Fig. 1. Quantum walk circuit diagrams. (a) Wave quantum walk; (b) a left and right conditional walking operation; (c) particle quantum walk; (d) wave-particle quantum walk

Fig. 2. Probability distribution calculated by even step position theory after 4-step evolution when ϕ=1.19π. (a) Classical random walk when θ=π/2 and quantum walk when θ=0; (b) wave-particle coherent and wave-particle mixed when θ=0.385π

Fig. 3. Change of position variance with the theoretical calculation of step t when ϕ=1.19πandθ=0.385π

Fig. 4. Changes of position variance with the theoretical calculation of auxiliary bit parameters. (a) When ϕ=1.19π, ν changes with θ; (b) when θ=0.385π, ν changes with ϕ

Fig. 5. Wave-particle quantum walk demonstrated in IBM quantum computer

Fig. 6. Probability distribution of even step position simulation after 4-step evolution when ϕ=1.19π. (a) Wave quantum walk when θ=0 and particle quantum walk when θ=π/2; (b) wave-particle coherent and wave-particle mixed quantum walk when θ=0.385π

Fig. 7. Changes of position variance with the simulation of step t when ϕ=1.19π,θ=0.385π

Fig. 8. Changes of position variance with the simulation of auxiliary bit parameters. (a) When ϕ=1.19π, ν changes with θ; (b) when θ=0.385π, ν changes with ϕ

Fig. 9. 4-step evolutionary probability distribution after adding noise when ϕ=1.19π and θ=0.385π. (a) Wave-particle mixed; (b) wave-particle coherent