Quantum circuit model for discrete-time three-state quantum walks on Cayley graphs

Abstract: We develop qutrit circuit models for discrete-time three-state quantum walks on Cayley graphs corresponding to Dihedral groups $D_N$ and the additive groups of integers modulo any positive integer $N$. The proposed circuits comprise of elementary qutrit gates such as qutrit rotation gates, qutrit-$X$ gates and two-qutrit controlled-$X$ gates. First, we propose qutrit circuit representation of special unitary matrices of order three, and the block diagonal special unitary matrices with $3\times 3$ diagonal blocks, which correspond to multi-controlled $X$ gates and permutations of qutrit Toffoli gates. We show that one-layer qutrit circuit model need $O(3nN)$ two-qutrit control gates and $O(3N)$ one-qutrit rotation gates for these quantum walks when $N=3^n$. Finally, we numerically simulate these circuits to mimic its performance such as time-averaged probability of finding the walker at any vertex on noisy quantum computers. The simulated results for the time-averaged probability distributions for noisy and noiseless walks are further compared using KL-divergence and total variation distance. These results show that noise in gates in the circuits significantly impacts the distributions than amplitude damping or phase damping errors.