The Super Parameter | PennyLane Challenges

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Challenge statement

This challenge is included in the QHack 2023 Flashback Badge Challenge event.

Within QML it is common to find the term expressivity, which refers to the size of all possible models that we can generate by varying our parameters. One way to increase the expressivity of our model family is usually by adding more parameters. However, this is not always a good thing, since increasing the number of parameters, and therefore the number of possible models, means that we have to perform our training on a very large set, making it more difficult to find the model that best suits our needs. Therefore, the real challenge of a good QML researcher is to find the smallest possible family of models that still contains the optimal solution. There is much more to the notion of expressivity, but in this challenge we are going to push the concept to its limits.

Suppose that we are in the situation where we have 3 qubits and we know that the solution to our problem is a computational basis state, i.e. an element of the set

\mathcal{B}=\left\lbrace |000\rangle, |001\rangle, |010\rangle,\dots, |111\rangle\right\rbrace.

We don't know exactly what the basis state is, so we would like to generate an ansatz expressive enough so that:

\begin{align*} U(\vec{\theta_0})|000\rangle & = |000\rangle \\ U(\vec{\theta_1})|000\rangle & = |001\rangle \\ U(\vec{\theta_2})|000\rangle & = |010\rangle \\ & \vdots \\ U(\vec{\theta_7})|000\rangle & = |111\rangle \end{align*}

for certain values of $\vec{\theta_i}$ . An example of ansatz that accomplishes this would be the following circuit:

drawing

This is the fundamental concept in Basis embedding, where you can see that by taking $\alpha$ , $\beta$ and $\gamma$ properly, we can generate any basis state. However, this challenge is not going to be this easy. You are asked to build an ansatz that, with only one parameter, is able to generate all the basis states. To judge your solution, we will ask you to provide us with a list of the 8 values of the parameter that generate each of them. Good luck!

Challenge code

You must complete the qnode model that will be in charge of obtaining different outputs. This model depends on a single parameter and you must ensure that it generates all the basis states. You must also define the function generate_coefficients, which will return a list with the 8 values of the parameter to generate these basis states.

Output

To judge this challenge, the generate_coefficients function will be called first. With the output of this function (the eight coefficients), we will call the model to ensure that the generated states are the desired ones. In addition, we will check that:

The model is continuous (small modifications of the parameter imply small modifications of the generated state). By putting the parameter inside rotation gates you will have no problems with this.
The generated coefficients are in the interval [0,10]. Solutions that do not fit this interval will be considered incorrect.

In this challenge, we will not work with public and private tests. We will simply check that all of the above is fulfilled. Good luck!