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arXiv:1908.11213 · Physical Review Letters

Grover Search as a Naturally Occurring Phenomenon

Mathieu Roget, Stéphane Guillet, Pablo J. Arrighi, Giuseppe Di Molfetta

Published May 4, 2020• 26 citations
arXivHTMLPDF

Abstract

We provide first evidence that under certain conditions, 1/2-spin fermions may naturally behave like a Grover search, looking for topological defects in a material. The theoretical framework is that of discrete-time quantum walks (QWs), i.e., local unitary matrices that drive the evolution of a single particle on the lattice. Some QWs are well known to recover the (2+1)-dimensional Dirac equation in continuum limit, i.e., the free propagation of the 1/2-spin fermion. We study two such Dirac QWs, one on the square grid and the other on a triangular grid reminiscent of graphenelike materials. The numerical simulations show that the walker localizes around the defects in O(sqrt[N]) steps with probability O(1/logN), in line with previous QW search on the grid. The main advantage brought by those of this Letter is that they could be implemented as "naturally occurring" freely propagating particles over a surface featuring topological defects-without the need for a specific oracle step. From a quantum computing perspective, however, this hints at novel applications of QW 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.

DOI
10.1103/physrevlett.124.180501

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We provide first evidence that under certain conditions, 1/2-spin fermions may naturally behave like a Grover search, looking for topological defects in a material. The theoretical framework is that of discrete-time quantum walks (QWs), i.e., local unitary matrices that drive the evolution of a single particle on the lattice. Some QWs are well known to recover the (2+1)-dimensional Dirac equation in continuum limit, i.e., the free propagation of the 1/2-spin fermion. We study two such Dirac QWs, one on the square grid and the other on a triangular grid reminiscent of graphenelike materials. The numerical simulations show that the walker localizes around the defects in O(sqrt[N]) steps with probability O(1/logN), in line with previous QW search on the grid. The main advantage brought by those of this Letter is that they could be implemented as "naturally occurring" freely propagating particles over a surface featuring topological defects-without the need for a specific oracle step. From a quantum computing perspective, however, this hints at novel applications of QW 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.
A discrete-time quantum walk (QW) is essentially an operator driving the evolution of a single particle on the lattice, through local unitaries. Some QWs admit a continuum limit, leading to well-known physics partial differential equations, such as the Dirac equation. We show that these simulation results need not rely on the grid: the Dirac equation in (2+1)(2+1)(2+1) dimensions can also be simulated, through local unitaries, on the honeycomb or the triangular lattice, both of interest in the study of quantum propagation on the nonrectangular grids, as in graphene-like materials. The latter, in particular, we argue, opens the door for a generalization of the Dirac equation to arbitrary discrete surfaces.
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(N1/4log⁡3/4N)O(N^{1/4} \log^{3/4} N)O(N1/4log3/4N) steps with probability O(1/log⁡N)O(1/\log N)O(1/logN), for an overall complexity of O(N1/4log⁡7/4N)O(N^{1/4}\log^{7/4}N)O(N1/4log7/4N). 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.
Discrete-time quantum walks (QWs) are transportation models of single quantum particles over a lattice. Their evolution is driven through causal and local unitary operators. QWs are a powerful tool for quantum simulation of fundamental physics, as some of them have a continuum limit converging to well-known physics partial differential equations, such as the Dirac or the Schr\"odinger equation. In this paper, we show how to recover the Dirac equation in (3+1)(3+1)(3+1) dimensions with a QW evolving in a tetrahedral space. This paves the way to simulate the Dirac equation on a curved space-time. This also suggests an ordered scheme for propagating matter over a spin network, of interest in loop quantum gravity, where matter propagation has remained an open problem.
We study the dynamical localization of discrete time evolution of topological split-step quantum random walk (QRW) on a single-site defect starting from a uniform distribution. Using analytical and numerical calculations, we determine the high localization probability regions in the parameter space of the quantum walker. These regions contain two or more pairs of trapped states, forming near a lattice defect. By investigating the spectral properties of the discrete-time evolution operators, we show that these trapped states have large overlap with the initial uniformly distributed state, thus offering a simple interpretation of the localization effect. As this localization scheme could be interpreted as a variation of spatial quantum search algorithm, we compare the localization probability and time with other types of two-dimensional quantum walks that do not have topological phases and realize localization time scaling similar to Grover's algorithm. Finally we show that mechanism of localization we identified is robust against the influence of disorder.
Quantum walk search on graphs is primarily limited to real evolution operators, although there is no restriction for using the complex unitary matrix as the evolution operator. In the context of quantum search, several studies on the Grover’s search utilize the complex unitary matrix as an evolution operator, which allows for efficient searching of marked elements. However, the generalization of the quantum walk search to complex space has largely been overlooked. In this paper, we extend existing discrete-time coined quantum walk search algorithms on one- and two-dimensional periodic lattices to complex space by introducing two phase parameters, [Formula: see text], and [Formula: see text], into the coin and oracle operators, respectively, for four different discrete-time coined quantum walk search algorithms. One of the advantages of this generalization is that it enables the Grover and SKW coins with specific values of the phase parameters [Formula: see text], and [Formula: see text] to search for the exceptional configurations on a two-dimensional periodic lattice with high success probability, which is otherwise impossible to search using the real Grover and SKW coins operators.
A discrete-time Quantum Walk (QW) is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs admit a continuum limit, leading to familiar PDEs (e.g. the Dirac equation), and thus provide us with discrete toy models of familiar particles (e.g. the electron). In this paper, we study the continuum limit of a wide class of QWs, and show that it leads to all those PDEs corresponding to the Hamiltonian form of the massive curved Dirac equation in (1 + 1) dimensions. Therefore a certain QW, which we make explicit, provides us with a unitary discrete toy model of the electron as a test particle in curved spacetime, in spite of the fixed background lattice. Mathematically we have introduced two novel ingredients for taking the continuum limit of a QW, but which apply to any quantum cellular automata: encoding and grouping.