quantum walk

#=
quantum_walk:
- Julia version: 1.1.1
- Author: shunpeik
- Date: 2019-11-30
=#
using Plots
using LinearAlgebra
gr()

let
    generate = (P, Q) -> function (φ)
        φ_next = hcat([0.0;0.0], zero(φ), [0.0;0.0])
        φ_next[:, 1:end-2] = mapslices(x ->  Q * x , φ, dims=1)
        φ_next[:, 3:end] .+=  mapslices(x ->  P * x , φ, dims=1)
        return φ_next
    end

    θ = π / 4
    @show P = [[cos(θ) sin(θ)];[0 0]]
    @show Q = [[0 0]; [sin(θ) -cos(θ)]]
    @show φ = [1.0/√2; 1.0im/√2]
    U = P + Q
    @show U'U
    @show φ'φ
    evolve = generate(P, Q)
    for i in 1:100
        φ = evolve(φ)
    end
    probs = mapslices(x -> convert(Float64, x'x), φ, dims=1)
    display(plot(1:length(probs), probs'))
    readline()
end