Search algorithm on strongly regular graphs based on scattering quantum walk

Abstract

Janmark, Meyer, and Wong showed that continuous-time quantum walk search on known families of strongly regular graphs (SRGs) with parameters (N, k, λ , µ) achieves full quantum speedup. The problem is reconsidered in terms of scattering quantum walk, a type of discrete-time quantum walks. Here, the search space is confined to a low-dimensional subspace corresponding to the collapsed graph of SRGs. To quantify the algorithm's performance, we leverage the fundamental pairing theorem, a general theory developed by Cottrell for quantum search of structural anomalies in star graphs. The search algorithm on the SRGs with k scales as N satisfies the theorem, and results can be immediately obtained, while search on the SRGs with k scales as √ N does not satisfy the theorem, and matrix perturbation theory is used to provide an analysis. Both these cases can be solved in O( √ N) time steps with a success probability close to 1. The analytical conclusions are verified by simulation results on two SRGs. These examples show that the formalism on star graphs can be applied more generally.

Finding more than one path through a simple maze with a quantum walk

“…Here we will present examples of the derivations of equations in equations ( 3) and (12). First let us look at U|ψ 2 in equation (3).…”

“…This has been done for a number of highly symmetric graphs, such as the hypercube [6,7], grids in different dimensions [8,9], and the complete graph [9,10]. The role of the symmetry of the graph in a quantum-walk search has been explored [11,12], and it has been shown that a quantum-walk search is optimal for almost all graphs [13]. The initial state of the walk cannot incorporate any knowledge of the distinguished vertex, and it is usually an equal superposition of all vertices, in the case of a coined walk, or an equal superposition of all edges, in the case of a scattering walk.…”

Quantum walk under coherence non-generating channels*

“…[16,18] Quantum walk (QW) is widely applied in physics, quantum computation, quantum information theory, and so on. [19][20][21][22][23][24][25][26][27][28][29][30] QW that we discuss is the discrete-time QW. Using the quantum coin and the "flip operations" instead of flipping the coin as in CW, the walker moves on a line in a discrete manner through the moving operations.…”