Fig. 1. Probability distribution of 9 non-orthogonal initial states after 25 steps of quantum walk. (a)‒(i) Probability distributions for states ψ1,ψ2,⋯,ψ9

Fig. 2. Quantum circuit diagram for preparing 9 initial states. (a)‒(i) Preparation process for states ψ1,ψ2,⋯,ψ9

Fig. 3. Controlled coin operations on position states with six qubits and simplified representation as C(x,t). (a) Perform corresponding coin operations C(x,t) based on the encoding of states at different positions; (b) simplified representation of C(x,t) in the circuit

Fig. 4. Performing the quantum walk operation on six qubits using the circuit and simplifying it to a Q gate. (a) Executing the walk operation using increment and decrement circuits; (b) simplified representation as a Q gate in the circuit

Fig. 5. Quantum circuit diagram for running steps 1 to 9

Fig. 6. Quantum circuit diagram for running steps 10 to 22

Fig. 7. Quantum circuit diagram for running steps 23 to 25

Fig. 8. Probability distribution charts of running 25 steps for 9 initial states. (a)-(i) Probability distributions for states ψ1,ψ2,⋯,ψ9