Quantum Coin

Tse Kai Chan

Quantum coin is an essential element of the discrete-time quantum walk, the quantum analog of random walk. Most existing papers relating to quantum coin discussed the implementation and performance of the coin on power-of-two states and degrees of graphs. In this paper, I proposed and analyzed three alternative coins–binary state coin, W state coin, and Walsh-Hadamard coin–that possessed the property to perform on all states and degrees of graphs. I discussed the implementation and the performance of the three coins in the application of Quantum Walk and Quantum Walk Search. I compared and discussed the advantages, disadvantages, and errors of the three coins on a quantum simulator and a NISQ computer (i.e. IBM Q). In the end, I presented further applications for the three coins in quantum-walk-based algorithms and other potential topics in quantum computing.

Ivens Carneiro, Meng Loo, Xibai Xu +3 more

Quantum walks, both discrete (coined) and continuous time, form the basis of several recent quantum algorithms. Here we use numerical simulations to study the properties of discrete, coined quantum walks. We investigate the variation in the entanglement between the coin and the position of the particle by calculating the entropy of the reduced density matrix of the coin. We consider both dynamical evolution and asymptotic limits for coins of dimensions from two to eight on regular graphs. For low coin dimensions, quantum walks which spread faster (as measured by the mean square deviation of their distribution from uniform) also exhibit faster convergence towards the asymptotic value of the entanglement between the coin and particle's position. For high-dimensional coins, the DFT coin operator is more efficient at spreading than the Grover coin. We study the entanglement of the coin on regular finite graphs such as cycles, and also show that on complete bipartite graphs, a quantum walk with a Grover coin is always periodic with period four. We generalize the 'glued trees' graph used by Childs et al (2003 Proc. STOC , pp 59–68) to higher branching rate (fan out) and verify that the scaling with branching rate and with tree depth is polynomial.

Takashi Mihara, Tetsuro Nishino

Louie Hong Yao, Sascha Wald

Disorder in coined quantum walks generally leads to localization. We investigate the influence of the localization on the entanglement properties of coined quantum walks. Specifically, we consider quantum walks on the line and explore the effects of quenched disorder in the coin operations. After confirming that our choice of disorder localizes the walker, we study how the localization affects the properties of the coined quantum walk. We find that the mixing properties of the walk are altered nontrivially with mixing being improved at short time scales. Special focus is given to the influence of coin disorder on the properties of the quantum state and the coin-walker entanglement. We find that disorder alters the quantum state significantly even when the walker probability distribution is still close to the nondisordered case. We observe that, generically, coin disorder decreases the coin-walker entanglement and that the localization leaves distinct traces in the entanglement entropy and the entanglement negativity of the coined quantum walk.