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)

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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!


1 Answer 1


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 : circuit for quantum walk 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 : walk step

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 :

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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


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