Nonreversal and nonrepeating quantum walks

Abstract

We introduce a variation of the discrete time quantum walk, the nonreversal quantum walk, which does not step back onto a position which it has just occupied. This allows us to simulate a dimer and we achieve it by introducing a new type of coin operator. The nonrepeating walk, which never moves in the same direction in consecutive time steps, arises by a permutation of this coin operator. We describe the basic properties of both walks and prove that the even-order joint moments of the nonrepeating walker are independent of the initial condition, being determined by five parameters derived from the coin instead. Numerical evidence suggests that the same is the case for the nonreversal walk. This contrasts strongly with previously studied coins, such as the Grover operator, where the initial condition can be used to control the standard deviation of the walker.

State transfer on two-fold Cayley trees via quantum walks*

“…Proctor et al introduced two previously unstudied types of coin operators for the DTQW, one of which is the nonrepeating coin, [31] which we will use in our perfect state transfer scheme on two-fold Cayley trees. Nonrepeating coin is the most general SU (4) operator with zeros on the diagonal, and the complete form can be found in Ref.…”

Quantum Walk Random Number Generation: Memory-based Models

“…Several memory-based quantum walk models have been introduced, including the quantum walk with memory [29]- [32], the non-repeating quantum walk [23], and the quantum walk with recycled coins [33]. We briefly review the quantum walk with recycled coins, called the history-dependent quantum walk (HD-QW), with additional memory space to store the coin-flip history [33].…”

Neighborhood-History Quantum Walk

“…It is noteworthy that physical realizations of QWs have emerged, and as they develop further [7,8], they are useful for demonstrating and manipulating quantum behavior. Altogether, strong interest continues to grow in understanding and designing their lattice traversal properties [9][10][11][12][13][14][15][16] to achieve faster and more accurate search algorithms.…”

“…From a search standpoint, to perform a more efficient search with a QW, a desirable feature may be self-avoidance: reduced probability of visiting previously visited sites in favor of unvisited sites [12,13]. With this goal, a model of QW proposed by Camilleri et al [12] expands the QW model to append a memory qubit to each lattice site, to maintain a record of particle visits.…”