Searching via Nonlinear Quantum Walk on the 2D-Grid

Abstract

We provide numerical evidence that the nonlinear searching algorithm introduced by Wong and Meyer, rephrased in terms of quantum walks with effective nonlinear phase, can be extended to the finite 2-dimensional grid, keeping the same computational advantage with respect to the classical algorithms. For this purpose, we have considered the free lattice Hamiltonian, with linear dispersion relation introduced by Childs and Ge The numerical simulations showed that the walker finds the marked vertex in O(N1/4log3/4N) steps, with probability O(1/logN), for an overall complexity of O(N1/4log5/4N), using amplitude amplification. We also proved that there exists an optimal choice of the walker parameters to avoid the time measurement precision affecting the complexity searching time of the algorithm.

Bright and dark solitons in a photonic nonlinear quantum walk: lessons from the continuum

“…Meyer and Gong study quantum search with the Gross-Pitaevskii equation [75,76] concluding that it solves the unstructured search problem more efficiently than does the Schrödinger equation, because it includes a cubic nonlinearity, and Chiew et al [77] demonstrate that the nonlinear quantum search can be more efficient than quantum search for graph comparison. Di Molfetta and Herzog [78] generalize the Meyer-Gross algorithm to two dimensions finding a clear advantage over the linear QW. Finally, in [79] the thresholds between modulational stability, rogue waves and soliton regimes are studied with coupled nonlinear Schrödinger equations with on site saturating nonlinearity.…”

“…where there is a dependence on the values of the walker in both dimensions. It was discussed in [78] that non-linear QWs, that introduce the nonlinearity in form of phases on the walker components, can be exploited to perform efficient search tasks on the two-dimensional grid. In line with those findings, we observed that the NLQW that introduces nonlinearities in the coin rotation operator angle produces ballistic dispersion, indicating that no soliton-like structures are formed in the two-dimensional case.…”

Fast Quantum State Discrimination with Nonlinear Positive Trace‐Preserving Channels

“…Quantum nonlinearity, beyond the stochastic nonlinearity provided by projective measurement, might be a powerful computational resource. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] However, there is no experimental evidence for physics beyond standard linear quantum mechanics. [18][19][20][21] It is well known to be challenging to even formulate a consistent, fundamentally nonlinear quantum theory in accordance with general principles.…”

Nonlinear and non-CP gates for Bloch vector amplification

“…Keywords: nonlinear channels, non-completely positive channels, Bloch vector amplification (Some figures may appear in colour only in the online journal) Several papers have explored the use of real or effective quantum nonlinearity for information processing [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Nonlinear master equations have also been frequently discussed in open systems theory [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32].…”