Visualizing quantum walks on small graphs

This page has all the Julia code necessary for creating the images; as such, it must begin with a couple of trivialities, as loading the dependencies and defining the adjacency matrix of a graph.

31.4 μs

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begin
    using LinearAlgebra
    using LightGraphs
    using Plots
    using Printf
end

47.7 s

adjacency_matrix (generic function with 2 methods)

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begin
    
    function adjacency_matrix(n :: Int64, e :: AbstractEdge)
        f(i :: Int64) = [ convert(Float64, i == j) for j in 1:n ]
        f(src(e)) .* f(dst(e))' + f(dst(e)) .* f(src(e))'
    end
    
    function adjacency_matrix(G :: SimpleGraph)
        f(e) = adjacency_matrix(nv(G), e)
        sum(f.(edges(G)))
    end
​
end

258 μs

Quantum Walk representation

I am representing a quantum walk as the adjacency matrix of the graph.

25.5 μs

QuantumWalk

 
begin
    struct QuantumWalk
        A :: Matrix
    end
    
    function QuantumWalk(G :: SimpleGraph)
        QuantumWalk(adjacency_matrix(G))
    end
end

9.8 ms

I also created a custom heatmap function for displaying the matrices.

25.9 μs

heatmap (generic function with 1 method)

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function heatmap(A :: Matrix; kargs...)
    Plots.heatmap(A;
            clim=(0.0, 1.0),
            framestyle = :none,
            ticks = false,
            aspectratio = :equal,
            c = :thermal,
            kargs...
        )
end

160 μs

Then it is easy to create an animation: Just iterate over a bunch of times and display the component-wise absolute value of $\exp (i t A)$ .

27.9 μs

animate (generic function with 1 method)

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function animate(walk :: QuantumWalk, t0 :: Float64, t1 :: Float64)
    step = (t1 - t0)/200
    animation = @animate for t in t0:step:t1
        heatmap(abs.(exp(im * t * walk.A)), title= @sprintf("t = %.2f", t),)
    end
    gif(animation, fps=10)
end

233 μs

Plots.GRBackend

14.4 ms

Visualizations

I can now display some quantum walks. I will show for $P_{2}$ and for $P_{3}$ at their respective periods.

34.1 μs

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animate(QuantumWalk(path_graph(2)), 0.0, 2.0 * pi)

31.5 s