time \(t=0\) to a final state \(|\psi_{f}\rangle\) in minimal time \(T = T_{min}\), i.e.

\[ U_{H}(T_{min},0) |\psi_{in}\rangle = \lambda |\psi_{f}\rangle \]

Where, \(|\psi_{in}\rangle\) and \(|\psi_{f}\rangle\), are:

\[\begin{split} |\psi_{in}\rangle = \left(\begin{array}{c} \cos \theta_{in}/2 \\ \sin \theta_{in}/2 \end{array}\right) , \quad |\psi_{f}\rangle = \left(\begin{array}{c} \sin \theta_{f}/2 \\ \cos \theta_{f}/2 \end{array}\right) \end{split}\]
\[ \theta_{in} = \pi/2 + \alpha, \quad \theta_{f} = \pi/2 - \alpha, \quad \alpha > 0 \]

What is the fastest time and the best driving protocol?

(Bonus)#

The general Hamiltonian of the qubit follows

\[ H = \Gamma(t) \sigma_{3} + \omega_{1} (t) \sigma_{1} + \omega_{2} (t) \sigma_{2} \]

Transform the initial state given in the previous section to the final state in minimal time \(T_{min}\), under the constrains:

  • You can only control \(|\Gamma(t)| < c\)

  • You can only control \(\Gamma(t)\) and \(\omega_{1}(t)\) without constraint.

  • You can only control \(\Gamma(t)\) and \(\omega_{1}(t)\) with constraint, \(|\Gamma (t)|^{2} + |\omega_{1}(t)|^{2} < c\)

In all cases all the other driving terms are constant. What is the optimal driving protocol and the minimal time now?

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