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

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, qutri...

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Bibliographic Details
Published in:arXiv.org 2024-01
Main Authors: Rohit Sarma Sarkar, Adhikari, Bibhas
Format: Article
Language:English
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Summary: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.
ISSN:2331-8422