Searching via nonlinear quantum walk on the 2D-grid

09/16/2020 ∙ by Basile Herzog, et al. ∙ 0

We provide numerical evidence that the nonlinear searching algorithm introduced by Wong and Meyer <cit.>, 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 <cit.>. The numerical simulations showed that the walker finds the marked vertex in O(N^1/4log^3/4 N) steps, with probability O(1/log N), for an overall complexity of O(N^1/4log^7/4N). 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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I Introduction

Searching an element among an unstructured database of size takes iterations, resulting in a linear complexity time. In 1996, L. Grover came up with a quantum algorithm it speeds up any brute force problem into a problem grover1996fast . The algorithm comes in many variants and has been rephrased in many ways, including in terms of quantum walks childs2004spatial . Quantum walks (QW) are essentially local unitary gates that drive the evolution of a particle on a graph venegas2012quantum , and although they may appear defined in a discrete and in a continuous time setting, recently it has been shown that a new family of ”plastic” QW unify and encompass both systems di2020quantum ; manighalam2020continuous . They have been used as a mathematical framework to express many quantum algorithms e.g. ambainis2007quantum ; childs2003exponential ; childs2009universal , but also many quantum simulation schemes e.g. hatifi2019quantum ; di2016quantum . In particular, it has been shown that many of these QW admit, as their continuum limit, the Dirac equation di2013quantum ; arrighi2018dirac , providing ‘quantum simulation schemes’, for the future quantum computers, to simulate all free spin-1/2 fermions. More interestingly, it has been recently proven by one of the authors, that the Grover algorithm is indeed a naturally occurring phenomenon, i.e. spontaneously implemented by some kind of particles in nature roget2020grover over arbitrary surfaces with topological defects. From a theoretical perspective, a Grover search on a graph, rephrased in terms of QW, is an alternation of the diffusion step and the oracle. The nodes of the graph, represent elements of the configuration space of a problem, and whose edges represent the existence of a local transformation between two configurations. So far, the QW search has only been used to look for ‘marked nodes’, i.e. good configurations within the configuration space, as specified by an oracle. In roget2020grover , it has been proved that 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. Here we will position ourselves in the same theoretical groove, but in a nonlinear framework.
The generalisation to an interacting multi-walkers scenario has never been explored in quantum algorithmics, although one unpublished result has shown that a nonlinear effective potential, embedded in the QW evolution operator may improve the searching time kahou2012spatial . Moreover, considering nonlinear effective terms to investigate the multi-particle case, makes the simulation feasible classically. Indeed, there exists many physical systems which may be described by a nonlinear effective equation, such as the Bose-Einstein condensates, making these models physically implementable kevrekidis2007emergent ; alberti2017quantum . From a more theoretical point of view, the growing interest for such systems started with Abrams and Lloyd, showing that nonlinearity in quantum computation, could make quantum systems solve NP-complete problems in polynomial time Abrams_1998 .
In particular, nonlinearity in Quantum Walks have been considered in several recent studies and may appear or under the form of nonlinear phases, (e.g. in Kerr medium molfetta2015nonlinear and Navarrete_Benlloch_2007 ) either via a feed-forward quantum diffusion operator shikano2014discrete . In this manuscript we will give numerical and analytical evidence that nonlinearity leads to a clear computational advantage on the two dimensional grid respect to the linear case, consistently with previous results on the hypercube wong2015nonlinear , paving the way to extend our nonlinear scheme in higher dimensional physical dimensions.

The manuscript is organized as follows : In Section II, we will introduce the QW in continuous time and discrete space; in Section III we will presents our numerical simulations for the linear and nonlinear algorithm and in Section IV we will derive analytically the probability peak recurrence time. Finally, in Section V, we discuss the results and conclude.

Ii Model

ii.1 The linear algorithm

Quantum Walks in continuous time are defined on a -vertices non directed graph with a set of vertices, where the corresponding Hilbert space is , and a set of edges linking two vertices : . A state , being the time parameter, is generally written in the basis , , where the complex amplitudes .
The evolution of obeys to the Schrödinger equation:


where are the coefficients of the Hamiltonian , governing the dynamics. The above evolution is formally close to a classical continuous time Markov process, in imaginary time, where the Hamiltonian plays the role of a discrete Laplacian.

In this framework, the search problem corresponds to find a vertex

, given a state vector

initialized as the uniform superposition over each vertex :


Although the above equation works for any arbitrary graph, in the following, we will choose a 2D-square lattice of vertices, labeled by , with , evolving in the Hilbert space


where the vector accounts the internal states of the walker, namely the coin states, as in Childs_2014 . The Hamiltonian reads :


and being the basis of the lattice with coordinates and .

Figure 1: The grid and its factorisation into cells (grey squares). The marked vertex (circled) is disconnected from its neighbors.

The 2D-grid is factorised into crystals of cells. Each cell is a square composed of four vertices (items). The new coordinates system is defined by , labelling the cell, such that with , and by , labelling the item inside each cell. In particular, we can write the following transformations, which map the vertices into cell coordinates :


denoting the floor function.

Then the searched vertex is :


with and and, the oracle Hamiltonian is defined by :


Notice that the above Hamiltonian, , added to , has the effect of disconnecting the marked vertex from his neighborhood, as in Fig. 1. Indeed, by the definition, , and it follows that . As consequence, one can not reach the neighbors of from itself. Moreover, one can not reach from its neighbors neither, because the total Hamiltonian is real and then it has to be symmetric.

More interestingly, one could remark that the isolated vertex introduces a topological defect in the grid, because the disconnected links reduce the connectivity of the neighborhood. Then, one may argue that any natural system described by , diffusing on a surface with a topological hole defect, can naturally implement a Grover search, as it has been recently proven by one of the authors, in the discrete-time setting roget2020grover .

In conclusion, searching the marked vertex formally coincides to search the state (10):


or more formally, by implementing the following algorithm:

Initialization :
while  do
end while
Algorithm 1 Searching algorithm for the -grid

where the initial state reads:


In order to justify , let us recall that, in the linear case, the system evolves approximately between the states and , where the states

are two eigenvectors of the Hamiltonian

, with respective eigenvalues


And notice that, in this basis, the Hamiltonian reads:


with eigenvalues . Then, the probability to be projected on the state is


which means that, the system will be sufficiently close to the state , in a time .

ii.2 Adding a nonlinearity

Now, let us consider a walker, driven by a nonlinear Hamiltonian. Among all possible choices, we decided to consider the most common, i.d., a nonlinear diagonal potential, which, physically speaking, corresponds to a Kerr nonlinearity. More formally, we add to the total Hamiltonian , the following operator:


where is a coupling real coefficient. The new Hamiltonian is now time-dependent and in general it is not spanned by the basis {, }. However, we can rescale the linear part of the Hamiltonian in such a way, the system, under certain conditions, still evolves between those two states.

First, let’s define and at time as follows :


The two-states system now reads:


By subtracting to the diagonal coefficients, the dynamics keeps unchanged and we get:




The new eingenvalues are now:


with eigenvectors, respectively:


Let us recall that, in several systems, it is possible to rescale the linear part of the global Hamiltonian by multiplying the linear part, , by a factor depending on the terms , and , to force the system to oscillate between and . As an example, in the hypercube graph, it is possible to factorize the main term of the nonlinearity in the oracle, as detailed in wong2015nonlinear . However, here, in , the diffusion term, , needs to have the same weight as the oracle, to ensure the disconnection of the marked vertex, which means that we cannot tune independently. At the same time, the oracle is not globally proportional to the projector , as in the hypercube case, so we cannot factorize in the same way.

The solution we propose here, is to change , keeping the perturbation sufficiently small. By multiplying the Hamiltonian by a term , with a real function of , the initial eigenvalue transforms as :


Now, in order to rescale the eigenvectors of the equation 22 as in the form , we need to impose , which implies that :


Also, the term must be different from zero, otherwise we would not have any diffusion of the walker.
For , we have , and as . This yields an upper bound on :


Now, keeping only the leading term in the lower bound for c, we end up with


In particular, the above inequality also implies that can not scale faster than .

Finally, we replace in the algorithm 1 the linear Hamiltonian by the rescaled one, we add the nonlinear part and we get :

Initialization :
while  do
end while
Algorithm 2 Searching with a nonlinear algorithm for the -grid

Contrary to the linear algorithm 1, an exact analytical treatment to explicitly calculate is not possible. In the following, first, we will provide strong numerical evidence that such a search scheme allows a clear temporal advantage over the linear case, deriving numerically and second, we will give an approximate analytical proof for it.

Iii Numerical results

Numerical simulations show in Fig. 2 the first peak probability time in function of at fixed and the corresponding probability peaks for several values of , in Fig. 3. In all cases, we have chosen , which ensures that . Notice that, for the running time is the same as the linear algorithm, , but the first probability peak is lower, as the initial eigenstate doesn’t rotate to the state , as well as in the linear case. In Fig 2, we can also remark that the probability peak becomes narrower, the more

increases, which would require more attempts to estimate the maximum of the probability curve, making the algorithm sub-optimal

wong2015nonlinear . These considerations suggest to choose a value for which, while ensuring good oscillatory behaviour of the system, leaves the peaks wide enough. A good enough example is , as one can see in Fig. 3. Let us keep this value for in the following for the data analysis.

Figure 2: Run time of the algorithm as a function of , .
Figure 3: Probability over for different values of -

Now, we proceed as follows : (i) Prepare, the initial state as in (11) ; (ii) Let the walker evolve with time; (iii) Quantify the number of steps before the walker reaches its probability peak of being localized in a ball of radius 1 around the center of the defect, namely the peak recurrence time, respectively for the linear and nonlinear search algorithm. Then, estimate this probability peak, at fixed ; (iv) Characterize and , i.e. the way the peak recurrence times and the probability peak depend upon the total number grid points. We run both algorithm, the linear and the nonlinear one and we derive from both, a numerical characterization for and . The peak probability in the linear and in the nonlinear case, behaves as , as it is shown in Fig. 4, approximately with the same pre-factor .

Figure 4: Probability peak as a function of for the nonlinear and linear algorithm (inset)

The Fig.5 shows the peak recurrence times, , and . As expected, in the linear model, we recover . The nonlinear case, instead, shows a significant advantage respect to the linear case, with a peak recurrence times , which yields an overall complexity of .

Figure 5: Peak recurrence times in the nonlinear and the linear algorithm (inset).

Iv Scale analysis

Although an exact analytical description is prohibitive, in this section we will provide some elements of analysis that will convince the reader of the robustness of the numerical results obtained in the previous section. Let’s start from the eigenvalues of Eq. 21, and rescale them using Eq. 23:


Fixing and , as in our numerical simulation, we can clearly distinguish two different behaviours for the probability :

  • A strictly linear regime, when , which evolves periodically with a period

  • A nonlinear regime, when is maximum (and ). In this case, we have and the period is constant:


Here, our aim will be to characterise how the system goes from the first regime to the second and under which conditions. Without a lack of generality, and because we are solely interested in the peak behaviour, we will focus on the first half period of the system dynamics. We propose the following ansatz:


where , and are three real constants.

Let us try to justify this choice. First we can notice that the above self-consistent equation recovers both behaviours for the probability : indeed , when , we get the Eq. (14), by choosing and , and when increases, the argument in the sinus increases as well, speeding-up the dynamics, as observed on Fig. 3. As and for large values of , there exists a critical time at which the Eq. (30) coincides with .
We may argue that the way scales with , may be the same one for the peak recurrence times. Numerical simulations shows that this is approximately true : in fact, in Fig. 6, the first probability peak is fitted, by several solutions of the Eq. (30), for several values of . Along the first half period, the ansatz works surprisingly well.

Figure 6: - Probability over for different values of and numerical resolution of eq. (30).

Moreover, if we compute the characteristic time at which the system transits from the first behavior with period to the second one with period , we derive exactly the same scaling laws as and . Say is such a characteristic time, then at , we have and it follows:


In conclusion, by keeping only the first term in the development of :


and using Eqs. (28) and (29), we get


which confirms the numerical results we had in Sec. III for the probability peak time in the nonlinear algorithm.

V Discussion

We proved numerically and analytically that nonlinear terms, embedded in the coined Quantum Walks lead to a strong computational advantage for spatial search algorithms, over the infinite square grid. The numerical simulations show that the walker finds the marked vertex in steps with probability , for an overall complexity of . These results are consistent with those ones proved by Wong and Meyer in wong2015nonlinear in the context of the nonlinear Schrödinger equation on the Hypercube. Our work presents numerous advantages respect the previous ones: (i) quantum walks are easily implementable in our labs by several physical systems; (ii) graphs having sets of vertices of constant size, are more natural and pave the way to a -dimensional generalisation; (iii) replacing the Grover oracle step by surface defects seems way more practical in terms of experimental realizations, whatever the substrate, possibly even in a biological setting. At a more abstract level, this suggests using QW to search, not just for ’good’ configurations within a space, but rather for topological properties of the configuration space itself.

Vi Acknowledgements

The authors acknowledge inspiring conversations with Thomas Wong. This work has been funded by the Pépinière d’Excellence 2018, AMIDEX fondation, project DiTiQuS and the ID 60609 grant from the John Templeton Foundation, as part of the “The Quantum Information Structure of Spacetime (QISS)” Project.


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