# Implementing Quantum Walks at IBM

a question about quantum walks, would this circuit be correct to start a quantum walk in a hypercube? I saw something about increment and decrement, but I didn't quite understand how they would work in a quantum walk (Image 1)

In addition to this circuit, would you recommend any articles related to the walks, whether in hypercubes or singular graphs or not? The theory I managed to understand, my problem is to pass the theory to a circuit, I'm a beginner and I would like to understand more how quantum walks would work in a quantum processor. Would there also be a way to plot the probability density graph of the quantum walks that were plotted on a computer? Is there any help material? Or the graph plotted by IBM computers would already be useful for this demonstration, it's just that I noticed that they are not very similar to those plotted in comparison of quantum and classical walks.... I would like to understand this better, thanks in advance!

Let's explain what this circuit do :

A coin quantum walk is a process in which a walker will evolve through the differents vertices of a graph. Its steps will be guided by a coin, a quantum register who will indicate him which direction to take.
We will consider a 4D-hypercube we want to perform a quantum walk on.
A 4D-hypercube count 16 vertices which will be labelled from 0 to 15 (all vertices labels can thus be stored in a 4-bits register). The vertices will be labelled as presented on this picture (here a 3D representation of an hypercube) :

Let's now consider your circuit : The 4 first qubits (q0-q3) of your register holds the position of your walker on the vertices of your hypercube (with q0 as the least significant bit).
The 2 last qubits (q5-q6) are the coin.

Starting from any point of this hypercube, we can take 4 different actions. For each of this actions, if the vertices are labelled as presented above, we can give a simple mathematical operation which transforms the label of the vertice to the vertice we get after the walk step.

• move on axis X : $$x \rightarrow x+1$$ or $$x \rightarrow x-1$$
• move on axis Y : $$x \rightarrow x+2$$ or $$x \rightarrow x-2$$
• move on axis Z : $$x \rightarrow x+4$$ or $$x \rightarrow x-4$$
• move on inner/outer cube : $$x \rightarrow x+8$$ or $$x \rightarrow x-8$$

The coin register which is a 2-qubits register can take 4 different states and we will associate for each of this state one of the actions presented above :

• $$|00\rangle$$ : move on axis X (+/-1)
• $$|01\rangle$$ : move on axis Y (+/-2)
• $$|10\rangle$$ : move on axis Z (+/-4)
• $$|11\rangle$$ : move on inner/outer cube (+/-8)

This is what is made in this part of the circuit :

For each possible state of the coin, we will flip appropriate bit of the walker register to perform the associated operation.

This part of the circuit is the coin flip :

It allows us to put the coin in adequate superposition of its 4 basis states.

Concerning papers and article, this paper talks about circuit-based implementation of quantum walks : https://journals.aps.org/pra/pdf/10.1103/PhysRevA.79.052335