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.

翻译：我们针对二面体群$D_N$及整数模任意正整数$N$的加法群对应的Cayley图，建立了离散时间三态量子游走的qutrit电路模型。所提出的电路由基本qutrit门组成，包括qutrit旋转门、qutrit-$X$门以及双qutrit受控-$X$门。首先，我们给出了三阶特殊酉矩阵的qutrit电路表示，以及具有$3\times 3$对角分块的分块对角特殊酉矩阵表示，这些对应于多控$X$门和qutrit Toffoli门的置换。我们证明，当$N=3^n$时，单层qutrit电路模型需要$O(3nN)$个双qutrit控制门和$O(3N)$个单qutrit旋转门来实现这些量子游走。最后，我们对这些电路进行数值模拟，以模拟其在噪声量子计算机上寻找游走器在任何顶点的时间平均概率等性能。进一步利用KL散度和全变差距离比较了噪声和无噪声游走的时间平均概率分布的模拟结果。这些结果表明，电路中门的噪声对分布的影响显著大于振幅阻尼或相位阻尼误差。