Moments of coinless quantum walks on lattices

Abstract

The properties of the coinless quantum-walk model have not been as thoroughly analyzed as those of the coined model. Both evolve in discrete time steps, but the former uses a smaller Hilbert space, which is spanned merely by the site basis. Besides, the evolution operator can be obtained using a process of lattice tessellation, which is very appealing. The moments of the probability distribution play an important role in the context of quantum walks. The ballistic behavior of the mean square displacement indicates that quantum-walk-based algorithms are faster than randomwalk-based ones. In this paper, we obtain analytical expressions for the moments of the coinless model on d-dimensional lattices by employing the methods of Fourier transforms and generating functions. The mean square displacement for large times is explicitly calculated for the one-and two-dimensional lattices, and using optimization methods, the parameter values that give the largest spread are calculated and compared with the equivalent ones of the coined model. Although we have employed asymptotic methods, our approximations are accurate even for small numbers of time steps.

Discrete-Time Quantum Walks on Oriented Graphs

“…To calculate x , we need to find the first derivative at k = 0 of the characteristic function ϕ X (k), which is the expected value of e ikX , where X is the position operator, that is, ϕ X (k) = ψ(t) e ikX ψ(t) . Using the methods outlined in [13] or [18,23], we obtain…”

Chirality from quantum walks without a quantum coin

“…By relation (46) in Proposition 2 combined with Theorem 3, for each fixed finite group Q, one can consider the equivalence classes of maps ϕ : Q → GL(d, Z) up to pre-composition with arbitrary β ∈ Aut(Q) and to conjugation by arbitrary α ∈ GL(d, Z). Lemma 4, using property (28) along with the fact that Inn(N ) is trivial, implies that ∀q ∈ Q ∃r q ∈ N : ϕ rq q (n) = n, ∀n ∈ N.…”

“…In this manuscript, we explore the second way, starting with the minimal coin dimension s = 1, and performing a systematic analysis of the Euclidean scenario. QWs with a one-dimensional coin are often referred to as scalar or coinless [38,[43][44][45][46]. Despite the algorithmic simplicity of the model, finding all scalar QWs for an arbitrary graph is not a straightforward task.…”

“…[38,39], and represents their completion. Here we construct the first examples of infinite scalar QWs with more than one space-dimension; moreover, we avoid partitions of the underlying graph or of the QW itself, as opposed to the literature where the partition is used [43,45,46] to circumvent the no-go theorem [38], leading to an inhomogeneous evolution. Furthermore, the scalar QWs constructed here are both non-Abelian and infinite, unlike those explored in Ref.…”

Path-sum solution of the Weyl quantum walk in 3 + 1 dimensions

“…The total Hilbert space of the walk is the tensor product of the above ones.Since their first appearance, QWs received an increasing attention in the literature, where their main application is the design of quantum algorithms [5][6][7][8]. As an example, efficient search algorithms [6,9,10] were devised exploiting the fact that the spread of a localized initial state after t steps is proportional to t, whereas for the classical random walk the spread is proportional to √ t. Remarkably, any quantum circuit can be implemented as a QW on a graph, proving that QWs can be used for universal quantum computation [11].Recently, in Ref.[12] the authors use QWs in a derivation of quantum field theory from principles, from which it follows that the graph of the QW is the Cayley graph of a finitely presented group G. The QWs reproducing free quantum field theory in Euclidean space correspond to group G = Z d that is Abelian, which is the most common case in the literature. In particular the simplest QW descending from the principles is the Weyl QW [12], reproducing the Weyl quantum field theory.…”