We provide first evidence that some families of nonlinear quantum systems , rephrased in terms of coined quantum walks with effective nonlinear phase, display a strong computational advantage for search algorithms, over graphs having sets of vertices of constant size, e. g. the infinite square grid.  The numerical simulations show that the walker finds the marked vertex in O(N^{1/4} \log^{3/4} N) steps  with probability O(1/\log N), for an overall complexity of O(N^{1/4}\log^{7/4}N). We also confirm this result analytically. Moreover the algorithm is implemented without the need for a specific oracle step, but by topological hole defect in the grid. From a quantum computing perspective, however, this hints at novel applications of quantum walk search: instead of using them to look for "good" solutions within the configuration space of a problem, we could use them to look for topological properties of the entire configuration space .
We provide numerical evidence that the nonlinear searching algorithm introduced by Wong and Meyer \mbox{%DIFAUXCMD meyer2013nonlinear , rephrased in terms of quantum walks with effective nonlinear phase, can be extended to the finite 2-dimensional grid, keeping the same computational advantage respect to the classical algorithms. For this purpose, we have considered the free lattice Hamiltonian, with linear dispersion relation introduced by Childs and Ge \mbox{%DIFAUXCMD Childs_2014 The numerical simulations showed that the walker finds the marked vertex in O(N^{1/4} \log^{3/4} N) steps , with probability O(1/\log N), for an overall complexity of O(N^{1/4}\log^{7/4}N). We also proved that there exists an optimal choice of the walker parameters to avoid that the time measurement precision affects the complexity searching time of the algorithm .
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first evidence that some families of nonlinear quantum systems
numerical evidence that the nonlinear searching algorithm introduced by Wong and Meyer \mbox{%DIFAUXCMD meyer2013nonlinear
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coined
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display a strong computational advantage for search algorithms, over graphs having sets of vertices of constant size, e. g. the infinite square grid.
can be extended to the finite 2-dimensional grid, keeping the same computational advantage
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respect to the classical algorithms. For this purpose, we have considered the free lattice Hamiltonian, with linear dispersion relation introduced by Childs and Ge \mbox{%DIFAUXCMD Childs_2014
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show
showed
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,
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confirm this result analytically. Moreover the algorithm is implemented without the need for a specific oracle step, but by topological hole defect in the grid. From a quantum computing perspective, however, this hints at novel applications of quantum walk search: instead of using them to look for "good" solutions within the configuration space of a problem, we could use them to look for topological properties of the entire configuration space
proved that there exists an optimal choice of the walker parameters to avoid that the time measurement precision affects the complexity searching time of the algorithm